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class="clamp-3"><span class="pr-sm"><span class="m-text text-sm"><div class="m-avatar flex items-center justify-center overflow-hidden rounded-full bg-brand-bold font-medium text-color-on-brand h-5 w-5 min-w-5 text-[0.575rem] !inline-block align-middle mr-xs" data-testid="m-avatar"><img src="/statics/img/design/default-user.png"><span class="sr-only"></span></div><span><a href="https://sciprofiles.com/profile/3073563" target="_blank" rel="noopener noreferrer">Konstantine Arkoudas</a></span><sup> *</sup></span></span></div><div class="invisible absolute"><span class="pr-sm"><span class="m-text text-sm"><div class="m-avatar flex items-center justify-center overflow-hidden rounded-full bg-brand-bold font-medium text-color-on-brand h-5 w-5 min-w-5 text-[0.575rem] !inline-block align-middle mr-xs" data-testid="m-avatar"><img src="/statics/img/design/default-user.png"><span class="sr-only"></span></div><span><a href="https://sciprofiles.com/profile/3073563" target="_blank" rel="noopener 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class="text-sm">This preprints belongs to the Collection</span><div class="absolute right-0 top-1/2 -translate-y-1/2"><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" data-testid="keyboard_arrow_down" class="origin-center transition duration-150 rotate-180" style="" width="16" height="16" viewBox="0 0 16 16"></svg></div></button><div id="accordion-panel-0" aria-labelledby="accordion-button-expand-0" role="region" class="m-accordion__body overflow-hidden transition-height duration-150"><div class="break-words pb-md"><div class="flex items-center"><p class="m-text text-sm text-color-link-bold cursor-pointer">Preprints.org 2023 Most Popular Preprints Award Winner Collection</p><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" data-testid="arrow_right_alt" class="text-color-link-bold cursor-pointer" style="" width="16" height="16" viewBox="0 0 16 16"></svg></div><div class="flex items-center"><p class="m-text text-sm text-color-link-bold cursor-pointer">Artificial Intelligence (AI) and Machine Learning</p><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" data-testid="arrow_right_alt" class="text-color-link-bold cursor-pointer" style="" width="16" height="16" viewBox="0 0 16 16"></svg></div></div></div></div></div></div><div class="py-md border-b border-color-default"><div class="m-select common-field bg-white"><div class="common-field__wrapper" data-headlessui-state><div class="relative w-full cursor-default"><div class="flex items-center"><button id="headlessui-listbox-button-mui-49605" type="button" aria-haspopup="listbox" aria-expanded="false" data-headlessui-state data-testid="select-activator" class="common-field__dropdown-input [&>span]:overflow-hidden h-10"><span class="flex w-full items-center gap-1 [&>span]:truncate"><span>Version 2</span></span><div class="flex"><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" data-testid="keyboard_arrow_down" class="duration-400 transition-transform" style="" width="24" height="24" viewBox="0 0 16 16"></svg></div></button></div></div></div></div><div class="pt-sm"><div class="flex"><p class="m-text text-xs">Submitted:</p><p class="m-text text-xs pl-xs">03 August 2023</p></div><div class="flex pt-sm"><p class="m-text text-xs">Posted: </p><p class="m-text text-xs pl-xs">07 August 2023</p></div></div><div class="pt-sm"><div><p class="m-text text-xs">You are already at the latest version </p></div></div><div></div></div><div class="py-md border-b border-color-default flex items-center justify-between" data-v-d8c082ee><div class="flex-none flex items-center" data-v-d8c082ee><span class="m-text text-xs pr-xs font-semibold flex-none" data-v-d8c082ee>Alerts</span><span data-v-d8c082ee></span></div><div class="m-switch flex items-center gap-2" data-v-d8c082ee><button class="relative inline-flex h-[1.4rem] w-[2.75rem] shrink-0 cursor-pointer items-center rounded-full border-2 border-color-transparent transition-colors duration-150 ease-out ui-focus-visible:ring-2 ui-focus-visible:ring-brand-bold ui-focus-visible:ring-opacity-75 bg-content-default hover:bg-content-bold active:bg-content-bolder" aria-label="Switch on" id="headlessui-switch-mui-49606" role="switch" type="button" tabindex="0" aria-checked="false" data-headlessui-state><span aria-hidden="true" class="pointer-events-none inline-block h-[1rem] w-[1rem] transform rounded-full bg-white shadow-lg ring-0 transition duration-200 ease-in-out translate-x-0.5"></span></button></div></div><div data-v-d8c082ee data-v-03f1a75d></div><div><h6 class="m-heading text-inherit m-h6 font-semibold pt-lg">Abstract</h6><div class="">GPT-4 was released in March 2023 to wide acclaim, marking a very substantial improvement across the board over GPT-3.5 (OpenAI's previously best model, which had powered the initial release of ChatGPT). Despite the genuinely impressive improvement, however, there are good reasons to be highly skeptical of GPT-4's ability to reason. This position paper discusses the nature of reasoning; criticizes the current formulation of reasoning problems in the NLP community and the way in which the reasoning performance of LLMs is currently evaluated; introduces a collection of 21 diverse reasoning problems; and performs a detailed qualitative analysis of GPT-4's performance on these problems. Based on the results of this analysis, the paper argues that, despite the occasional flashes of analytical brilliance, GPT-4 at present is utterly incapable of reasoning.</div></div><div data-v-3ce94018><div class="pt-lg" data-v-3ce94018><span class="m-text text-body font-semibold" data-v-3ce94018>Keywords: </span><span class="m-text text-body pt-sm" data-v-3ce94018></span></div></div><div class="flex-1 pt-lg mr-lg lg:pt-0" data-v-6afb3413><div class="pt-md" data-v-6afb3413><div class="flex flex-wrap lg:flex-nowrap items-center" data-v-6afb3413><span class="m-text text-body font-semibold" data-v-6afb3413>Subject: </span><span class="m-text text-body" data-v-6afb3413>Computer Science and Mathematics</span><span class="" data-v-6afb3413> - </span><span class="m-text text-body" data-v-6afb3413>Artificial Intelligence and Machine Learning</span></div></div><div class="content-container" id="articleRef" data-v-6afb3413><script type="text/x-mathjax-config"> MathJax.Hub.Config({ menuSettings: { CHTMLpreview: false }, "CHTML-preview":{ disabled: true }, "HTML-CSS": { scale: 90, availableFonts: [], preferredFont: null, preferredFonts: null, webFont:"Gyre-Pagella", imageFont:'TeX', undefinedFamily:"'Arial Unicode MS',serif", linebreaks: { automatic: false } }, "TeX": { extensions: ["noErrors.js"], noErrors: { inlineDelimiters: ["",""], multiLine: true, style: { "font-size": "90%", "text-align": "left", "color": "black", "padding": "1px 3px", "border": "1px solid" } } } }); </script><script type="text/javascript" async="" src="https://www.mdpi.com/bundles/mathjax/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <section id="sec1-preprints-81518" type="intro"><h2 data-nested="1" id="preprints-h2-1"> 1. Introduction</h2> <div class="html-p">In early January I wrote <a href="https://medium.com/@konstantine_45825/chatgpt-is-no-stochastic-parrot-but-it-also-claims-that-1-is-greater-than-1-e3cd1fc303e0" target="_blank">a commentary</a><a href="#fn001-preprints-81518" class="html-fn">1</a> presenting an informal evaluation of ChatGPT across a broad range of subject areas: conventional NLU, folk physics, information retrieval, pragmatics, theory of mind, spatial inference, simple logical reasoning, and math. The key takeaways were that ChatGPT was a seminal breakthrough; that LLM-based systems are not mere stochastic parrots but build genuine abstractions and can exhibit creativity; that such systems will enable a large array of new and exciting applications; and that, despite all of the above, these systems are still severely limited when it comes to reasoning.</div> <div class="html-p">GPT-4 was released a couple of months after that, delivering very substantial improvements across the board. I remain impressed and excited by the general capabilities and potential of LLMs, and I have little doubt that their performance will continue to improve in the near future. Nevertheless, there are increasing grounds for skepticism concerning their reasoning abilities. In this position paper I will argue that the best LLM at this time, GPT-4, is utterly incapable of reasoning, in spite of its sporadic displays of ingenuity.</div> <div class="html-p">I will largely steer clear of the much broader—and more vague—debate about whether LLMs <span class="html-italic">in general</span> are capable of (consistently robust) reasoning, but a few brief remarks will help to set the stage and clarify why it makes sense to restrict attention to a specific LLM. On one side of that broader debate, rosy predictions by LLM enthusiasts rely excessively on ever-changing scaling “laws” that rest on flimsy empirical evidence and on a host of questionable modeling assumptions, ill-understood concepts (such as “emergent” LLM properties<a href="#fn002-preprints-81518" class="html-fn">2</a>), and a somewhat dogmatic belief that minimizing cross-entropy loss on next-token prediction over a huge corpus will deliver a general reasoning engine via the magic of transfer learning and the construction of generic higher-level representations.</div> <div class="html-p">On the other side of the debate, while LLM skeptics have serious arguments to make, those arguments are mostly a priori and somewhat vague (for instance, that LLMs lack “a model of the world”), and I do not think they settle the question. In my view, the most compelling a priori considerations against the plausibility of reliably robust LLM reasoning turn on computational complexity results. Reasoning is a (very) computationally hard problem. In fact, in the general case (first-order or higher-order logic), it is algorithmically undecidable, i.e., every bit as unsolvable as the halting problem. Thus, by Church’s thesis, we cannot expect <span class="html-italic">any</span> algorithm, LLMs included, to solve arbitrary reasoning problems in a sound and complete way.<a href="#fn003-preprints-81518" class="html-fn">3</a> But even “easier” classes of reasoning problems<a href="#fn004-preprints-81518" class="html-fn">4</a> typically have either exponential or at least nontrivial polynomial-time complexity profiles. Problem classes that have linear-time inference algorithms, such as Horn clauses over literals, are rarely expressive enough. This tradeoff between generality and expressivity on the one hand and tractability on the other means that no LLM, no matter how large or how extensively and cleverly trained and tuned, will ever be able to crack an <span class="html-italic">arbitrary</span> reasoning problem. And this is consistent with the famous “no free lunch” theorem of machine learning, which points to a similar inverse relationship between model generality and performance.</div> <div class="html-p">But LLM advocates can make a couple of cogent counterpoints, while granting that there will never be an AI oracle that can essentially solve the halting problem. First, they can point out that even though a problem might have high worst-case asymptotic complexity, it might still be solvable well enough <span class="html-italic">in practice</span>. Unlike random instances, real-world instances of reasoning problems (and indeed real-world instances of most computationally hard problems) appear to have structure that allows clever algorithms to tackle them effectively.<a href="#fn005-preprints-81518" class="html-fn">5</a> There are many examples here, from the simplex algorithm for linear programming and SAT solvers to term unification algorithms and even automatic theorem provers for full first-order logic. All of these problems are hard (having at least exponential-time worst-case complexity), yet somehow we have algorithms for them that seem to work successfully on a wide variety of inputs.</div> <div class="html-p">Second, and perhaps more important, we need not aim for an oracle anyway. Humans are not oracles either, nor do they seem to follow any particular algorithm that captures any one specific class of reasoning problems. The ability of humans to reason is much more fluid and messy, but impressive nevertheless. Is it impossible to build something like an LLM-based system with the reasoning ability of a well-trained engineer of average intelligence (which perhaps can then become even more intelligent and better trained by an endless process of learning and improvement)?</div> <div class="html-p">I don’t think that building such a system can be ruled out on a priori grounds (and here I differ from hard-core AI skeptics). I think it’s implausible, for a number of reasons,<a href="#fn006-preprints-81518" class="html-fn">6</a> but ultimately this strikes me as an empirical question that must be decided on a case-by-case basis, by subjecting a specific system to testing, i.e., by interrogating it, probing it, and analyzing its responses. And the case I will consider here is that of GPT-4, which appears, by all accounts, to be the most capable LLM at present.</div> <div class="html-p">There are two questions that must be addressed before we proceed. First, we must agree on what reasoning is, and second, we must say something about methodology. The next section contains a brief discussion of reasoning, but for those who wish to skip that section and dive right into the problems, the upshot is that we’ll focus on (a liberal conception of) deductive reasoning. Regarding methodology, just like the January piece, my evaluation here is not based on a corpus or set of corpora. Instead, I present a detailed qualitative analysis of GPT-4’s performance on 21 simple reasoning problems across a wide range of areas, most of which have been made up from scratch, while the rest (such as Wason’s selection task) have been manually tweaked so as to make them less recognizable to the model.</div> <div class="html-p">This is done partly to avoid data contamination, which is a serious problem affecting corpus-based evaluations. Given how little we know about the training regimen of ChatGPT, it is impossible to know for sure whether any existing dataset or problem has effectively been “seen” by the model during its pretraining or subsequent alignment, whether we’re talking about NLP datasets, medical licensing exams, Python programming problems, LSAT or bar-entrance exams, SAT or GRE tests, and so on.<a href="#fn007-preprints-81518" class="html-fn">7</a> The qualification “effectively” is important, because even though a specific problem might not have been seen in its <span class="html-italic">exact</span> form (in a string-matching sense), an essentially equivalent variant with a different surface formulation might well have been. Hence, simple contamination tests based on substring checks, such as those carried out by OpenAI in their <a href="https://arxiv.org/abs/2303.08774" target="_blank">GPT-4 Technical Report</a> [<a href="#B8-preprints-81518" class="html-bibr">8</a>] (posted in March 2023), are not sufficient to guarantee lack of contamination.<a href="#fn008-preprints-81518" class="html-fn">8</a> </div> <div class="html-p">The absence of a large corpus makes the discussion more qualitative rather than quantitative. However, the results are arguably more informative than a numeric metric computed over a corpus, for a number of reasons. First, because contamination can be ruled out conclusively; second, because the problems span a large gamut of areas; and third, because a qualitative discussion of a problem allows for greater depth of analysis and more context in which to interpret the results. By contrast, the only way to perform a truly informative quantitative evaluation is to come up with a brand new corpus that satisfies all of the following criteria: (a) originality; (b) uniformly high quality; (c) sufficiently large size; and (d) diversity (not being limited to one type of task only). This is a very challenging undertaking. Even then, a few simple numeric metrics on a brand new dataset might not be particularly illuminating. Are the numbers measuring the right things? Do we even know the right things to measure? Is there an appropriate backdrop in which the numbers can be understood? For deeper insight, we need to put individual examples under a magnifying glass.</div> <div class="html-p">This is particularly important because we need to scrutinize the explanations (“chains of thought”) generated by a reasoner. Unfortunately, almost all reasoning corpora comprise either multiple-choice questions or binary classification problems (e.g., “Does sentence <math display="inline"><semantics> <msub> <mi>p</mi> <mn>2</mn> </msub> </semantics></math> follow from premise <math display="inline"><semantics> <msub> <mi>p</mi> <mn>1</mn> </msub> </semantics></math>, yes or no?”). Why? Mostly because it is easy to mechanically evaluate model performance on such datasets. But even in the absence of contamination, this type of test set runs the serious risk that the LLM will manage to pick the right answers by latching on to spurious statistical regularities, i.e., to arrive at the right answers for the wrong reasons [<a href="#B6-preprints-81518" class="html-bibr">6</a>,<a href="#B10-preprints-81518" class="html-bibr">10</a>].<a href="#fn009-preprints-81518" class="html-fn">9</a> Adversarial augmentation of an existing dataset might help, especially if we know what we are trying to guard against, but unless an adversarial version restores near-random performance, this can quickly devolve into a game of whac-a-mole, where we detect a new round of bogus regularities exploited by the model and must undertake a new round of adversarial interventions.</div> <div class="html-p">Ultimately, there is really no proper way to assess the reasoning ability of a system unless we ask it to explain its output. This is an essential part of reasoning, which is not about producing the right answer by hook or by crook but about <span class="html-italic">deriving</span> the right answer <span class="html-italic">for the right reasons</span>. And rote metrics like ROUGE-L are not fit for purpose here. We need to roll up our sleeves and analyze LLM explanations and proof attempts manually. We also need to gauge their performance in a dialog setting (e.g., what happens when a reasoning error is pointed out to them?). This is the sort of analysis undertaken in this paper. I believe the results show unequivocally that GPT-4 cannot reason. The errors are too pervasive and too egregious. GPT-4 doesn’t solve even one of the 21 problems discussed here. But much more concerning are the fundamentally flawed explanations and proof attempts it produces along the way.</div> <div class="html-p">LLM believers will probably demur: <span class="html-italic">But humans also make mistakes, and surely we’re not prepared to say that humans can’t reason just because they make mistakes</span>? First, it is not accurate to say without qualification that “humans can reason,” certainly not in the sense that we can randomly pluck any person from the street and expect them to reliably perform normatively correct reasoning. Most neurobiologically normal humans have the <span class="html-italic">capacity</span> to become proficient in reasoning, but actually attaining such proficiency takes significant training and discipline. Humans are known to be susceptible to a large assortment of cognitive biases, which can only be overcome by rigorous instruction. Focusing on the reasoning skills of untrained people is a bit like focusing on the singing skills of the general population. Everybody sings in the shower, but without formal training (or at least exceptional talent) the results are usually regrettable.</div> <div class="html-p">Of course, even sophisticated human reasoners make mistakes, just like trained singers can hit false notes. But if a human made <span class="html-italic">these</span> mistakes, the ones reported in this article, then I would conclude without any hesitation that they cannot reason. Even if they went on to list a large number of other examples demonstrating impeccable reasoning, I would suspect that other factors (such as rote memorization or cheating) were behind the performance discrepancy. For the mistakes reported here are not performance mistakes, the sort of innocuous errors that humans might make—and promptly correct—when they are careless or tired. If a human made these mistakes, and made them consistently under repeated questioning, that would indicate without doubt that they don’t have the necessary logical <span class="html-italic">competence</span>, that they lack fundamental concepts that are part and parcel of the fabric of reasoning, such as logical entailment and set membership. And I would certainly not entrust that person with generating reams of Python or Javascript code for an enterprise. Nor would I start organizing international conferences to investigate how their reasoning prowess might threaten humanity with extinction.</div></section><section id="sec2-preprints-81518" type><h2 data-nested="1" id="preprints-h2-2"> 2. What is Reasoning?</h2> <div class="html-p">Reasoning is not quite the same thing as intelligence, but it’s a necessary ingredient for it. Broadly put, reasoning is the process of drawing and evaluating <span class="html-italic">conclusions</span> from a given body of information. More precisely, it is the process of making and—more importantly—<span class="html-italic">justifying</span> arguments. An argument consists of a conclusion (the argument’s upshot, so to speak) and a set of <span class="html-italic">premises</span> from which the conclusion is derived. Premises represent information that is taken as given, if only provisionally, for the purposes of the argument. The conclusion and the premises are typically declarative sentences (expressed either in natural language or in the notation of a symbolic logic) that can be true or false, but they may also be represented by alternative notational devices, such as diagrams. We say that a set of premises <span class="html-italic">S</span> logically <span class="html-italic">entails</span> (or logically <span class="html-italic">implies</span>) a conclusion <span class="html-italic">p</span> iff <span class="html-italic">p</span> is true whenever all the sentences in <span class="html-italic">S</span> are true, in which case the argument is said to be <span class="html-italic">valid</span>. This means that it’s logically impossible to have a state of affairs in which every element of <span class="html-italic">S</span> holds but <span class="html-italic">p</span> does not. This key logical relationship is a linchpin of human reasoning.<a href="#fn010-preprints-81518" class="html-fn">10</a> </div> <div class="html-p">Valid deductive arguments (whose conclusions are entailed by the premises) are said to be <span class="html-italic">analytical</span> (or sometimes <span class="html-italic">tautological</span>), insofar as, <span class="html-italic">technically speaking</span>, they convey no information.<a href="#fn011-preprints-81518" class="html-fn">11</a> This idea is also sometimes expressed by calling such arguments <span class="html-italic">non-ampliative</span>, meaning that there is no information contained in the conclusion that is not already contained—if only latently—in the premises. Deduction is the process of making and justifying non-ampliative arguments.</div> <div class="html-p">Deductive arguments are typically justified by <span class="html-italic">proofs</span>, which are sequences of inference steps, each of which applies an <span class="html-italic">inference rule</span> to a number of premises and/or results of previous steps and derives a new result. The last step derives the final conclusion of the proof. An inference rule may be low-level and easy to apply or higher-level and computationally expensive. But all inference rules are required to be <span class="html-italic">sound</span> (or <span class="html-italic">truth-preserving</span>), that is, they must ensure that if the inputs are true then so is the output. All mathematical proofs are deductive, and mathematical reasoning in general is predominantly deductive.<a href="#fn012-preprints-81518" class="html-fn">12</a> </div> <div class="html-p">The conventional view is that some arguments are <span class="html-italic">ampliative</span>, meaning that the conclusion is not quite entailed by the premises. In other words, it is possible for the premises to be true while the conclusion is false. These are typically subdivided into <span class="html-italic">inductive</span> and <span class="html-italic">abductive</span> arguments,<a href="#fn013-preprints-81518" class="html-fn">13</a> although some authors view induction as a species of abduction, and even more authors view abduction as a species of induction. There is no rigorous definition of either, but roughly, the premises of a good inductive argument make its conclusion <span class="html-italic">likely</span>, though never quite certain (in contrast to deduction, where the truth of the premises guarantees the truth of the conclusion). Induction can generate specific conclusions from all kinds of premises (specific or general), but often it proceeds from specific individual observations <math display="inline"><semantics> <mrow> <msub> <mi>o</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>o</mi> <mi>n</mi> </msub> </mrow> </semantics></math> to a more general hypothesis <span class="html-italic">H</span> that subsumes the individual <math display="inline"><semantics> <msub> <mi>o</mi> <mi>i</mi> </msub> </semantics></math> in some sense (for instance, <span class="html-italic">H</span> may be a universally quantified sentence and the <math display="inline"><semantics> <msub> <mi>o</mi> <mi>i</mi> </msub> </semantics></math> could be instances of that sentence). Much of what ML algorithms do can be viewed as inductive reasoning. For instance, a linear-regression algorithm might take as input <span class="html-italic">n</span> datapoints about car models, where each datapoint is of the form <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mi>i</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo>(</mo> <msub> <mi>c</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>h</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>m</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mrow> </semantics></math>, where <math display="inline"><semantics> <msub> <mi>c</mi> <mi>i</mi> </msub> </semantics></math> is the number of cylinders for the <math display="inline"><semantics> <msup> <mi>i</mi> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msup> </semantics></math> car model, <math display="inline"><semantics> <msub> <mi>h</mi> <mi>i</mi> </msub> </semantics></math> is the horsepower, <math display="inline"><semantics> <msub> <mi>y</mi> <mi>i</mi> </msub> </semantics></math> is the model year, and the dependent variable <math display="inline"><semantics> <msub> <mi>m</mi> <mi>i</mi> </msub> </semantics></math> is the mpg (miles per gallon). And it might produce as output a formula like <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>·</mo> <mi>c</mi> <mo>+</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>·</mo> <mi>h</mi> <mo>+</mo> <msub> <mi>w</mi> <mn>3</mn> </msub> <mo>·</mo> <mi>y</mi> <mo>+</mo> <mi>b</mi> </mrow> </semantics></math>, which predicts the mpg of a car model from its number of cylinders, horsepower, and model year.<a href="#fn014-preprints-81518" class="html-fn">14</a> Here <math display="inline"><semantics> <msub> <mi>w</mi> <mn>1</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>w</mi> <mn>2</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>w</mi> <mn>3</mn> </msub> </semantics></math>, and <span class="html-italic">b</span> are specific numbers (weights) representing a hyperplane that minimizes the mean squared error for the input data (meaning that the hyperplane determined by these weights might not fit the <span class="html-italic">n</span> datapoints perfectly, but it does so better than the hyperplane determined by any other set of weights).<a href="#fn015-preprints-81518" class="html-fn">15</a> </div> <div class="html-p">The main distinguishing feature of abductive reasoning is a strong emphasis on explanation. Abduction consists mostly in making and justifying arguments that explain a set of facts. If one day I come home early from work and I see a plumber’s van parked in my neighbors’ driveway, I might conclude that my neighbors are having some plumbing work done in their house. The premise here is “There is a plumbing van parked in my neighbors’ driveway” and the conclusion is “My neighbors are having plumbing work done in their house.” This is sometimes called “inference to the best explanation,” because the conclusion serves to explain the premise(s). This is also a form of ampliative reasoning—the conclusion does not follow logically from the premises. There are many alternative explanations of a given set of facts or observations (perhaps a plumber parked there temporarily, or the neighbors bought the van, or the neighbors have a plumber friend who is making a social visit, and so on). A <span class="html-italic">good</span> abductive inference will yield a hypothesis that has more explanatory value than competing hypotheses. But how exactly to measure the quality of an abductive piece of reasoning is an open question.<a href="#fn016-preprints-81518" class="html-fn">16</a> Note that it doesn’t take a large leap of imagination to view induction as a form of abduction. Observing a large number of black (and only black) swans and then conjecturing that all swans are black could be seen as abductive reasoning, as the conclusion <math display="inline"><semantics> <mrow> <mo>∀</mo> <mspace width="0.222222em" /> <mi>x</mi> <mspace width="0.222222em" /> <mo>.</mo> <mspace width="0.222222em" /> <mrow> <mi>s</mi> <mi>w</mi> <mi>a</mi> <mi>n</mi> </mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>⇒</mo> <mrow> <mi>c</mi> <mi>o</mi> <mi>l</mi> <mi>o</mi> <mi>r</mi> </mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>=</mo> <mrow> <mi>b</mi> <mi>l</mi> <mi>a</mi> <mi>c</mi> <mi>k</mi> </mrow> </mrow> </semantics></math> would explain all the observed data. Linear regression can also be seen as the making of an abductive hypothesis, as can (much more generally) Maximum Likelihood Estimation, a principle that underlies many ML algorithms and is often associated with induction.</div> <div class="html-p">All of the above is received wisdom, but it’s worth mentioning that there have been thinkers, called “deductivists” (ranging from philosophers such as Popper and Musgrave to statisticians such as Fisher), who contend that deduction is the only real form of reasoning there is, insofar as it’s the only one for which we have a rigorous and properly understood formal notion of validity; and that other (ampliative) arguments are best understood as reconstructed deductions, typically as enthymemes (arguments that omit tacitly understood premises). I find that position congenial,<a href="#fn017-preprints-81518" class="html-fn">17</a> but venturing into that discussion would take us too far afield. For present purposes it suffices to say that we will focus on deduction, because it is the type of reasoning that underpins most logico-mathematical thought and for which we have clear normative standards of evaluation.</div> <div class="html-p">An important note: I view the discovery and justification of particular <span class="html-italic">models</span> (including counterexamples and countermodels in general) as part and parcel of reasoning. This is not a controversial view; some cognitive scientists view models and associated cognitive processes as the fundamental ingredients of human reasoning [<a href="#B11-preprints-81518" class="html-bibr">11</a>]. In addition, however, I view model-based reasoning as at least partly deductive, because even though the actual process of discovering models might not be a process of deduction<a href="#fn018-preprints-81518" class="html-fn">18</a>, its outcome is a claim (namely, that a given interpretation satisfies a set of premises) that can be verified or falsified deductively, taking as premises the definition of the model itself and possibly other general knowledge about the model’s domain. Indeed, I will consider even computation as a form of deduction, because a particular computation can be naturally regarded as a deductive derivation of a conclusion of the form <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>(</mo> <msub> <mi>e</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mi>n</mi> </msub> <mo>)</mo> <mo>=</mo> <mi>v</mi> </mrow> </semantics></math>, where <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>(</mo> <msub> <mi>e</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </semantics></math> is the application of an arbitrary function <span class="html-italic">f</span> to arbitrary argument expressions <math display="inline"><semantics> <mrow> <msub> <mi>e</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mi>n</mi> </msub> </mrow> </semantics></math>, ultimately yielding value <span class="html-italic">v</span> as the result. The premises for the derivation consist of the definition of <span class="html-italic">f</span> and possibly other auxiliary functions, along with the usual equational axioms (reflexivity, symmetry, transitivity, and functional/relational congruence).<a href="#fn019-preprints-81518" class="html-fn">19</a> </div></section><section id="sec3-preprints-81518" type><h2 data-nested="1" id="preprints-h2-3"> 3. Test Problems</h2> <div class="html-p">This section will start with the usual caveat: GPT-4 is a nondeterministic system that might produce different answers on different runs, even with the same parameter settings. All of the following exchanges with GPT-4 have been transcribed verbatim, and in my experience the errors discussed here tend to be robust, but it’s conceivable that for a given example GPT-4 might generate a different output even in response to the exact same prompt.<a href="#fn020-preprints-81518" class="html-fn">20</a> </div> <section id="sec3dot1-preprints-81518" type><h4 class="html-italic" data-nested="2"> 3.1. Simple Arithmetic</h4> <div class="html-p">The ability to perform basic arithmetic is a necessary ingredient for reasoning. GPT-4 is still unable to reliably perform elementary arithmetic operations such as addition and multiplication. To ensure that GPT-4 isn’t falling back on rote memorization, we can ask it to first select two random integers in a range of our choice and then perform the operation on the selected values:</div> <div class="html-p"> <b>KA</b><span class="html-italic">: Select two random numbers between 1381 and 1453 and multiply them together, reporting the result.</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">Sure, let’s select two random numbers in the range of 1381 to 1453. Let’s say these numbers are 1405 and 1421.</span> </div> <div class="html-p"><span class="html-italic">To get the product, we simply multiply these two numbers together:</span></div> <div class="html-p"><span class="html-italic">1405 * 1421 = 1996025</span></div> <div class="html-p">Alas, the correct answer is <math display="inline"><semantics> <mrow> <mn>1405</mn> <mo>·</mo> <mn>1421</mn> <mo>=</mo> <mn>1996505</mn> </mrow> </semantics></math>.</div> <div class="html-p">One response here from the LLM camp has been to concede that GPT-4 is not a Turing-complete framework and cannot perform general-purpose computation: It can’t add, multiply or divide, it can’t sort lists, it can’t invert matrices or solve systems of differential equations, it can’t compile arbitrary programs to machine code, it can’t derive theorems, and so on <span class="html-italic">ad infinitum</span>. Nevertheless, the response goes, these computational “gaps” can be filled by plug-ins, along the lines of the <a href="https://arxiv.org/pdf/2210.03629.pdf" target="_blank">ReAct</a> (short for “Reasoning and Acting”) [<a href="#B4-preprints-81518" class="html-bibr">4</a>] or <a href="https://arxiv.org/pdf/2205.00445.pdf" target="_blank">MRKL</a> (short for “Miracle”) [<a href="#B3-preprints-81518" class="html-bibr">3</a>] frameworks. It is this line of work that has led to systems such as <a href="https://python.langchain.com/docs/get_started/introduction.html" target="_blank">LangChain</a> and <a href="https://autogpt.net/" target="_blank">AutoGPT</a>. The idea here is to use the LLM as a planner to recursively decompose a complex problem into a sequence of subproblems that can be dispatched to appropriate specialized <span class="html-italic">agents</span> via appropriate APIs.</div> <div class="html-p">The problem with this approach is its circularity. <span class="html-italic">Planning itself requires reasoning</span>, so this is a Catch-22 situation: We can’t solve reasoning by delegating to appropriate agents, because figuring out the delegation (how the problem should be decomposed, which agents to call and how, and how to compose the results) is itself computationally infeasible. It not only requires the ability to understand natural language, but also the ability to reason about preconditions and effects. And this is assuming a fixed collection of agents with clear-cut APIs.<a href="#fn021-preprints-81518" class="html-fn">21</a> Even under these overly simplistic assumptions, planning is very computationally expensive (PSPACE-complete), and radical simplifications are needed to scale down the complexity even to the level of NP-completeness. Under more realistic assumptions, planning is performed under incomplete knowledge and it is mechanically undecidable. As we understand them today, expecting generic LLMs to tackle planning with any degree of success is naive.</div></section><section id="sec3dot2-preprints-81518" type><h4 class="html-italic" data-nested="2"> 3.2. Simple Counting</h4> <div class="html-p">While concrete counting is not necessarily a reasoning activity,<a href="#fn022-preprints-81518" class="html-fn">22</a> it is surely a requirement for any generally capable reasoning system. Here I give GPT-4 a propositional variable with 27 negation signs in front of it and ask it to count the number of negations. For a human this would be an easy task, especially because the negation signs are written in five blocks with five tildes each, followed by a final pair of negation signs.</div> <div class="html-p"> <b>KA</b><span class="html-italic">: How many times is p negated in the following formula:</span> </div> <div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math>p?</span></div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">The formula you’ve given includes 28 negation symbols (“∼”) in front of the propositional variable “p”. In logic, the negation symbol changes the truth value of a statement to its opposite. If a statement is true, its negation is false, and vice versa. Double negation, like “<math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math>p”, is equivalent to “p”, because negating a proposition twice returns it to its original truth value.</span> </div> <div class="html-p"><span class="html-italic">Given this, to find the effective number of negations, we can simply divide the total number of negation symbols by 2. If there’s a remainder, then there is a net negation. If not, then there isn’t.</span></div> <div class="html-p"><span class="html-italic">28 divided by 2 is 14, with no remainder, meaning there’s no net negation. So effectively, the variable “p” is not negated in the given formula. It’s equivalent to just “p”.</span></div> <div class="html-p">A difference of one negation sign might not seem like a serious matter, until you realize that it makes all the difference in logical import,<a href="#fn023-preprints-81518" class="html-fn">23</a> as GPT-4’s own (unsolicited) explanation stresses. Repeating this experiment multiple times gave widely different results (including 50 and 54), even after explicitly telling GPT-4 to take its time and count carefully.</div> <div class="html-p">Here is another example. I greet GPT-4 five times in a row and then I ask it a simple question:</div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">Hi</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">Hello! How can I assist you today?</span> </div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">Hello</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">How can I assist you today?</span> </div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">Greetings</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">Greetings! How can I assist you today?</span> </div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">Hi there.</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">Hi there! How can I help you today?</span> </div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">Hullo!</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">Hello! How can I assist you today?</span> </div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">You’ve been greeted a few times so far. Three of those greetings were actually from my girlfriend, the rest from me. How many times did I greet you?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">If your girlfriend greeted me three times and all the other greetings were from you, then you greeted me four times. This is assuming that all seven greetings (“Hi”, “Hello”, “Greetings”, “Hi there”, “Hullo!”) were included in the count.</span> </div> <div class="html-p">In this case the arithmetic is correct (<math display="inline"><semantics> <mrow> <mn>7</mn> <mo>−</mo> <mn>3</mn> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>) but the count—and ultimately the resulting answer—is hallucinated.</div></section><section id="sec3dot3-preprints-81518" type><h4 class="html-italic" data-nested="2"> 3.3. (Medical) Common Sense</h4> <div class="html-p">In the present setting we may regard commonsensical arguments as straightforward enthymematic deductions of conclusions from given information <span class="html-italic">plus</span> unstated premises that constitute tacit, generally accepted background knowledge. In this particular case, such common-sense knowledge would be propositions like “A person is alive until they die, after which they do not become alive again.”</div> <div class="html-p"> <b>KA</b><span class="html-italic">: Mable’s heart rate at 9 AM was 75 bpm and her blood pressure at 7 PM was 120/80. She died at 11 PM. Was she alive at noon?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">Based on the information provided, it’s impossible to definitively say whether Mable was alive at noon. The data points provided (heart rate at 9 AM and blood pressure at 7 PM) both indicate that she was alive at those times. However, without additional information about her condition between 7 PM and 11 PM, or at noon specifically, we can’t make a certain determination about her status at noon.</span> </div> <div class="html-p"><span class="html-italic">Remember, medical information can be complex and nuanced. Understanding a person’s health status often requires a comprehensive examination and should be performed by a healthcare professional. If this is a hypothetical question, keep in mind that real-life situations would require more complete information.</span></div> <div class="html-p">No comment is needed here, though the gratuitous—but by now de rigueur—piece of legalese issuing the vague disclaimer at the end (“this-is-a-complex-issue-and-make-sure-to-consult-with-a-professional”) seems ironic.</div></section><section id="sec3dot4-preprints-81518" type><h4 class="html-italic" data-nested="2"> 3.4. Elementary Logic</h4> <div class="html-p">If <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> implies <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </semantics></math> does not hold then we can infer, by modus tollens, that <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </semantics></math> does not hold either (because if it did then <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </semantics></math> would too). This is as elementary of a tautology as can be, yet GPT-4 is perfectly willing to produce a countermodel:</div> <div class="html-p"> <b>KA</b><span class="html-italic">: Find a model in which P(x) implies Q(x), Q(a) does not hold, and P(a) holds.</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">To find a model in which P(x) implies Q(x), Q(a) does not hold, and P(a) holds, let’s consider the following interpretations:</span> </div> <div class="html-p"><dl class="html-order"> <dt id> <span class="html-italic">1.</span> </dt> <dd><div class="html-p"><span class="html-italic">P(x): x is an even number</span></div></dd> <dt id> <span class="html-italic">2.</span> </dt> <dd><div class="html-p"><span class="html-italic">Q(x): x is a positive number</span></div></dd> <dt id> <span class="html-italic">3.</span> </dt> <dd><div class="html-p"><span class="html-italic">a: <math display="inline"><semantics> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math></span></div></dd> </dl></div> <div class="html-p"><span class="html-italic">Now, let’s check if these interpretations satisfy the given conditions:</span></div> <div class="html-p"><dl class="html-order"> <dt id> <span class="html-italic">1.</span> </dt> <dd><div class="html-p"><span class="html-italic">P(x) implies Q(x): If x is an even number, then x is a positive number. This statement is not universally true, as we can have negative even numbers. However, it does not preclude the existence of a model with the other given conditions.</span></div></dd> <dt id> <span class="html-italic">2.</span> </dt> <dd><div class="html-p"><span class="html-italic">Q(a) does not hold: Q(<math display="inline"><semantics> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math>) means <math display="inline"><semantics> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math> is a positive number, which is false. So, this condition is satisfied.</span></div></dd> <dt id> <span class="html-italic">3.</span> </dt> <dd><div class="html-p"><span class="html-italic">P(a) holds: P(<math display="inline"><semantics> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math>) means <math display="inline"><semantics> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math> is an even number, which is true. So, this condition is also satisfied.</span></div></dd> </dl></div> <div class="html-p"><span class="html-italic">Thus, the model with the interpretations P(x): x is an even number, Q(x): x is a positive number, and a: <math display="inline"><semantics> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math> satisfies the conditions P(x) implies Q(x), Q(a) does not hold, and P(a) holds.</span></div> <div class="html-p">Observe that GPT-4 recognizes that <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> does <span class="html-italic">not</span> in fact imply <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> under the interpretation it gave, but it nevertheless dismisses this inconvenient consideration because “it does not preclude the existence of a model with the other given conditions,” as if the satisfaction of the other conditions was enough (it isn’t; a countermodel must satisfy all of the given premises while falsifying the conclusion). Moreover, just a few sentences later GPT-4 claims that <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math><span class="html-italic">does</span> imply <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> under the given interpretation, contradicting its own earlier claim. We will see that <span class="html-italic">internal inconsistency</span> is a recurrent theme.</div> <div class="html-p">When we ask it to elaborate on why it thinks that <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> implies <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>, GPT-4’s response is revealing:</div> <div class="html-p"> <b>KA</b><span class="html-italic">: Can you explain why this model satisfies the first constraint (namely, that <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> implies <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>)?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">You are correct that my previous explanation was not clear enough for the first constraint. I apologize for that. In the given model, <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> does not necessarily imply <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>. Let’s reconsider the model and find one that satisfies all the constraints.</span> </div> <div class="html-p"><dl class="html-order"> <dt id> <span class="html-italic">1.</span> </dt> <dd><div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>: x is a multiple of 4</span></div></dd> <dt id> <span class="html-italic">2.</span> </dt> <dd><div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>: x is an even number</span></div></dd> <dt id> <span class="html-italic">3.</span> </dt> <dd><div class="html-p"><math display="inline"><semantics> <mrow> <mi>a</mi> <mo>:</mo> <mn>4</mn> </mrow> </semantics></math></div></dd> </dl></div> <div class="html-p" style="text-indent: 0;"><span class="html-italic">Now let’s check if these interpretations satisfy the given conditions:</span></div> <div class="html-p"><dl class="html-order"> <dt id> <span class="html-italic">1.</span> </dt> <dd><div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> implies <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>: If x is a multiple of 4, then x is an even number. This statement is universally true because every multiple of 4 is also a multiple of 2, making it an even number.</span></div></dd> <dt id> <span class="html-italic">2.</span> </dt> <dd><div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </semantics></math> does not hold: <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </semantics></math> means 4 is an even number, which is true. So, this condition is not satisfied.</span></div></dd> </dl></div> <div class="html-p" style="text-indent: 0;"><span class="html-italic">To satisfy all three conditions, we can modify the interpretations as follows:</span></div> <div class="html-p"><dl class="html-order"> <dt id> <span class="html-italic">1.</span> </dt> <dd><div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>: x is a positive multiple of 4</span></div></dd> <dt id> <span class="html-italic">2.</span> </dt> <dd><div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>: x is a positive even number</span></div></dd> <dt id> <span class="html-italic">3.</span> </dt> <dd><div class="html-p"><math display="inline"><semantics> <mrow> <mi>a</mi> <mo>:</mo> <mo>−</mo> <mn>4</mn> </mrow> </semantics></math></div></dd> </dl></div> <div class="html-p" style="text-indent: 0;"><span class="html-italic">Now let’s check if these interpretations satisfy the given conditions:</span></div> <div class="html-p"><dl class="html-order"> <dt id> <span class="html-italic">1.</span> </dt> <dd><div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> implies <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>: If x is a positive multiple of 4, then x is a positive even number. This statement is universally true because every positive multiple of 4 is also a multiple of 2, making it a positive even number.</span></div></dd> <dt id> <span class="html-italic">2.</span> </dt> <dd><div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </semantics></math> does not hold: <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mo>−</mo> <mn>4</mn> <mo>)</mo> </mrow> </semantics></math> means <math display="inline"><semantics> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </semantics></math> is a positive even number, which is false. So, this condition is satisfied.</span></div></dd> <dt id> <span class="html-italic">3.</span> </dt> <dd><div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </semantics></math> holds: <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mo>−</mo> <mn>4</mn> <mo>)</mo> </mrow> </semantics></math> means <math display="inline"><semantics> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </semantics></math> is a positive multiple of 4, which is false. So, this condition is not satisfied.</span></div></dd> </dl></div> <div class="html-p" style="text-indent: 0;"><span class="html-italic">It seems that finding a model that satisfies all three conditions is challenging. I must correct my previous claim: there may not be a model that satisfies all three conditions, <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> implies <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </semantics></math> does not hold, and <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </semantics></math> holds, simultaneously.</span></div> <div class="html-p">Thus, after multiple unsuccessful attempts to find a countermodel, GPT-4 acknowledges that such a countermodel might not exist. But, remarkably, when GPT-4 is asked to settle the matter one way or the other, it jumps back to the position that a countermodel does exist after all.</div> <div class="html-p"> <b>KA</b><span class="html-italic">: Can you prove that one exists or that one does not exist?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">I apologize for the confusion earlier. Let me provide a formal proof to show that a model that satisfies all three conditions <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> implies <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </semantics></math> does not hold, and <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </semantics></math> holds does indeed exist. Let’s consider a model with the following interpretations:</span> </div> <div class="html-p"><dl class="html-order"> <dt id> <span class="html-italic">1.</span> </dt> <dd><div class="html-p"><span class="html-italic">P(x): x is an element of set A</span></div></dd> <dt id> <span class="html-italic">2.</span> </dt> <dd><div class="html-p"><span class="html-italic">Q(x): x is an element of set B</span></div></dd> <dt id> <span class="html-italic">3.</span> </dt> <dd><div class="html-p"><span class="html-italic">a: a specific element, a</span></div></dd> </dl></div> <div class="html-p"><span class="html-italic">Now, let’s define the sets A and B as follows:</span></div> <div class="html-p"><span class="html-italic">Set <math display="inline"><semantics> <mrow> <mi>A</mi> <mo>=</mo> <mo>{</mo> <mi>a</mi> <mo>}</mo> </mrow> </semantics></math>.</span></div> <div class="html-p"><span class="html-italic">Set <math display="inline"><semantics> <mrow> <mi>B</mi> <mo>=</mo> <mo>{</mo> <mi>x</mi> <mspace width="0.222222em" /> <mo>|</mo> <mspace width="0.222222em" /> <mrow> <mi>x</mi> <mi>i</mi> <mi>s</mi> <mi>a</mi> <mi>n</mi> <mi>e</mi> <mi>l</mi> <mi>e</mi> <mi>m</mi> <mi>e</mi> <mi>n</mi> <mi>t</mi> <mi>o</mi> <mi>f</mi> <mi>A</mi> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mi>x</mi> <mo>≠</mo> <mi>a</mi> </mrow> <mo>}</mo> </mrow> </semantics></math>.</span></div> <div class="html-p"><span class="html-italic">Now, let’s check if these interpretations satisfy the given conditions:</span></div> <div class="html-p"><ul class="html-order"> <li><div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> implies <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>: If x is an element of set A, then x is an element of set B. In this case, since the only element in set A is a, and set B is defined such that it contains all elements of A except a, this implication is vacuously true. This is because there are no other elements in set A for which the implication would be false.</span></div></li> <li><div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </semantics></math> does not hold: <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </semantics></math> means a is an element of set B. By definition, set B does not include a, so this condition is satisfied.</span></div></li> <li><div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </semantics></math> holds: <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </semantics></math> means a is an element of set A. Since set A is defined to contain only a, this condition is also satisfied.</span></div></li> </ul></div> <div class="html-p"><span class="html-italic">Thus, the model with the interpretations</span></div> <div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>: x is an element of set A,</span></div> <div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>: x is an element of set B,</span></div> <div class="html-p"><span class="html-italic">and a: a specific element,</span></div> <div class="html-p"><span class="html-italic">satisfies the conditions <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> implies <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </semantics></math> does not hold, and <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </semantics></math> holds.</span></div> <div class="html-p">The response verges on nonsensical. GPT-4 has defined <span class="html-italic">B</span> (in a roundabout way) as the empty set <span class="html-italic">∅</span>, so that <span class="html-italic">Q</span> is always false. This means that <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> cannot possibly imply <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> as long as there is a single value of <span class="html-italic">x</span> for which <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> holds—and GPT-4 itself has provided such a value: <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mi>a</mi> </mrow> </semantics></math>. Yet somehow GPT-4 claims that <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> implies <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>.</div> <div class="html-p"> <b>KA</b><span class="html-italic">: Can you again explain why <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> implies <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> in this model?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">You are correct. My previous explanation for <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> implies <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> was not accurate. I apologize for the confusion. I will now provide a correct model that satisfies all three conditions. Let’s consider a model with the following interpretations:</span> </div> <div class="html-p"><dl class="html-order"> <dt id> <span class="html-italic">1.</span> </dt> <dd><div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>: x is divisible by 3</span></div></dd> <dt id> <span class="html-italic">2.</span> </dt> <dd><div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>: x is divisible by 2</span></div></dd> <dt id> <span class="html-italic">3.</span> </dt> <dd><div class="html-p"><span class="html-italic">a: 3</span></div></dd> </dl></div> <div class="html-p"><span class="html-italic">Now, let’s check if these interpretations satisfy the given conditions:</span></div> <div class="html-p"><dl class="html-order"> <dt id> <span class="html-italic">1.</span> </dt> <dd><div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> implies <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>: If x is divisible by 3, then x is divisible by 2. This statement is not universally true. However, we can create a domain D that contains only one element, <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, which is divisible by 3 but not by 2. By restricting the domain, the implication <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> implies <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> becomes vacuously true within the domain D because there are no elements in D for which the implication would be false.</span></div></dd> <dt id> <span class="html-italic">2.</span> </dt> <dd><div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </semantics></math> does not hold: <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics></math> means 3 is divisible by 2. Since 3 is not divisible by 2, this condition is satisfied.</span></div></dd> <dt id> <span class="html-italic">3.</span> </dt> <dd><div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </semantics></math> holds: <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics></math> means 3 is divisible by 3. Since 3 is divisible by 3, this condition is also satisfied.</span></div></dd> </dl></div> <div class="html-p"><span class="html-italic">Thus, the model with the interpretations</span></div> <div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>: x is divisible by 3,</span></div> <div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>: x is divisible by 2, and</span></div> <div class="html-p"><span class="html-italic">a: 3,</span></div> <div class="html-p"><span class="html-italic">and the domain <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> satisfies the conditions <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> implies <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </semantics></math> does not hold, and <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </semantics></math> holds.</span></div></section><section id="sec3dot5-preprints-81518" type><h4 class="html-italic" data-nested="2"> 3.5. Simple Quantifier Semantics</h4> <div class="html-p">Here we give GPT-4 two easy problems to test its understanding of quantifiers. Here is the first problem:</div> <div class="html-p"> <b>KA</b><span class="html-italic">: Consider the following three sentences:</span> </div> <div class="html-p"><dl class="html-order"> <dt id> <span class="html-italic">1.</span> </dt> <dd><div class="html-p"><tt>[forall x.P(x) ==> Q(x)]</tt></div></dd> <dt id> <span class="html-italic">2.</span> </dt> <dd><div class="html-p"><tt>[exists x.P(x)]</tt></div></dd> <dt id> <span class="html-italic">3.</span> </dt> <dd><div class="html-p"><tt>[exists x.<math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math>Q(x)]</tt></div></dd> </dl></div> <div class="html-p"><span class="html-italic">Either disprove or prove the following claim: These three sentences are jointly satisfiable.</span></div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">The three sentences are not jointly satisfiable. Here’s why:</span> </div> <div class="html-p"><dl class="html-order"> <dt id> <span class="html-italic">1.</span> </dt> <dd><div class="html-p"> <tt>[forall x.P(x) ==> Q(x)]</tt><span class="html-italic">says that for all x, if P(x) is true then Q(x) is true.</span> </div></dd> <dt id> <span class="html-italic">2.</span> </dt> <dd><div class="html-p"> <tt>[exists x.P(x)]</tt><span class="html-italic">says that there exists at least one x such that P(x) is true.</span> </div></dd> <dt id> <span class="html-italic">3.</span> </dt> <dd><div class="html-p"> <tt>[exists x.<math display="inline"><semantics> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> </semantics></math>Q(x)]</tt><span class="html-italic">says that there exists at least one x such that Q(x) is not true.</span> </div></dd> </dl></div> <div class="html-p"><span class="html-italic">If we consider sentence 2, there exists at least one x such that P(x) is true. According to sentence 1, for this x (or these x’s), Q(x) must also be true.</span></div> <div class="html-p"><span class="html-italic">However, sentence 3 states that there exists at least one x for which Q(x) is not true. Given sentences 1 and 2, this cannot be the case - there cannot exist an x for which P(x) is true and yet Q(x) is not true. Therefore, the three sentences are not jointly satisfiable.</span></div> <div class="html-p">The sentences are clearly jointly satisfiable, a trivial model being the domain <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>}</mo> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>¬</mo> <mi>P</mi> <mo>(</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mo>¬</mo> <mi>Q</mi> <mo>(</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math>.</div> <div class="html-p">The second problem concerns this biconditional: <div class="html-disp-formula-info" id="FD1-preprints-81518"> <div class="f"> <math display="block"><semantics> <mrow> <mo>[</mo> <mo>∀</mo> <mspace width="0.222222em" /> <mi>x</mi> <mrow> <mspace width="0.277778em" /> <mo>.</mo> <mspace width="0.277778em" /> </mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>⇔</mo> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>]</mo> <mo>⇔</mo> <mo>[</mo> <mo>(</mo> <mo>∀</mo> <mspace width="0.222222em" /> <mi>x</mi> <mrow> <mspace width="0.277778em" /> <mo>.</mo> <mspace width="0.277778em" /> </mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>)</mo> <mo>⇔</mo> <mo>(</mo> <mo>∀</mo> <mspace width="0.222222em" /> <mi>x</mi> <mrow> <mspace width="0.277778em" /> <mo>.</mo> <mspace width="0.277778em" /> </mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>)</mo> <mo>]</mo> </mrow> </semantics></math> </div> <div class="l"> <label>(1)</label> </div> </div> The left-to-right implication holds, but the right-to-left direction fails. Counterexamples are easy to find, for example, take the domain to be integers, <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> to mean <span class="html-italic">x</span> is even and <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> to mean <span class="html-italic">x</span> is odd. Then the equivalence on the right-hand side of (<a href="#FD1-preprints-81518" class="html-disp-formula">1</a>) holds, but clearly it’s not true that every integer is even iff it is odd.</div> <div class="html-p"> <b>KA</b><span class="html-italic">: Prove or disprove the following:</span> <tt>(forall x.P(x) <==> Q(x))</tt><span class="html-italic">holds if and only if the following biconditional holds:</span> <tt>(forall x.P(x)) <==> (forall x.Q(x))</tt> . </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">We will prove this statement by showing that both directions of the main biconditional hold.</span> </div> <div class="html-p"><ul class="html-bullet"> <li> <div class="html-p"> <tt>(forall x.P(x)<==>Q(x))</tt>  <span class="html-italic">implies</span>  <tt>(forall x.P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">Assume that for all x,</span>  <tt>P(x)<==>Q(x)</tt>  <span class="html-italic">holds.</span> </div> <div class="html-p"> <span class="html-italic">Direction 1: If forall x,</span>  <tt>P(x)</tt>  <span class="html-italic">holds, we need to show that forall x, Q(x) holds. Since</span>  <tt>P(x)<==>Q(x)</tt>  <span class="html-italic">for all x, if</span>  <tt>P(x)</tt>  <span class="html-italic">holds for all x, then</span>  <tt>Q(x)</tt>  <span class="html-italic">must also hold for all x, as they are equivalent.</span> </div> <div class="html-p"> <span class="html-italic">Direction 2: If forall x,</span>  <tt>Q(x)</tt>  <span class="html-italic">holds, we need to show that forall x,</span>  <tt>P(x)</tt>  <span class="html-italic">holds. Since</span>  <tt>P(x)<==>Q(x)</tt>  <span class="html-italic">for all x, if</span>  <tt>Q(x)</tt>  <span class="html-italic">holds for all x, then</span>  <tt>P(x)</tt>  <span class="html-italic">must also hold for all x, as they are equivalent.</span> </div> <div class="html-p"> <span class="html-italic">Thus, we have shown that</span>  <tt>(forall x.P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">holds when</span>  <tt>(forall x.P(x)<==>Q(x))</tt>  <span class="html-italic">holds.</span> </div> </li> <li> <div class="html-p"> <tt>(forall x.P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">implies</span>  <tt>(forall.P(x)<==>Q(x))</tt>  <span class="html-italic">Assume that</span>  <tt>(forall x.P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">holds.</span> </div> <div class="html-p"> <span class="html-italic">We need to show that for all x,</span> <tt>P(x)<==>Q(x)</tt> . </div> <div class="html-p"> <span class="html-italic">Let</span>  <tt>x</tt>  <span class="html-italic">be an arbitrary element from the domain of discourse. We will consider the two directions of the biconditional</span>  <tt>P(x)<==>Q(x)</tt>  <span class="html-italic">:</span> </div> <div class="html-p"> <span class="html-italic">Direction 1: If</span>  <tt>P(x)</tt>  <span class="html-italic">holds, we need to show that</span>  <tt>Q(x)</tt>  <span class="html-italic">holds. If</span>  <tt>P(x)</tt>  <span class="html-italic">holds for our arbitrary</span>  <tt>x</tt>  <span class="html-italic">, then it must be the case that forall</span>  <tt>x</tt>  <span class="html-italic">,</span>  <tt>P(x)</tt>  <span class="html-italic">holds. Since</span>  <tt>(forall x.P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">holds, it follows that forall</span>  <tt>x</tt>  <span class="html-italic">,</span>  <tt>Q(x)</tt>  <span class="html-italic">holds. Therefore,</span>  <tt>Q(x)</tt>  <span class="html-italic">must hold for our arbitrary</span>  <tt>x</tt>  <span class="html-italic">as well.</span> </div> <div class="html-p"> <span class="html-italic">Direction 2: If</span>  <tt>Q(x)</tt>  <span class="html-italic">holds, we need to show that</span>  <tt>P(x)</tt>  <span class="html-italic">holds. If</span>  <tt>Q(x)</tt>  <span class="html-italic">holds for our arbitrary</span>  <tt>x</tt>  <span class="html-italic">, then it must be the case that forall</span>  <tt>x</tt>  <span class="html-italic">,</span>  <tt>Q(x)</tt>  <span class="html-italic">holds. Since</span>  <tt>(forall x.P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">holds, it follows that forall</span>  <tt>x</tt>  <span class="html-italic">,</span>  <tt>P(x)</tt>  <span class="html-italic">holds. Therefore,</span>  <tt>P(x)</tt>  <span class="html-italic">must hold for our arbitrary</span>  <tt>x</tt>  <span class="html-italic">as well.</span> </div> </li> </ul></div> <div class="html-p" style="text-indent: 0;"> <span class="html-italic">Thus, we have shown that for all</span>  <tt>x</tt>  <span class="html-italic">,</span>  <tt>P(x) <==> Q(x)</tt>  <span class="html-italic">holds when</span>  <tt>(forall x.P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">holds.</span> </div> <div class="html-p"> <span class="html-italic">In conclusion, we have shown that</span>  <tt>(forall x.P(x)<==>Q(x))</tt>  <span class="html-italic">holds if and only if</span>  <tt>(forall x.P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">holds.</span> </div> <div class="html-p">Notice how pernicious the fallacy is here: <span class="html-italic">“If</span> <tt>P(x)</tt><span class="html-italic">holds for our arbitrary</span> <tt>x</tt><span class="html-italic">, then it must be the case that forall</span> <tt>x</tt><span class="html-italic">,</span> <tt>P(x)</tt><span class="html-italic">holds.”</span> It sounds plausible, but of course it is quite wrong.</div> <div class="html-p"> <b>KA</b><span class="html-italic">: Can you elaborate on your claim “If P(x) holds for our arbitrary x, then it must be the case that forall x, P(x) holds”?</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: You are correct to question that claim. It was a mistake in my response. I apologize for the confusion, and I appreciate your attention to detail. Here is the corrected proof.</span> </div> <div class="html-p"> <tt>(forall x.P(x)<==>Q(x))</tt>  <span class="html-italic">implies</span>  <tt>(forall x.P(x))<==>(forall x . Q(x))</tt> </div> <div class="html-p"> <span class="html-italic">Assume that for all</span>  <tt>x</tt>  <span class="html-italic">,</span>  <tt>P(x)<==>Q(x)</tt>  <span class="html-italic">holds.</span> </div> <div class="html-p"> <span class="html-italic">Direction 1: If forall</span>  <tt>x</tt>  <span class="html-italic">,</span>  <tt>P(x)</tt>  <span class="html-italic">holds, we need to show that forall</span>  <tt>x</tt>  <span class="html-italic">,</span>  <tt>Q(x)</tt>  <span class="html-italic">holds. Since</span>  <tt>P(x)<==>Q(x)</tt>  <span class="html-italic">for all</span>  <tt>x</tt>  <span class="html-italic">, if</span>  <tt>P(x)</tt>  <span class="html-italic">holds for all</span>  <tt>x</tt>  <span class="html-italic">, then</span>  <tt>Q(x)</tt>  <span class="html-italic">must also hold for all</span>  <tt>x</tt>  <span class="html-italic">, as they are equivalent.</span> </div> <div class="html-p"> <span class="html-italic">Direction 2: If forall</span>  <tt>x</tt>  <span class="html-italic">,</span>  <tt>Q(x)</tt>  <span class="html-italic">holds, we need to show that forall</span>  <tt>x</tt>  <span class="html-italic">,</span>  <tt>P(x)</tt>  <span class="html-italic">holds. Since</span>  <tt>P(x)<==>Q(x)</tt>  <span class="html-italic">for all</span>  <tt>x</tt>  <span class="html-italic">, if</span>  <tt>Q(x)</tt>  <span class="html-italic">holds for all</span>  <tt>x</tt>  <span class="html-italic">, then</span>  <tt>P(x)</tt>  <span class="html-italic">must also hold for all</span>  <tt>x</tt>  <span class="html-italic">, as they are equivalent.</span> </div> <div class="html-p"> <span class="html-italic">Thus, we have shown that</span>  <tt>(forall x.P(x))<==>(forall x . Q(x))</tt>  <span class="html-italic">holds when</span>  <tt>(forall x.P(x)<==>Q(x))</tt>  <span class="html-italic">holds.</span> </div> <div class="html-p"> <tt>(forall x.P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">implies</span>  <tt>(forall x.P(x)<==>Q(x))</tt> </div> <div class="html-p"> <span class="html-italic">Assume that</span>  <tt>(forall x.P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">holds.</span> </div> <div class="html-p"> <span class="html-italic">We need to show that for all</span> <tt>x</tt><span class="html-italic">,</span> <tt>P(x)<==>Q(x)</tt> . </div> <div class="html-p"> <span class="html-italic">Let</span>  <tt>x</tt>  <span class="html-italic">be an arbitrary element from the domain of discourse. We will consider the two directions of the biconditional</span>  <tt>P(x)<==>Q(x)</tt>  <span class="html-italic">:</span> </div> <div class="html-p"> <span class="html-italic">Direction 1: If</span> <tt>P(x)</tt><span class="html-italic">holds, we need to show that</span> <tt>Q(x)</tt><span class="html-italic">holds. If</span> <tt>P(x)</tt><span class="html-italic">holds for our arbitrary</span> <tt>x</tt><span class="html-italic">, we cannot directly conclude that forall</span> <tt>x</tt><span class="html-italic">,</span> <tt>P(x)</tt><span class="html-italic">holds. However, we can proceed as follows: Since</span> <tt>(forall x.P(x))<==>(forall x.Q(x))</tt><span class="html-italic">holds, either both sides of the biconditional are true or both sides are false. If forall</span> <tt>x</tt><span class="html-italic">,</span> <tt>P(x)</tt><span class="html-italic">holds, then by the biconditional, forall</span> <tt>x</tt><span class="html-italic">,</span> <tt>Q(x)</tt><span class="html-italic">must hold as well, and thus</span> <tt>Q(x)</tt><span class="html-italic">holds for our arbitrary</span> <tt>x</tt><span class="html-italic">. If it is not the case that forall</span> <tt>x</tt><span class="html-italic">,</span> <tt>P(x)</tt><span class="html-italic">holds, then there must be some</span> <tt>x</tt><span class="html-italic">where</span> <tt>P(x)</tt><span class="html-italic">does not hold. But, for our arbitrary</span> <tt>x</tt><span class="html-italic">,</span> <tt>P(x)</tt><span class="html-italic">holds, which means our arbitrary</span> <tt>x</tt><span class="html-italic">cannot be one of those instances. In this case, it is not guaranteed that</span> <tt>Q(x)</tt><span class="html-italic">holds for our arbitrary</span> <tt>x</tt> . </div> <div class="html-p"> <span class="html-italic">Direction 2: If</span> <tt>Q(x)</tt><span class="html-italic">holds, we need to show that</span> <tt>P(x)</tt><span class="html-italic">holds. Similarly, if</span> <tt>Q(x)</tt><span class="html-italic">holds for our arbitrary</span> <tt>x</tt><span class="html-italic">, we cannot directly conclude that forall</span> <tt>x</tt><span class="html-italic">,</span> <tt>Q(x)</tt><span class="html-italic">holds. We can use the same approach as in Direction 1. Since</span> <tt>(forall x.P(x))<==>(forall x.Q(x))</tt><span class="html-italic">holds, either both sides of the biconditional are true or both sides are false. If forall</span> <tt>x</tt><span class="html-italic">,</span> <tt>Q(x)</tt><span class="html-italic">holds, then by the biconditional, forall</span> <tt>x</tt><span class="html-italic">,</span> <tt>P(x)</tt><span class="html-italic">must hold as well, and thus</span> <tt>P(x)</tt><span class="html-italic">holds for our arbitrary</span> <tt>x</tt><span class="html-italic">. If it is not the case that forall</span> <tt>x</tt><span class="html-italic">,</span> <tt>Q(x)</tt><span class="html-italic">holds, then there must be some</span> <tt>x</tt><span class="html-italic">where</span> <tt>Q(x)</tt><span class="html-italic">does not hold. But, for our arbitrary</span> <tt>x</tt><span class="html-italic">,</span> <tt>Q(x)</tt><span class="html-italic">holds, which means our arbitrary</span> <tt>x</tt><span class="html-italic">cannot be one of those instances. In this case, it is not guaranteed that</span> <tt>P(x)</tt><span class="html-italic">holds for our arbitrary</span> <tt>x</tt> . </div> <div class="html-p"> <span class="html-italic">As we cannot guarantee that</span>  <tt>P(x)<==>Q(x)</tt>  <span class="html-italic">for our arbitrary</span> </div> <div class="html-p"> <b>KA</b><span class="html-italic">: Please continue.</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: My apologies for the premature ending in my previous response. Let me provide an alternative approach to the proof for the second part.</span>  <tt>(forall x.P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">implies</span>  <tt>(forall x.P(x)<==>Q(x))</tt>  <span class="html-italic">Assume that</span>  <tt>(forall x.P(x))</tt>  <span class="html-italic">holds.</span> </div> <div class="html-p"> <span class="html-italic">We need to show that for all</span> <tt>x</tt><span class="html-italic">,</span> <tt>P(x)<==>Q(x)</tt> . </div> <div class="html-p"> <span class="html-italic">Let</span>  <tt>x</tt>  <span class="html-italic">be an arbitrary element from the domain of discourse. We will consider the two directions of the biconditional</span>  <tt>P(x)<==>Q(x)</tt>  <span class="html-italic">:</span> </div> <div class="html-p"> <span class="html-italic">Direction 1: If</span>  <tt>P(x)</tt>  <span class="html-italic">holds, we need to show that</span>  <tt>Q(x)</tt>  <span class="html-italic">holds. We will use proof by contradiction. Assume that</span>  <tt>Q(x)</tt>  <span class="html-italic">does not hold for our arbitrary</span>  <tt>x</tt>  <span class="html-italic">. Now, since</span>  <tt>Q(x)</tt>  <span class="html-italic">does not hold, it cannot be the case that forall</span>  <tt>x</tt>  <span class="html-italic">,</span>  <tt>Q(x)</tt>  <span class="html-italic">holds. Since</span>  <tt>(forall x<==>P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">holds, this implies that it cannot be the case that forall</span>  <tt>x</tt>  <span class="html-italic">,</span>  <tt>P(x)</tt>  <span class="html-italic">holds. However, we assumed that</span>  <tt>P(x)</tt>  <span class="html-italic">holds for our arbitrary</span>  <tt>x</tt>  <span class="html-italic">, which leads to a contradiction. Thus, our assumption that</span>  <tt>Q(x)</tt>  <span class="html-italic">does not hold must be incorrect, and</span>  <tt>Q(x)</tt>  <span class="html-italic">must hold.</span> </div> <div class="html-p"> <span class="html-italic">Direction 2: If</span>  <tt>Q(x)</tt>  <span class="html-italic">holds, we need to show that</span>  <tt>P(x)</tt>  <span class="html-italic">holds. We will use proof by contradiction again. Assume that</span>  <tt>P(x)</tt>  <span class="html-italic">does not hold for our arbitrary</span>  <tt>x</tt>  <span class="html-italic">. Now, since</span>  <tt>P(x)</tt>  <span class="html-italic">does not hold, it cannot be the case that forall</span>  <tt>x</tt>  <span class="html-italic">,</span>  <tt>P(x)</tt>  <span class="html-italic">holds. Since</span>  <tt>(forall x.P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">holds, this implies that it cannot be the case that forall</span>  <tt>x</tt>  <span class="html-italic">,</span>  <tt>Q(x)</tt>  <span class="html-italic">holds. However, we assumed that</span>  <tt>Q(x)</tt>  <span class="html-italic">holds for our arbitrary</span>  <tt>x</tt>  <span class="html-italic">, which leads to a contradiction. Thus, our assumption that</span>  <tt>P(x)</tt>  <span class="html-italic">does not hold must be incorrect, and</span>  <tt>P(x)</tt>  <span class="html-italic">must hold.</span> </div> <div class="html-p"> <span class="html-italic">In both directions, we have shown that</span>  <tt>P(x)<==>Q(x)</tt>  <span class="html-italic">holds for our arbitrary</span>  <tt>x</tt>  <span class="html-italic">. Since</span>  <tt>x</tt>  <span class="html-italic">was arbitrary, this holds for all</span>  <tt>x</tt>  <span class="html-italic">. Therefore,</span><div class="html-disp-formula-info" id> <div class="f"> <tt>(forall x.P(x)<==>Q(x))</tt> </div> <div class="l"> </div> </div> <span class="html-italic">holds when</span>  <tt>(forall x.P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">holds.</span> </div> <div class="html-p"> <span class="html-italic">In conclusion, we have shown that</span>  <tt>(forall x.P(x)<==>Q(x))</tt>  <span class="html-italic">holds if and only if</span>  <tt>(forall x.P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">holds.</span> </div> <div class="html-p">While the reasoning appears to proceed along the right general lines, it is, again, insidiously wrong. The reductio ad absurdum given for the two directions hinges on the fallacy that if a property <span class="html-italic">T</span> is assumed to hold for a <span class="html-italic">particular but arbitrarily chosenx</span>, then we must have <math display="inline"><semantics> <mrow> <mo>∀</mo> <mspace width="0.222222em" /> <mi>x</mi> <mspace width="0.222222em" /> <mo>.</mo> <mspace width="0.222222em" /> <mi>T</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>, leading to the erroneous conclusion that <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>¬</mo> <mo>∀</mo> <mspace width="0.222222em" /> <mi>x</mi> <mspace width="0.222222em" /> <mo>.</mo> <mspace width="0.222222em" /> <mi>T</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> are jointly inconsistent. It is as if GPT-4 knows what needs to be done but does not know how to do it properly, and instead of saying so, it proceeds to generate a bogus but plausible-looking “proof.”</div> <div class="html-p"> <b>KA</b><span class="html-italic">: Please clarify your claim that “However, we assumed that P(x) holds for our arbitrary x, which leads to a contradiction”.</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: You are correct to question that claim, as the reasoning was unclear. I apologize for the confusion. Let me provide a clearer explanation for the second part of the proof.</span><div class="html-disp-formula-info" id> <div class="f"> <tt>(forall x.P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">implies</span>  <tt>(forall x.P(x)<==>Q(x))</tt> </div> <div class="l"> </div> </div> <span class="html-italic">Assume that</span>  <tt>(forall x.P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">holds.</span> </div> <div class="html-p"> <span class="html-italic">We need to show that for all</span> <tt>x</tt><span class="html-italic">,</span> <tt>P(x)<==>Q(x)</tt> . </div> <div class="html-p"> <span class="html-italic">Let</span>  <tt>x</tt>  <span class="html-italic">be an arbitrary element from the domain of discourse. We will consider the two directions of the biconditional</span>  <tt>P(x)<==>Q(x)</tt>  <span class="html-italic">:</span> </div> <div class="html-p"> <span class="html-italic">Direction 1: If</span> <tt>P(x)</tt><span class="html-italic">holds, we need to show that</span> <tt>Q(x)</tt><span class="html-italic">holds. If</span> <tt>P(x)</tt><span class="html-italic">holds for our arbitrary</span> <tt>x</tt><span class="html-italic">, we can make the following argument: If it is not the case that forall</span> <tt>x</tt><span class="html-italic">,</span> <tt>P(x)</tt><span class="html-italic">holds, then there must exist some</span> <tt>x</tt><span class="html-italic">for which</span> <tt>P(x)</tt><span class="html-italic">does not hold. However, if forall</span> <tt>x</tt><span class="html-italic">,</span> <tt>P(x)</tt><span class="html-italic">does hold, then by the given biconditional</span> <tt>(forall x.P(x))<==>(forall x.Q(x))</tt><span class="html-italic">forall</span> <tt>x</tt><span class="html-italic">,</span> <tt>Q(x)</tt><span class="html-italic">must also hold. Since</span> <tt>P(x)</tt><span class="html-italic">holds for our arbitrary</span> <tt>x</tt><span class="html-italic">, either it holds for all</span> <tt>x</tt><span class="html-italic">or there exists some other</span> <tt>x</tt><span class="html-italic">where</span> <tt>P(x)</tt><span class="html-italic">does not hold. In either case, we can’t deduce anything about</span> <tt>Q(x)</tt><span class="html-italic">for our arbitrary</span> <tt>x</tt> . </div> <div class="html-p"> <span class="html-italic">Direction 2: If</span> <tt>Q(x)</tt><span class="html-italic">holds, we need to show that</span> <tt>P(x)</tt><span class="html-italic">holds. If</span> <tt>Q(x)</tt><span class="html-italic">holds for our arbitrary</span> <tt>x</tt><span class="html-italic">, we can make a similar argument as in Direction 1: If it is not the case that forall</span> <tt>x</tt><span class="html-italic">,</span> <tt>Q(x)</tt><span class="html-italic">holds, then there must exist some</span> <tt>x</tt><span class="html-italic">for which</span> <tt>Q(x)</tt><span class="html-italic">does not hold. However, if forall</span> <tt>x</tt><span class="html-italic">,</span> <tt>Q(x)</tt><span class="html-italic">does hold, then by the given biconditional</span> <tt>(forall x.P(x))<==>(forall x.Q(x))</tt><span class="html-italic">, forall</span> <tt>x</tt><span class="html-italic">,</span> <tt>P(x)</tt><span class="html-italic">must also hold. Since</span> <tt>Q(x)</tt><span class="html-italic">holds for our arbitrary</span> <tt>x</tt><span class="html-italic">, either it holds for all</span> <tt>x</tt><span class="html-italic">or there exists some other</span> <tt>x</tt><span class="html-italic">where</span> <tt>Q(x)</tt><span class="html-italic">does not hold. In either case, we can’t deduce anything about</span> <tt>P(x)</tt><span class="html-italic">for our arbitrary</span> <tt>x</tt> . </div> <div class="html-p"> <span class="html-italic">Since we can’t make any definitive conclusions about the relationship between</span>  <tt>P(x)</tt>  <span class="html-italic">and</span>  <tt>Q(x)</tt>  <span class="html-italic">for our arbitrary</span>  <tt>x</tt>  <span class="html-italic">based solely on the given biconditional</span>  <tt>(forall x.P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">, it is incorrect to claim that</span><div class="html-disp-formula-info" id> <div class="f"> <tt>(forall x.P(x))<==>(forall x.Q(x))</tt> </div> <div class="l"> </div> </div> <span class="html-italic">implies</span> <tt>(forall x.P(x)<==>Q(x))</tt> . </div> <div class="html-p">So where does that leave us? After the problematic inference step was pointed out to GPT-4 , it proceeded to “clarify” its reasoning, but ultimately it ended up admitting that its reasoning was wrong. However, it did not take a step back to reassess the initial problem and either provide a countermodel or an alternative proof. When it is explicitly asked to take a final position one way or another, it again chooses the wrong answer and provides yet another bogus “proof” for it:</div> <div class="html-p"> <b>KA</b><span class="html-italic">: So what does that prove?</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: My previous response was not clear, and I apologize for the confusion.</span> </div> <div class="html-p"><math display="inline"><semantics> <mrow> <mo>[</mo> <mo>⋯</mo> <mrow> <mi>G</mi> <mi>P</mi> <mi>T</mi> <mo>−</mo> <mn>4</mn> <mi>r</mi> <mi>e</mi> <mi>h</mi> <mi>a</mi> <mi>s</mi> <mi>h</mi> <mi>e</mi> <mi>s</mi> <mi>i</mi> <mi>t</mi> <mi>s</mi> <mi>p</mi> <mi>r</mi> <mi>e</mi> <mi>v</mi> <mi>i</mi> <mi>o</mi> <mi>u</mi> <mi>s</mi> <mi>a</mi> <mi>n</mi> <mi>s</mi> <mi>w</mi> <mi>e</mi> <mi>r</mi> <mi>h</mi> <mi>e</mi> <mi>r</mi> <mi>e</mi> </mrow> <mo>⋯</mo> <mo>]</mo> </mrow> </semantics></math></div> <div class="html-p"> <span class="html-italic">In conclusion, we have only shown that</span>  <tt>(forall x.P(x)<==>Q(x))</tt>  <span class="html-italic">implies</span><div class="html-disp-formula-info" id> <div class="f"> <tt>(forall x.P(x))<==>(forall x.Q(x))</tt> </div> <div class="l"> </div> </div> <span class="html-italic">but we have not proven the reverse implication. The statement</span>  <tt>(forall x.P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">implies</span>  <tt>(forall x.P(x)<==>Q(x))</tt>  <span class="html-italic">remains unproven.</span> </div> <div class="html-p"> <b>KA</b><span class="html-italic">: So can you prove it, or else disprove it?</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: I apologize for the confusion in my previous responses. I will now provide a correct proof for the second part of the question. The goal is to show that</span> <tt>(forall x.P(x))<==>(forall x.Q(x))</tt><span class="html-italic">implies</span> <tt>(forall x.P(x)<==>Q(x))</tt> . </div> <div class="html-p"> <tt>(forall x.P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">implies</span>  <tt>(forall x.P(x)<==>Q(x))</tt>  <span class="html-italic">. Assume that</span>  <tt>(forall x.P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">holds.</span> </div> <div class="html-p"> <span class="html-italic">We need to show that for all</span> <tt>x</tt><span class="html-italic">,</span> <tt>P(x)<==>Q(x)</tt> . </div> <div class="html-p"> <span class="html-italic">Let</span>  <tt>x</tt>  <span class="html-italic">be an arbitrary element from the domain of discourse. We will consider the two directions of the biconditional</span>  <tt>P(x)<==>Q(x)</tt>  <span class="html-italic">:</span> </div> <div class="html-p"> <span class="html-italic">Direction 1: If</span>  <tt>P(x)</tt>  <span class="html-italic">holds, we need to show that</span>  <tt>Q(x)</tt>  <span class="html-italic">holds.</span> </div> <div class="html-p"> <span class="html-italic">Since</span>  <tt>(forall x.P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">, there are two possible scenarios:</span> </div> <div class="html-p"> <span class="html-italic">Both sides of the biconditional are true: forall</span> <tt>x</tt><span class="html-italic">,</span> <tt>P(x)</tt><span class="html-italic">holds and forall</span> <tt>x</tt><span class="html-italic">,</span> <tt>Q(x)</tt><span class="html-italic">holds. Both sides of the biconditional are false: it is not the case that forall</span> <tt>x</tt><span class="html-italic">,</span> <tt>P(x)</tt><span class="html-italic">holds, and it is not the case that forall</span> <tt>x</tt><span class="html-italic">,</span> <tt>Q(x)</tt><span class="html-italic">holds. If scenario 1 is true, then</span> <tt>Q(x)</tt><span class="html-italic">holds for all</span> <tt>x</tt><span class="html-italic">, including our arbitrary x. If scenario 2 is true, then there must exist some element in the domain where</span> <tt>P(x)</tt><span class="html-italic">does not hold. However, for our arbitrary</span> <tt>x</tt><span class="html-italic">, we assumed that</span> <tt>P(x)</tt><span class="html-italic">holds. Thus, scenario 2 cannot be true for our arbitrary</span> <tt>x</tt><span class="html-italic">. Consequently, scenario 1 must be true, and</span> <tt>Q(x)</tt><span class="html-italic">holds for our arbitrary</span> <tt>x</tt> . </div> <div class="html-p"> <span class="html-italic">Direction 2: If</span>  <tt>Q(x)</tt>  <span class="html-italic">holds, we need to show that</span>  <tt>P(x)</tt>  <span class="html-italic">holds.</span> </div> <div class="html-p"> <span class="html-italic">The same analysis used in Direction 1 can be applied to Direction 2. If scenario 1 is true, then</span> <tt>P(x)</tt><span class="html-italic">holds for all</span> <tt>x</tt><span class="html-italic">, including our arbitrary</span> <tt>x</tt><span class="html-italic">. If scenario 2 is true, then there must exist some element in the domain where</span> <tt>Q(x)</tt><span class="html-italic">does not hold. However, for our arbitrary</span> <tt>x</tt><span class="html-italic">, we assumed that</span> <tt>Q(x)</tt><span class="html-italic">holds. Thus, scenario 2 cannot be true for our arbitrary</span> <tt>x</tt><span class="html-italic">. Consequently, scenario 1 must be true, and</span> <tt>P(x)</tt><span class="html-italic">holds for our arbitrary</span> <tt>x</tt> . </div> <div class="html-p"> <span class="html-italic">In both directions, we have shown that</span>  <tt>P(x)<==>Q(x)</tt>  <span class="html-italic">holds for our arbitrary</span>  <tt>x</tt>  <span class="html-italic">. Since x was arbitrary, this holds for all</span>  <tt>x</tt>  <span class="html-italic">. Therefore,</span>  <tt>(forall x.P(x)<==>Q(x))</tt>  <span class="html-italic">holds when</span><div class="html-disp-formula-info" id> <div class="f"> <tt>(forall x.P(x))<==>(forall x.Q(x))</tt> </div> <div class="l"> </div> </div> <span class="html-italic">holds.</span> </div> <div class="html-p"> <span class="html-italic">In conclusion, we have shown that</span>  <tt>(forall x.P(x)<==>Q(x))</tt>  <span class="html-italic">holds if and only if</span>  <tt>(forall x.P(x))<==>(forall x.Q(x))</tt>  <span class="html-italic">holds.</span> </div> <div class="html-p">Regrettably, the error is the same one that GPT-4 already made before (and which it supposedly fixed before). The model seems fundamentally confused about the role of the so-called <span class="html-italic">eigenvariable x</span> (the “arbitrarily chosen” <span class="html-italic">x</span>), believing that <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> is inconsistent with <math display="inline"><semantics> <mrow> <mo>¬</mo> <mo>∀</mo> <mspace width="0.222222em" /> <mi>x</mi> <mspace width="0.222222em" /> <mo>.</mo> <mspace width="0.222222em" /> <mi>T</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>.</div></section><section id="sec3dot6-preprints-81518" type><h4 class="html-italic" data-nested="2"> 3.6. Simple Graph Coloring</h4> <div class="html-p">We first consider a graph-coloring problem that does not have a solution. It is trivial to see that two colors do not suffice for the graph described in this problem (e.g., vertices 0, 2, and 4 form a clique and hence require at least 3 colors).</div> <div class="html-p"> <b>KA</b><span class="html-italic">: Consider an undirected graph with 6 vertices (0 through 5) and the following set of edges:</span><div class="html-disp-formula-info" id> <div class="f"> <math display="block"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>3</mn> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>5</mn> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>5</mn> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>)</mo> <mo>}</mo> <mo>.</mo> </mrow> </semantics></math> </div> <div class="l"> </div> </div> <span class="html-italic">Color every vertex either red or green, so that no two adjacent vertices receive the same color.</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">This undirected graph is complete, meaning that every vertex is connected to every other vertex by an edge. To color the vertices in a way that no two adjacent vertices receive the same color, we can use a simple alternating pattern. Here’s one possible solution:</span> </div> <div class="html-p"><ul class="html-bullet"> <li><div class="html-p"><span class="html-italic">Vertex 0: Red</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 1: Green</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 2: Red</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 3: Green</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 4: Red</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 5: Green</span></div></li> </ul></div> <div class="html-p" style="text-indent: 0;"><span class="html-italic">This coloring ensures that no two adjacent vertices have the same color. For example, vertex 0 is connected to vertices 1, 2, 3, 4, and 5. Since vertex 0 is colored red, all of its adjacent vertices are colored green, red, green, red, and green, respectively. The same goes for all other vertices.</span></div> <div class="html-p">The sheer number of egregious errors in this short output is shocking. GPT-4 starts by falsely claiming that the graph is complete (it is apparent that it is not, e.g., there is no edge between vertices 2 and 3). Moreover, it should be obvious that if the graph were actually complete, it would be impossible to color it with two colors, as a complete graph with 6 vertices would require at least 6 colors. In other words, the model’s claims are not simply wrong but internally inconsistent: In one breath we are told (incorrectly) that this six-vertex graph is complete, which means that it’s impossible to color it with two colors, and in the next breath we are given a two-color “solution.”</div> <div class="html-p">Remarkably, this dismal display is not because GPT-4 does not have sufficient knowledge or data about graphs (which are, after all, pervasive mathematical entities with copious information about them on the Internet, as well as in countless articles and books), and even about complete graphs in particular. When I asked GPT-4 to tell me what it knows about complete graphs it rattled off the correct definition of complete graphs along with a long list of results about <math display="inline"><semantics> <msub> <mi>K</mi> <mi>n</mi> </msub> </semantics></math> (the complete graphs with <span class="html-italic">n</span> vertices), such as the number of edges (<math display="inline"><semantics> <mrow> <mi>n</mi> <mo>·</mo> <mo>(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>)</mo> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>), the degree of each vertex (<math display="inline"><semantics> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>), the connectivity (maximal), the number of subgraphs (<math display="inline"><semantics> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>·</mo> <mo>(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>)</mo> <mo>/</mo> <mn>2</mn> </mrow> </msup> </semantics></math>), the fact that they’re Hamiltonian and Eulerian, their planarity, and a whole lot of other information, including their chromatic number: <span class="html-italic">Chromatic Number: The chromatic number of a complete graph <math display="inline"><semantics> <msub> <mi>K</mi> <mi>n</mi> </msub> </semantics></math> (i.e., the smallest number of colors needed to color the vertices of <math display="inline"><semantics> <msub> <mi>K</mi> <mi>n</mi> </msub> </semantics></math> so that no two adjacent vertices share the same color) is n</span> (this italicized sentence is a verbatim quotation from GPT-4). It is clear that GPT-4 has memorized all this information but is unable to use it in a new setting.</div> <div class="html-p">When I pointed out that the solution is wrong, GPT-4 went on to make the exact same type of mistake:</div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">But vertices 0 and 2 are adjacent and you’ve colored both of them red.</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">You are right, my previous response had an error. I apologize for that. Let’s try another coloring:</span> </div> <div class="html-p"><ul class="html-bullet"> <li><div class="html-p"><span class="html-italic">Vertex 0: Red</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 1: Green</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 2: Green</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 3: Red</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 4: Green</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 5: Red</span></div></li> </ul></div> <div class="html-p" style="text-indent: 0;"><span class="html-italic">In this coloring, no two adjacent vertices have the same color.</span></div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">But 0 and 3 are adjacent and again you’ve colored both of them red.</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">You are correct again, and I apologize for the mistake. Given the complete graph, we can use a bipartite coloring strategy. Let’s try this coloring:</span> </div> <div class="html-p"><ul class="html-bullet"> <li><div class="html-p"><span class="html-italic">Vertex 0: Red</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 1: Green</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 2: Red</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 3: Green</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 4: Red</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 5: Green</span></div></li> </ul></div> <div class="html-p" style="text-indent: 0;"><span class="html-italic">In this coloring, vertices 0, 2, and 4 are red, and vertices 1, 3, and 5 are green. As you can see, no two adjacent vertices share the same color.</span></div> <div class="html-p">At this point it was clear that asking the model to produce a solution with 2 colors was a dead end. A solution with three colors surely would be easier (such solutions do exist and are easy to find):</div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">Again, 0 and 2 are adjacent vertices. You’ve colored both of them red. Let’s try with 3 colors. Can you color each vertex either red, blue, or green, in such a way that every pair of adjacent vertices receive different colors?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">I apologize for the repeated mistakes in my previous responses. Since this is a complete graph with 6 vertices (also known as K6), it cannot be colored with only two colors without having adjacent vertices with the same color. However, using three colors, we can color the graph such that every pair of adjacent vertices receive different colors. Here’s a possible coloring with red, blue, and green:</span> </div> <div class="html-p"><ul class="html-bullet"> <li><div class="html-p"><span class="html-italic">Vertex 0: Red</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 1: Green</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 2: Blue</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 3: Red</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 4: Green</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 5: Blue</span></div></li> </ul></div> <div class="html-p" style="text-indent: 0;"><span class="html-italic">In this coloring, no two adjacent vertices share the same color, and the graph is properly colored using three colors.</span></div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">No, this is not correct either. 0 and 3 are adjacent and you’ve made both of them red.</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">I apologize once again for my mistake. I appreciate your patience. Here’s the correct coloring using three colors:</span> </div> <div class="html-p"><ul class="html-bullet"> <li><div class="html-p"><span class="html-italic">Vertex 0: Red</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 1: Green</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 2: Blue</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 3: Green</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 4: Red</span></div></li> <li><div class="html-p"><span class="html-italic">Vertex 5: Blue</span></div></li> </ul></div> <div class="html-p" style="text-indent: 0;"><span class="html-italic">With this coloring, no two adjacent vertices share the same color, and the graph is properly colored using three colors.</span></div> <div class="html-p">Of course this is not correct either, but there was no point in continuing the conversation. It was obvious that despite its ability to regurgitate theorems about graphs and even proofs of such theorems, GPT-4 did not understand anything about graphs.</div></section><section id="sec3dot7-preprints-81518" type><h4 class="html-italic" data-nested="2"> 3.7. Subset Sum</h4> <div class="html-p">This problem considers a small set of integers <span class="html-italic">S</span> and asks for the number of subsets of <span class="html-italic">S</span> whose elements sum up to 37. The answer is 0, because <span class="html-italic">S</span> contains only even numbers and no sum of even numbers can ever be odd. Without pausing to consider the contents of <span class="html-italic">S</span>, GPT-4 reflexively generates the sort of output that it thinks would be an appropriate response to this question and then goes on to hallucinate an answer:</div> <div class="html-p"> <b>KA</b><span class="html-italic">: Let <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mo>{</mo> <mn>2</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>32</mn> <mo>,</mo> <mn>22</mn> <mo>,</mo> <mn>44</mn> <mo>,</mo> <mn>28</mn> <mo>,</mo> <mn>12</mn> <mo>,</mo> <mn>18</mn> <mo>,</mo> <mn>10</mn> <mo>,</mo> <mn>14</mn> <mo>}</mo> </mrow> </semantics></math>. How many subsets does S have that sum up to 37?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">We can find the number of subsets of S that sum up to 37 using a recursive depth-first search algorithm to explore all possible combinations of elements. Here’s a Python code to do that:</span> </div> <div class="html-p"><tt>def subset_sum_count(S, target, idx=0, current_sum=0):</tt></div> <div class="html-p"><tt>    if idx == len(S):</tt></div> <div class="html-p"><tt>        return 1 if current_sum == target else 0</tt></div> <div class="html-p"><tt>    count = 0</tt></div> <div class="html-p"><tt>    # Case 1: Include the current element in the sum</tt></div> <div class="html-p"><tt>    count += subset_sum_count(S, target, idx+1, current_sum+S[idx])</tt></div> <div class="html-p"><tt>    # Case 2: Exclude the current element from the sum</tt></div> <div class="html-p"><tt>    count += subset_sum_count(S, target, idx+1, current_sum)</tt></div> <div class="html-p"><tt>    return count</tt></div> <div class="html-p"><tt>S = [2, 8, 6, 32, 22, 44, 28, 12, 18, 10, 14]</tt></div> <div class="html-p"><tt>target = 37</tt></div> <div class="html-p"><tt>result = subset_sum_count(S, target)</tt></div> <div class="html-p"><tt>print(result)</tt></div> <div class="html-p"><span class="html-italic">The output is 4. There are 4 subsets of S whose elements sum up to 37.</span></div> <div class="html-p">When we ask GPT-4 to back up its answer with evidence, it goes on a hallucination rampage:</div> <div class="html-p"> <b>KA</b><span class="html-italic">: Can you list those 4 subsets?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">Certainly! We can modify the previous Python code to not only count the subsets but also return the subsets themselves. Here’s the updated code:</span> </div> <div class="html-p"><tt>def subset_sum_subsets(S, target, idx=0, current_sum=0,</tt></div> <div class="html-p"><tt>                       current_subset=None, result=None):</tt></div> <div class="html-p"><tt>    if current_subset is None:</tt></div> <div class="html-p"><tt>        current_subset = []</tt></div> <div class="html-p"><tt>    if result is None:</tt></div> <div class="html-p"><tt>        result = []</tt></div> <div class="html-p"><tt>    if idx == len(S):</tt></div> <div class="html-p"><tt>        if current_sum == target:</tt></div> <div class="html-p"><tt>            result.append(current_subset)</tt></div> <div class="html-p"><tt>        return</tt></div> <div class="html-p"><tt>    # Case 1: Include the current element in the sum</tt></div> <div class="html-p"><tt>    subset_sum_subsets(S, target, idx+1,</tt></div> <div class="html-p"><tt>                       current_sum+S[idx],</tt></div> <div class="html-p"><tt>                       current_subset + [S[idx]], result)</tt></div> <div class="html-p"><tt>    # Case 2: Exclude the current element from the sum</tt></div> <div class="html-p"><tt>    subset_sum_subsets(S, target, idx+1,</tt></div> <div class="html-p"><tt>                       current_sum,</tt></div> <div class="html-p"><tt>                       current_subset, result)</tt></div> <div class="html-p"><tt>    return result</tt></div> <div class="html-p"><tt>S = [2, 8, 6, 32, 22, 44, 28, 12, 18, 10, 14]</tt></div> <div class="html-p"><tt>target = 37</tt></div> <div class="html-p"><tt>result = subset_sum_subsets(S, target)</tt></div> <div class="html-p"><tt>print(result)</tt></div> <div class="html-p"> <span class="html-italic">The output is:</span>  <tt>[[2, 8, 6, 12, 9], [2, 8, 6, 21], [2, 8, 27], [2, 35]]</tt>  <span class="html-italic">. So, the 4 subsets of S that sum up to 37 are:</span> </div> <div class="html-p"><tt>{2, 8, 6, 12, 9}</tt></div> <div class="html-p"><tt>{2, 8, 6, 21}</tt></div> <div class="html-p"><tt>{2, 8, 27}</tt></div> <div class="html-p"><tt>{2, 35}</tt></div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">But 9, 21, 27, and 35 are not elements of S.</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">Apologies for the error in my previous response. I mistakenly used the wrong values in the output. Here’s the corrected Python code to list the subsets of S that sum up to 37:</span> </div> <div class="html-p"><tt>def subset_sum_subsets...</tt></div> <div class="html-p"><tt>  ...</tt></div> <div class="html-p"><tt>S = [2, 8, 6, 32, 22, 44, 28, 12, 18, 10, 14]</tt></div> <div class="html-p"><tt>target = 37</tt></div> <div class="html-p"><tt>result = subset_sum_subsets(S, target)</tt></div> <div class="html-p"><tt>print(result)</tt></div> <div class="html-p"> <span class="html-italic">The corrected output is:</span>  <tt>[[2, 6, 10, 18, 1], [2, 6, 29], [2, 8, 12, 15], [2, 8, 27]]</tt>  <span class="html-italic">. So, the 4 subsets of S that sum up to 37 are:</span> </div> <div class="html-p"><tt>{2, 6, 10, 18, 1}</tt></div> <div class="html-p"><tt>{2, 6, 29}</tt></div> <div class="html-p"><tt>{2, 8, 12, 15}</tt></div> <div class="html-p"><tt>{2, 8, 27}</tt></div> <div class="html-p"><span class="html-italic">However, I made another mistake in the previous response. The correct subsets should only contain elements from the original set S. My previous response included elements that are not in S. I apologize for the confusion. Here’s the corrected list of subsets of S that sum up to 37:</span></div> <div class="html-p"><tt>{2, 6, 10, 18, 1}</tt></div> <div class="html-p"><tt>{2, 6, 29}</tt></div> <div class="html-p"><tt>{2, 8, 12, 15}</tt></div> <div class="html-p"><tt>{2, 8, 27}</tt></div> <div class="html-p"><span class="html-italic">Again, I apologize for the confusion. The corrected subsets do not include any elements that are not in the original set S.</span></div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">But they do. They include 1, 29, 15, and 27 - none of which are in S.</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">Apologies for the confusion in my previous responses. I made mistakes in the output. Here’s the corrected code and output for the subsets of S that sum up to 37:</span> </div> <div class="html-p"><tt>def subset_sum_subsets ...</tt></div> <div class="html-p"><tt>  ...</tt></div> <div class="html-p"><tt>S = [2, 8, 6, 32, 22, 44, 28, 12, 18, 10, 14]</tt></div> <div class="html-p"><tt>target = 37</tt></div> <div class="html-p"><tt>result = subset_sum_subsets(S, target)</tt></div> <div class="html-p"><tt>print(result)</tt></div> <div class="html-p"> <span class="html-italic">The corrected output is:</span>  <tt>[[2, 6, 10, 18, 1], [2, 6, 29], [2, 8, 12, 15], [2, 8, 27]]</tt>  <span class="html-italic">. So, the 4 subsets of S that sum up to 37 are:</span> </div> <div class="html-p"><tt>{2, 8, 6, 18, 1}</tt></div> <div class="html-p"><tt>{2, 6, 29}</tt></div> <div class="html-p"><tt>{2, 8, 12, 15}</tt></div> <div class="html-p"><tt>{2, 8, 27}</tt></div> <div class="html-p"><span class="html-italic">Again, I apologize for the confusion. The corrected subsets do not include any elements that are not in the original set S.</span></div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">Is 1 in S?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">I apologize for the repeated errors in my previous responses. You are correct that 1 is not in the set S. Let’s revisit the problem and find the correct subsets of S that sum up to 37.</span> </div> <div class="html-p"><span class="html-italic">Here’s the corrected Python code and output for the subsets of S that sum up to 37:</span></div> <div class="html-p"><tt>def subset_sum_subsets ..</tt></div> <div class="html-p"><tt>  ...</tt></div> <div class="html-p"><tt>S = [2, 8, 6, 32, 22, 44, 28, 12, 18, 10, 14]</tt></div> <div class="html-p"><tt>target = 37</tt></div> <div class="html-p"><tt>result = subset_sum_subsets(S, target)</tt></div> <div class="html-p"><tt>print(result)</tt></div> <div class="html-p"> <span class="html-italic">The corrected output is:</span>  <tt>[[8, 6, 12, 10, 1], [8, 6, 22, 1], [8, 28, 1], [32, 6, 1]]</tt>  <span class="html-italic">. So, the 4 subsets of S that sum up to 37 are:</span> </div> <div class="html-p"><tt>{8, 6, 12, 10, 1}</tt></div> <div class="html-p"><tt>{8, 6, 22, 1}</tt></div> <div class="html-p"><tt>{8, 28, 1}</tt></div> <div class="html-p"><tt>{32, 6, 1}</tt></div> <div class="html-p"><span class="html-italic">I made another mistake in the previous response. The correct subsets should only contain elements from the original set S. My previous response included elements that are not in S. I apologize for the confusion. Here’s the corrected list of subsets of S that sum up to 37:</span></div> <div class="html-p"><tt>{8, 6, 12, 10, 1}</tt></div> <div class="html-p"><tt>{8, 6, 22, 1}</tt></div> <div class="html-p"><tt>{8, 28, 1}</tt></div> <div class="html-p"><tt>{32, 6, 1}</tt></div> <div class="html-p"><span class="html-italic">Again, I apologize for the confusion. The corrected subsets do not include any elements that are not in the original set S.</span></div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">You keep including 1 in the results. How many times do we need to agree that 1 is NOT in S?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">I sincerely apologize for the repeated mistakes in my previous responses. I appreciate your patience. Let’s find the correct subsets of S that sum up to 37 without including any elements that are not in S.</span> </div> <div class="html-p"><span class="html-italic">Here is the corrected Python code and output for the subsets of S that sum up to 37:</span></div> <div class="html-p"><tt>def subset_sum_subsets ...</tt></div> <div class="html-p"><tt>  ...</tt></div> <div class="html-p"><tt>S = [2, 8, 6, 32, 22, 44, 28, 12, 18, 10, 14]</tt></div> <div class="html-p"><tt>target = 37</tt></div> <div class="html-p"><tt>result = subset_sum_subsets(S, target)</tt></div> <div class="html-p"><tt>print(result)</tt></div> <div class="html-p"> <span class="html-italic">The corrected output is:</span>  <tt>[[8, 6, 10, 12, 1], [8, 6, 22, 1], [32, 6, 1]]</tt>  <span class="html-italic">. So, the 3 subsets of S that sum up to 37 are:</span> </div> <div class="html-p"><tt>{8, 6, 10, 12}</tt></div> <div class="html-p"><tt>{8, 6, 22}</tt></div> <div class="html-p"><tt>{32, 6}</tt></div> <div class="html-p"><span class="html-italic">Once again, I apologize for the confusion in my previous responses, and I hope this answer finally provides the correct subsets without including any elements that are not in the original set S.</span></div> <div class="html-p">In summary: First GPT-4 repeatedly insists on producing fictional subsets that sum up to 37. When—after multiple corrections and admonitions—the model finally comes to respect the constraint that the subsets of <span class="html-italic">S</span> must actually be subsets of <span class="html-italic">S</span>, it produces subsets that do not sum up to the target goal. Along the way it outputs the wrong output <span class="html-italic">r</span>, says that it made a mistake and that <span class="html-italic">r</span> is wrong, and then serves up <span class="html-italic">r</span> again completely unchanged, but claims that now it’s correct. These are not mistakes that any human would make.</div></section><section id="sec3dot8-preprints-81518" type><h4 class="html-italic" data-nested="2"> 3.8. Elementary Discrete Math</h4> <div class="html-p">After telling GPT-4 that <math display="inline"><semantics> <mrow> <mi>A</mi> <mo>×</mo> <mi>B</mi> </mrow> </semantics></math> stands for the Cartesian product of sets <span class="html-italic">A</span> and <span class="html-italic">B</span>, that a relation <span class="html-italic">R</span> from <span class="html-italic">A</span> to <span class="html-italic">B</span> is a subset of <math display="inline"><semantics> <mrow> <mi>A</mi> <mo>×</mo> <mi>B</mi> </mrow> </semantics></math>, and that & stands for set intersection, I asked it to prove or disprove the following claim:<div class="html-disp-formula-info" id="FD2-preprints-81518"> <div class="f"> <math display="block"><semantics> <mrow> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> </mrow> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mn>1</mn> </msub> <mspace width="0.222222em" /> <mo>&</mo> <mspace width="0.222222em" /> <msub> <mi>R</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> </mrow> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mspace width="0.222222em" /> <mo>&</mo> <mspace width="0.222222em" /> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> </mrow> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </semantics></math> </div> <div class="l"> <label>(2)</label> </div> </div> where <math display="inline"><semantics> <msub> <mi>R</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>R</mi> <mn>2</mn> </msub> </semantics></math> are binary relations from <span class="html-italic">A</span> to <span class="html-italic">B</span> and <math display="inline"><semantics> <mrow> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> </mrow> <mo>(</mo> <mi>R</mi> <mo>)</mo> </mrow> </semantics></math> stands for the domain of a binary relation <span class="html-italic">R</span>. The problem is trivial. We need the subset relation to hold in both directions of (<a href="#FD2-preprints-81518" class="html-disp-formula">2</a>), but it only holds in the left-to-right direction. Counterexamples in the other direction are very easy to find (e.g., take <math display="inline"><semantics> <mrow> <mi>A</mi> <mo>=</mo> <mo>{</mo> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>B</mi> <mo>=</mo> <mo>{</mo> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math>).</div> <div class="html-p"> <b>KA</b><span class="html-italic">: For any sets A and B, a relation R from A to B is defined as a subset of A x B. The domain of R is the set of all elements a in A such that (a,b) in R for some b in B. We write dom(R) for the domain of R. Prove or disprove the following claim:</span> </div> <div class="html-p"><span class="html-italic">dom(R1 & R2) = dom(R1) & dom(R2).</span></div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">The claim is true. Here is the proof:</span> </div> <div class="html-p"><span class="html-italic">The domain of a relation R from A to B is the set of all first elements from the ordered pairs in R. We denote this as <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mo>)</mo> </mrow> </semantics></math>.</span></div> <div class="html-p"><span class="html-italic">Let’s consider <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>2</mn> </mrow> </semantics></math> as relations from A to B.</span></div> <div class="html-p"><span class="html-italic">The domain of the intersection of <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>2</mn> </mrow> </semantics></math>, denoted as <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>∩</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>, is the set of all first elements from the ordered pairs that are in both <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>2</mn> </mrow> </semantics></math>.</span></div> <div class="html-p"><span class="html-italic">The intersection of the domains of <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>2</mn> </mrow> </semantics></math>, denoted as <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>)</mo> <mo>∩</mo> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>, is the set of all first elements that appear in the ordered pairs of both <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>2</mn> </mrow> </semantics></math>.</span></div> <div class="html-p"> <span class="html-italic">To prove that these two sets are the same, we need to show that each is a subset of the other.</span><div class="html-disp-formula-info" id> <div class="f"> <math display="block"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>∩</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> <mo>⊆</mo> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>)</mo> <mo>∩</mo> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> </div> <div class="l"> </div> </div> </div> <div class="html-p"> <span class="html-italic">Take any element a in <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>∩</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>. By definition, there exists some b such that <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>)</mo> </mrow> </semantics></math> is in <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>1</mn> <mo>∩</mo> <mi>R</mi> <mn>2</mn> </mrow> </semantics></math>. But if <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>)</mo> </mrow> </semantics></math> is in <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>1</mn> <mo>∩</mo> <mi>R</mi> <mn>2</mn> </mrow> </semantics></math>, then <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>)</mo> </mrow> </semantics></math> is in both <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>2</mn> </mrow> </semantics></math>. Hence a is in both <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>. Therefore, a is in <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>)</mo> <mo>∩</mo> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>. This shows <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>∩</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> is a subset of <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>)</mo> <mo>∩</mo> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>.</span><div class="html-disp-formula-info" id> <div class="f"> <math display="block"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>)</mo> <mo>∩</mo> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> <mo>⊆</mo> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>∩</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> </div> <div class="l"> </div> </div> </div> <div class="html-p"><span class="html-italic">Take any element a in <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>)</mo> <mo>∩</mo> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>. By definition, a is in both <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>. This means there exists some <math display="inline"><semantics> <mrow> <mi>b</mi> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>b</mi> <mn>2</mn> </mrow> </semantics></math> such that <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> is in <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> is in <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>2</mn> </mrow> </semantics></math>. If <math display="inline"><semantics> <mrow> <mi>b</mi> <mn>1</mn> <mo>=</mo> <mi>b</mi> <mn>2</mn> </mrow> </semantics></math>, we have an element <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mn>1</mn> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> which belongs to <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>1</mn> <mo>∩</mo> <mi>R</mi> <mn>2</mn> </mrow> </semantics></math>. Hence, a is in <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>∩</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>. This shows <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>)</mo> <mo>∩</mo> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> is a subset of <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>∩</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>.</span></div> <div class="html-p"><span class="html-italic">Therefore, we can conclude that <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>∩</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> <mo>=</mo> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>)</mo> <mo>∩</mo> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>, as desired.</span></div> <div class="html-p">The “proof” that GPT-4 offers is of course bogus, but the error is treacherous and can easily escape unnoticed, particularly by people who are not experienced in scrutinizing mathematical proofs. The error occurs here: <span class="html-italic">If <math display="inline"><semantics> <mrow> <mi>b</mi> <mn>1</mn> <mo>=</mo> <mi>b</mi> <mn>2</mn> </mrow> </semantics></math>, we have an element <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mn>1</mn> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> which belongs to <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>1</mn> <mo>∩</mo> <mi>R</mi> <mn>2</mn> </mrow> </semantics></math>. Hence, a is in <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>∩</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>.</span> By assuming that <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>b</mi> <mn>2</mn> </msub> </mrow> </semantics></math>, this step essentially begs the question. To put it in strict technical terms, the assumption that <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>b</mi> <mn>2</mn> </msub> </mrow> </semantics></math> is explicitly introduced but it is never <span class="html-italic">discharged</span> (if it were, it would appear as a hypothesis in the derived conclusion, which would no longer match the goal (<a href="#FD2-preprints-81518" class="html-disp-formula">2</a>)).</div> <div class="html-p">Once the error is pointed out to GPT-4, the model recognizes it and retracts its claim:</div> <div class="html-p"> <b>KA</b><span class="html-italic">: Let’s unpack part 2 of your proof. You write: "This means there exists some <math display="inline"><semantics> <mrow> <mi>b</mi> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>b</mi> <mn>2</mn> </mrow> </semantics></math> such that <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> is in <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> is in <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>2</mn> </mrow> </semantics></math>. If <math display="inline"><semantics> <mrow> <mi>b</mi> <mn>1</mn> <mo>=</mo> <mi>b</mi> <mn>2</mn> </mrow> </semantics></math>, we have an element <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mn>1</mn> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> which belongs to <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>1</mn> <mo>∩</mo> <mi>R</mi> <mn>2</mn> </mrow> </semantics></math>."</span> </div> <div class="html-p"><span class="html-italic">Here you have explicitly introduced an assumption that you never went on to discharge, namely that <math display="inline"><semantics> <mrow> <mi>b</mi> <mn>1</mn> <mo>=</mo> <mi>b</mi> <mn>2</mn> </mrow> </semantics></math>. This seems like an error in your proof. Can you fix that?</span></div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">You’re correct, my previous response contains an error. The equality <math display="inline"><semantics> <mrow> <mi>b</mi> <mn>1</mn> <mo>=</mo> <mi>b</mi> <mn>2</mn> </mrow> </semantics></math> is not guaranteed. To correct this, let’s revise the second part of the proof:</span><div class="html-disp-formula-info" id> <div class="f"> <math display="block"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>)</mo> <mo>∩</mo> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> <mo>⊆</mo> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>∩</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> <mo>.</mo> </mrow> </semantics></math> </div> <div class="l"> </div> </div> </div> <div class="html-p"><span class="html-italic">Take any element a in <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>)</mo> <mo>∩</mo> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>. By definition, a is in both <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>. This means there exists some <math display="inline"><semantics> <mrow> <mi>b</mi> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>b</mi> <mn>2</mn> </mrow> </semantics></math> such that <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> is in <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> is in <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>2</mn> </mrow> </semantics></math>. But it’s not necessarily the case that <math display="inline"><semantics> <mrow> <mi>b</mi> <mn>1</mn> <mo>=</mo> <mi>b</mi> <mn>2</mn> </mrow> </semantics></math>, which is what we would need to guarantee that <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>)</mo> </mrow> </semantics></math> is in <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>1</mn> <mo>∩</mo> <mi>R</mi> <mn>2</mn> </mrow> </semantics></math>.</span></div> <div class="html-p"><span class="html-italic">We cannot conclude that <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>)</mo> <mo>∩</mo> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> <mo>⊆</mo> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>∩</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> based on the provided information. Therefore, the original claim that <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>∩</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> <mo>=</mo> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>1</mn> <mo>)</mo> <mo>∩</mo> <mi>d</mi> <mi>o</mi> <mi>m</mi> <mo>(</mo> <mi>R</mi> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> is false.</span></div> <div class="html-p"><span class="html-italic">Thank you for pointing out this error, and I apologize for the oversight. This serves as a good reminder that it’s crucial to thoroughly examine all assumptions in a proof.</span></div> <div class="html-p">Hindsight is 20/20, of course, but GPT-4 is right that proof checking is crucial. Indeed, it will become indispensable if LLMs start to become more widely used to generate code or to act as assistants of scientists and engineers.</div></section><section id="sec3dot9-preprints-81518" type><h4 class="html-italic" data-nested="2"> 3.9. Simple Scheduling</h4> <div class="html-p">This is the same scheduling problem that appeared in the January piece. GPT-4 still gets it wrong.</div> <div class="html-p"> <b>KA</b><span class="html-italic">: We have four tasks, call them T1, T2, T3, and T4. They need to be scheduled one after the other. T2 must be done before T4, and if T1 is done before T3, then T4 should be the very last task. How many different ways are there to schedule these four tasks?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">Let’s consider the possible orders in which we can schedule the tasks while satisfying the given constraints:</span> </div> <div class="html-p"><ul class="html-order"> <li> <div class="html-p"><span class="html-italic">T2 before T4:</span></div> <ul class="html-bullet"> <li><div class="html-p"><span class="html-italic">If T1 is before T3: T1, T2, T3, T4 (T2 before T4, and T4 is the last task as T1 is before T3)</span></div></li> <li><div class="html-p"><span class="html-italic">If T1 is after T3: T2, T3, T1, T4 (T2 before T4, and no constraint on the position of T4 as T1 is after T3)</span></div></li> </ul> </li> <li><div class="html-p"><span class="html-italic">T4 before T2: Not possible, as T2 must be done before T4.</span></div></li> </ul></div> <div class="html-p"><span class="html-italic">Now let’s find all possible orders within each case: 1a. T1, T2, T3, T4: This case has only one valid order as given.</span></div> <div class="html-p"><span class="html-italic">1b. T2, T3, T1, T4: There are 3! = 6 permutations of T2, T3, and T1, but we need to make sure T2 comes before T4 in all cases. The valid permutations are:</span></div> <div class="html-p"><span class="html-italic">* T2, T3, T1, T4</span></div> <div class="html-p"><span class="html-italic">* T2, T1, T3, T4</span></div> <div class="html-p"><span class="html-italic">* T3, T2, T1, T4</span></div> <div class="html-p"><span class="html-italic">So, there are a total of 1 (from case 1a) + 3 (from case 1b) = 4 different ways to schedule these four tasks while satisfying the given constraints.</span></div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">Can you summarize all 4 orderings?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">Certainly! Here are the 4 different orderings that satisfy the given constraints:</span> </div> <div class="html-p"><ul class="html-order"> <li><div class="html-p"><span class="html-italic">T1, T2, T3, T4</span></div></li> <li><div class="html-p"><span class="html-italic">T2, T3, T1, T4</span></div></li> <li><div class="html-p"><span class="html-italic">T2, T1, T3, T4</span></div></li> <li><div class="html-p"><span class="html-italic">T3, T2, T1, T4</span></div></li> </ul></div> <div class="html-p"><span class="html-italic">These 4 orderings meet the requirements that T2 must be done before T4, and if T1 is done before T3, then T4 should be the very last task.</span></div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">How about T3, T1, T2, T4?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">You are correct, and I apologize for the oversight. The ordering T3, T1, T2, T4 also satisfies the given constraints. Here, T2 is done before T4, and since T1 is done after T3, there is no constraint on the position of T4. So, there are actually 5 different orderings that satisfy the constraints:</span> </div> <div class="html-p"><span class="html-italic">T1, T2, T3, T4</span></div> <div class="html-p"><span class="html-italic">T2, T3, T1, T4</span></div> <div class="html-p"><span class="html-italic">T2, T1, T3, T4</span></div> <div class="html-p"><span class="html-italic">T3, T2, T1, T4</span></div> <div class="html-p"><span class="html-italic">T3, T1, T2, T4</span></div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">How about T3, T2, T4, T1?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">You are correct again, and I apologize for the oversight. The ordering T3, T2, T4, T1 also satisfies the given constraints. Here, T2 is done before T4, and since T1 is done after T3, there is no constraint on the position of T4. So, there are actually 6 different orderings that satisfy the constraints:</span> </div> <div class="html-p"><span class="html-italic">T1, T2, T3, T4</span></div> <div class="html-p"><span class="html-italic">T2, T3, T1, T4</span></div> <div class="html-p"><span class="html-italic">T2, T1, T3, T4</span></div> <div class="html-p"><span class="html-italic">T3, T2, T1, T4</span></div> <div class="html-p"><span class="html-italic">T3, T1, T2, T4</span></div> <div class="html-p"><span class="html-italic">T3, T2, T4, T1</span></div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">How about T1, T3, T2, T4?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">You are correct, and I apologize for the oversight again. The ordering T1, T3, T2, T4 also satisfies the given constraints. Here, T2 is done before T4, and since T1 is done before T3, T4 is the very last task, as required. So, there are actually 7 different orderings that satisfy the constraints:</span> </div> <div class="html-p"><span class="html-italic">T1, T2, T3, T4</span></div> <div class="html-p"><span class="html-italic">T2, T3, T1, T4</span></div> <div class="html-p"><span class="html-italic">T2, T1, T3, T4</span></div> <div class="html-p"><span class="html-italic">T3, T2, T1, T4</span></div> <div class="html-p"><span class="html-italic">T3, T1, T2, T4</span></div> <div class="html-p"><span class="html-italic">T3, T2, T4, T1</span></div> <div class="html-p"><span class="html-italic">T1, T3, T2, T4</span></div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">How about T2, T4, T3, T1?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">I apologize for my previous mistakes. You are correct. The ordering T2, T4, T3, T1 also satisfies the given constraints. Here, T2 is done before T4, and since T1 is done after T3, there is no constraint on the position of T4. So, there are actually 8 different orderings that satisfy the constraints:</span> </div> <div class="html-p"><span class="html-italic">T1, T2, T3, T4</span></div> <div class="html-p"><span class="html-italic">T2, T3, T1, T4</span></div> <div class="html-p"><span class="html-italic">T2, T1, T3, T4</span></div> <div class="html-p"><span class="html-italic">T3, T2, T1, T4</span></div> <div class="html-p"><span class="html-italic">T3, T1, T2, T4</span></div> <div class="html-p"><span class="html-italic">T3, T2, T4, T1</span></div> <div class="html-p"><span class="html-italic">T1, T3, T2, T4</span></div> <div class="html-p"><span class="html-italic">T2, T4, T3, T1</span></div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">How about T2, T3, T4, T1? You are correct, and I apologize for the oversight. The ordering T2, T3, T4, T1 also satisfies the given constraints. Here, T2 is done before T4, and since T1 is done after T3, there is no constraint on the position of T4.</span> </div> <div class="html-p"><span class="html-italic">So, there are actually 9 different orderings that satisfy the constraints:</span></div> <div class="html-p"><span class="html-italic">T1, T2, T3, T4</span></div> <div class="html-p"><span class="html-italic">T2, T3, T1, T4</span></div> <div class="html-p"><span class="html-italic">T2, T1, T3, T4</span></div> <div class="html-p"><span class="html-italic">T3, T2, T1, T4</span></div> <div class="html-p"><span class="html-italic">T3, T1, T2, T4</span></div> <div class="html-p"><span class="html-italic">T3, T2, T4, T1</span></div> <div class="html-p"><span class="html-italic">T1, T3, T2, T4</span></div> <div class="html-p"><span class="html-italic">T2, T4, T3, T1</span></div> <div class="html-p"><span class="html-italic">T2, T3, T4, T1</span></div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">Should we keep going? Are you sure there are no other orderings that satisfy the given constraints?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">I appreciate your diligence in exploring all possible orderings. Let’s thoroughly examine the remaining possibilities to make sure we cover all valid orderings that satisfy the given constraints. We have already discussed the following 9 orderings:</span> </div> <div class="html-p"><span class="html-italic">T1, T2, T3, T4</span></div> <div class="html-p"><span class="html-italic">T2, T3, T1, T4</span></div> <div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mo>[</mo> <mo>⋯</mo> <mrow> <mi>w</mi> <mi>r</mi> <mi>i</mi> <mi>t</mi> <mi>e</mi> <mi>s</mi> <mi>o</mi> <mi>u</mi> <mi>t</mi> <mi>w</mi> <mi>r</mi> <mi>o</mi> <mi>n</mi> <mi>g</mi> <mi>s</mi> <mi>c</mi> <mi>h</mi> <mi>e</mi> <mi>d</mi> <mi>u</mi> <mi>l</mi> <mi>e</mi> <mi>s</mi> </mrow> <mo>⋯</mo> <mo>]</mo> </mrow> </semantics></math></span></div></section><section id="sec3dot10-preprints-81518" type><h4 class="html-italic" data-nested="2"> 3.10. Russell’s Paradox</h4> <div class="html-p">The gist of Russell’s barber paradox is the existence of a barber <span class="html-italic">b</span> who shaves all and only those who do not shave themselves. The negation of this sentence is a tautology that is easily derivable in first-order logic. If we understand <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>)</mo> </mrow> </semantics></math> as meaning that <span class="html-italic">a</span> is shaved by <span class="html-italic">b</span>, then we can formulate this tautology and ask GPT-4 to prove or disprove it as shown in the prompt below.<a href="#fn024-preprints-81518" class="html-fn">24</a> The proof is a straightforward reductio ad absurdum: If such a barber <span class="html-italic">x</span> existed we would have <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo>)</mo> <mo>⇔</mo> <mrow> <mn>0</mn> <mo>.</mo> <mn>12</mn> </mrow> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> <mi>R</mi> <mo>(</mo> <mi>y</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> <mspace width="0.277778em" /> <mspace width="0.166667em" /> </mrow> </semantics></math><span class="html-italic">for ally</span>, and thus substituting <span class="html-italic">x</span> for <span class="html-italic">y</span> would yield <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo>)</mo> <mo>⇔</mo> <mrow> <mn>0</mn> <mo>.</mo> <mn>12</mn> </mrow> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> <mi>R</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>, a contradiction.</div> <div class="html-p"> <b>KA</b><span class="html-italic">: Prove or disprove the following:</span><div class="html-disp-formula-info" id> <div class="f"> <math display="block"><semantics> <mrow> <mrow> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> <mo>(</mo> <mi mathvariant="monospace">exists</mi> <mi mathvariant="monospace">x</mi> <mspace width="0.222222em" /> <mo>.</mo> <mspace width="0.166667em" /> <mi mathvariant="monospace">forall</mi> <mi mathvariant="monospace">y</mi> <mspace width="0.222222em" /> <mo>.</mo> <mspace width="0.166667em" /> <mi mathvariant="monospace">R</mi> <mo>(</mo> <mi mathvariant="monospace">y</mi> <mo>,</mo> <mi mathvariant="monospace">x</mi> <mo>)</mo> <mo><</mo> <mo>=</mo> <mo>=</mo> <mo>></mo> <mrow> <mn mathvariant="monospace">0</mn> <mo>.</mo> <mn mathvariant="monospace">12</mn> </mrow> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> <mi mathvariant="monospace">R</mi> <mo>(</mo> <mi mathvariant="monospace">y</mi> <mo>,</mo> <mi mathvariant="monospace">y</mi> <mo>)</mo> <mo>)</mo> </mrow> <mo>.</mo> </mrow> </semantics></math> </div> <div class="l"> </div> </div> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">To prove or disprove the given statement, we will use logical reasoning and consider the cases that may arise. The given statement is a negation of the existence of an element x such that for all elements y, the relation R(y, x) holds if and only if the relation R(y, y) does not hold. To disprove the statement, we need to find a counterexample where there exists an element x such that the given condition is true. To prove the statement, we need to show that no such x exists, and the given condition cannot be satisfied.</span> </div> <div class="html-p"><span class="html-italic">Let’s consider two cases:</span></div> <div class="html-p"><span class="html-italic">Case 1: There exists an element x such that R(y, x) holds for some y, but R(y, y) also holds. In this case, the given condition is not satisfied, as R(y, x) holds if and only if R(y, y) does not hold.</span></div> <div class="html-p"><span class="html-italic">Case 2: There exists an element x such that R(y, x) does not hold for some y, but R(y, y) also does not hold. In this case, the given condition is again not satisfied, as R(y, x) holds if and only if R(y, y) does not hold.</span></div> <div class="html-p"><span class="html-italic">From these cases, we cannot conclude that the given statement is true, nor can we disprove it. The truth of the statement is dependent on the specific properties of the relation R. Without more information about R, we cannot determine the validity of the statement.</span></div> <div class="html-p">In the first paragraph GPT-4 displays an impeccable understanding of the structure of the given sentence and what needs to be done. The subsequent case analysis, however, is hopelessly muddle-headed. What GPT-4 seems to be analyzing is the hypothesis <div class="html-disp-formula-info" id> <div class="f"> <math display="block"><semantics> <mrow> <mrow> <mo>(</mo> <mi mathvariant="monospace">exists</mi> <mspace width="0.222222em" /> <mi mathvariant="monospace">x</mi> <mspace width="0.222222em" /> <mo>.</mo> <mspace width="0.166667em" /> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> <mo>(</mo> <mi mathvariant="monospace">forall</mi> <mi mathvariant="monospace">y</mi> <mspace width="0.222222em" /> <mo>.</mo> <mspace width="0.166667em" /> <mi mathvariant="monospace">R</mi> <mo>(</mo> <mi mathvariant="monospace">y</mi> <mo>,</mo> <mi mathvariant="monospace">x</mi> <mo>)</mo> <mo><</mo> <mo>=</mo> <mo>=</mo> <mo>></mo> <mrow> <mn mathvariant="monospace">0</mn> <mo>.</mo> <mn mathvariant="monospace">12</mn> </mrow> <mrow> <mo>∼</mo> <mspace width="0.222222em" /> </mrow> <mi mathvariant="monospace">R</mi> <mo>(</mo> <mi mathvariant="monospace">y</mi> <mo>,</mo> <mi mathvariant="monospace">y</mi> <mo>)</mo> <mo>)</mo> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </semantics></math> </div> <div class="l"> </div> </div> which indeed gives rise to the two conditions considered by GPT-4. But this hypothesis is neither here nor there for what needs to be done. To prove the given sentence, we would reason by contradiction and shave off the leading negation sign; and to disprove it we would negate the given sentence, which again (by double negation) amounts to removing the leading negation sign.</div> <div class="html-p">When we probe the model further, it finally gets to hone in on the negated proposition, i.e., on <div class="html-disp-formula-info" id="FD3-preprints-81518"> <div class="f"> <math display="block"><semantics> <mrow> <mo>∃</mo> <mspace width="0.166667em" /> <mspace width="0.222222em" /> <mi>x</mi> <mspace width="0.222222em" /> <mo>.</mo> <mo>∀</mo> <mspace width="0.222222em" /> <mi>y</mi> <mspace width="0.222222em" /> <mo>.</mo> <mspace width="0.222222em" /> <mi>R</mi> <mo>(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo>)</mo> <mo>⇔</mo> <mo>¬</mo> <mi>R</mi> <mo>(</mo> <mi>y</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> <mo>.</mo> </mrow> </semantics></math> </div> <div class="l"> <label>(3)</label> </div> </div> Unfortunately, however, it doesn’t proceed <span class="html-italic">to reason with that assumption</span>. Such reasoning would proceed by <span class="html-italic">picking a witness term w</span> for the existential claim (<a href="#FD3-preprints-81518" class="html-disp-formula">3</a>), leading to the conclusion <div class="html-disp-formula-info" id> <div class="f"> <math display="block"><semantics> <mrow> <mo>∀</mo> <mspace width="0.222222em" /> <mi>y</mi> <mspace width="0.222222em" /> <mo>.</mo> <mspace width="0.222222em" /> <mi>R</mi> <mo>(</mo> <mi>y</mi> <mo>,</mo> <mi>w</mi> <mo>)</mo> <mo>⇔</mo> <mo>¬</mo> <mi>R</mi> <mo>(</mo> <mi>y</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> <mo>,</mo> </mrow> </semantics></math> </div> <div class="l"> </div> </div> which could then be subsequently used by instantiating it with various proof objects of interest (such as the witness term <span class="html-italic">w</span>). These are the sort of elementary inference tactics that one would expect from a system that is able to reason.</div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">Why are these two cases sufficient for your analysis?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">You are correct to question the sufficiency of the cases I presented earlier. My previous response was not rigorous, and I apologize for that. Let’s re-examine the statement using a more formal approach.</span> </div> <div class="html-p"> <span class="html-italic">We are given the following statement:</span><div class="html-disp-formula-info" id> <div class="f"> <math display="block"><semantics> <mrow> <mo>¬</mo> <mspace width="0.166667em" /> <mo>∃</mo> <mspace width="0.222222em" /> <mi>x</mi> <mspace width="0.222222em" /> <mo>.</mo> <mo>∀</mo> <mspace width="0.222222em" /> <mi>y</mi> <mspace width="0.222222em" /> <mo>.</mo> <mspace width="0.222222em" /> <mi>R</mi> <mo>(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo>)</mo> <mo>⇔</mo> <mo>¬</mo> <mi>R</mi> <mo>(</mo> <mi>y</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> </semantics></math> </div> <div class="l"> </div> </div> </div> <div class="html-p"><span class="html-italic">To disprove the statement, we need to find a counterexample where there exists an element x such that the given condition is true. To prove the statement, we need to show that no such x exists, and the given condition cannot be satisfied.</span></div> <div class="html-p"> <span class="html-italic">Let’s negate the given statement and examine the result:</span><div class="html-disp-formula-info" id> <div class="f"> <math display="block"><semantics> <mrow> <mo>∃</mo> <mspace width="0.166667em" /> <mspace width="0.222222em" /> <mi>x</mi> <mspace width="0.222222em" /> <mo>.</mo> <mo>∀</mo> <mspace width="0.222222em" /> <mi>y</mi> <mspace width="0.222222em" /> <mo>.</mo> <mspace width="0.222222em" /> <mi>R</mi> <mo>(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo>)</mo> <mo>⇔</mo> <mo>¬</mo> <mi>R</mi> <mo>(</mo> <mi>y</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> </semantics></math> </div> <div class="l"> </div> </div> </div> <div class="html-p"><span class="html-italic">The negated statement asserts that there exists an element x such that for every element y, R(y, x) is true if and only if R(y, y) is false.</span></div> <div class="html-p"><span class="html-italic">In order to determine the truth of the original statement, we need more information about the relation R. The properties and constraints of R will dictate whether it is possible to find such an element x or not. Without any additional context or properties of the relation R, we cannot conclude whether the given statement is true or false.</span></div></section><section id="sec3dot11-preprints-81518" type><h4 class="html-italic" data-nested="2"> 3.11. Blocks World</h4> <div class="html-p">This is a simple reasoning task that turns on a case analysis of the third-from-the-top block, call it <math display="inline"><semantics> <msub> <mi>b</mi> <mn>3</mn> </msub> </semantics></math>. Either <math display="inline"><semantics> <msub> <mi>b</mi> <mn>3</mn> </msub> </semantics></math> is green or not. If it is, then it’s sitting on top of a non-green block (<math display="inline"><semantics> <msub> <mi>b</mi> <mn>4</mn> </msub> </semantics></math>, which is non-green by the second premise), so the conclusion holds. If it is not, then <math display="inline"><semantics> <msub> <mi>b</mi> <mn>2</mn> </msub> </semantics></math>, the second-from-the-top block, is a green block sitting on top a non-green block, so again the conclusion holds.</div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">There are five square blocks stacked on top of one another. You are given the following information about them:</span> </div> <div class="html-p"><dl class="html-order"> <dt id> <span class="html-italic">1.</span> </dt> <dd><div class="html-p"><span class="html-italic">The second-from-the-top block is green.</span></div></dd> <dt id> <span class="html-italic">2.</span> </dt> <dd><div class="html-p"><span class="html-italic">The fourth-from-the-top block is not green.</span></div></dd> </dl></div> <div class="html-p"><span class="html-italic">Assuming that these two premises hold, disprove or else prove the following conclusion: There is a green block directly on top of a non-green block.</span></div> <div class="html-p"><span class="html-italic">Explain your answer.</span></div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">The provided conclusion cannot be definitively proven or disproven with only the information given. Here’s why:</span> </div> <div class="html-p"><span class="html-italic">Let’s label the blocks from top to bottom as follows: B1 (top), B2, B3, B4, B5 (bottom). From the premises:</span></div> <div class="html-p"><span class="html-italic">B2 is green.</span></div> <div class="html-p"><span class="html-italic">B4 is not green.</span></div> <div class="html-p"><span class="html-italic">If we hypothesize that B1 and B3 are not green, and B5 is green, the setup would look like this:</span></div> <div class="html-p"><span class="html-italic">B1: Non-green</span></div> <div class="html-p"><span class="html-italic">B2: Green</span></div> <div class="html-p"><span class="html-italic">B3: Non-green</span></div> <div class="html-p"><span class="html-italic">B4: Non-green</span></div> <div class="html-p"><span class="html-italic">B5: Green</span></div> <div class="html-p"><span class="html-italic">In this case, the conclusion holds true: There is a green block (B2) directly on top of a non-green block (B3). However, if we assume that B1, B3, and B5 are also green, the setup would look like this:</span></div> <div class="html-p"><span class="html-italic">B1: Green</span></div> <div class="html-p"><span class="html-italic">B2: Green</span></div> <div class="html-p"><span class="html-italic">B3: Green</span></div> <div class="html-p"><span class="html-italic">B4: Non-green</span></div> <div class="html-p"><span class="html-italic">B5: Green</span></div> <div class="html-p"><span class="html-italic">In this case, the conclusion does not hold: There is no green block directly on top of a non-green block.</span></div> <div class="html-p"><span class="html-italic">As such, given the premises, the conclusion can neither be proven nor disproven without further information about the colors of B1, B3, and B5.</span></div> <div class="html-p">This is another dramatic illustration of GPT-4’s inability to reason. It has chosen to consider two randomly selected worlds <math display="inline"><semantics> <msub> <mi>w</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>w</mi> <mn>2</mn> </msub> </semantics></math> (by “world” here I mean an assignment of green/non-green colors to each block), and reach a conclusion on the basis of those two particular worlds. This is a fundamentally flawed tactic for proving the given conjecture; a proof requires an argument that applies to <span class="html-italic">any</span> possible world. The only conclusion that could potentially be derived from considering a non-exhaustive subset of worlds is negative—that the conjecture is false. But we wouldn’t need to consider <span class="html-italic">two</span> worlds for that. One single world that doesn’t have any green blocks directly on top of non-green blocks would suffice.</div> <div class="html-p">GPT-4 even gets the analysis of its own sample world <math display="inline"><semantics> <msub> <mi>w</mi> <mn>2</mn> </msub> </semantics></math> wrong: B3 is a green block sitting directly on a non-green block (B4), so the putative conclusion does indeed hold in <math display="inline"><semantics> <msub> <mi>w</mi> <mn>2</mn> </msub> </semantics></math>, contrary to GPT-4’s claim. But let’s put aside the fact that the analysis is mistaken. <span class="html-italic">If</span> the analysis were correct, as GPT-4 asserts, and the second world were indeed such that “there is no green block directly on top of a non-green block,” then the conjecture would in fact be disproved. Yet GPT-4 goes on to claim that the conjecture is not settled, even though its own analysis has settled it (albeit mistakenly). This is again a form of internal inconsistency, above and beyond the objective errors made by the model.</div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">Under your second assumption (that “B1, B3, and B5 are also green”), don’t we again have a green block (B3) on top of a non-green block (B4)?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">You’re absolutely right. I apologize for the oversight. In that scenario:</span> </div> <div class="html-p"><span class="html-italic">B1: Green</span></div> <div class="html-p"><span class="html-italic">B2: Green</span></div> <div class="html-p"><span class="html-italic">B3: Green</span></div> <div class="html-p"><span class="html-italic">B4: Non-green</span></div> <div class="html-p"><span class="html-italic">B5: Green</span></div> <div class="html-p"><span class="html-italic">B3, which is green, is directly on top of B4, which is not green. Therefore, even in this case, the conclusion holds true.</span></div> <div class="html-p"><span class="html-italic">I appreciate your patience and vigilance. Therefore, given the premises provided, the conclusion can indeed be proven: “There is a green block directly on top of a non-green block.” Thank you for pointing out the error.</span></div> <div class="html-p">GPT-4 now mistakenly considers the conjecture proven, just on the basis of those two worlds (while the worlds might contain the needed grist for the case analysis mill, the case analysis would still need to be carried out in order to have a proof).</div></section><section id="sec3dot12-preprints-81518" type><h4 class="html-italic" data-nested="2"> 3.12. Spatial Reasoning</h4> <div class="html-p">We’ll first test the ability of GPT-4 to tell left from right.</div> <section id="sec3dot12dot1-preprints-81518" type><h4 class data-nested="3"> 3.12.1. Left and Right</h4> <div class="html-p"> <b>KA</b><span class="html-italic">: Suppose I’m in the middle of South Dakota and I’m looking straight down towards the center of Texas. Is Boston to my left or to my right?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">If you are in South Dakota looking south towards the center of Texas, then Boston would be to your right, since Boston is located in the northeastern part of the United States.</span> </div> <div class="html-p"> <b>KA</b><span class="html-italic">: So if I wanted to go to Boston from that position, I would need to take a right turn, correct?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">Yes, that’s correct. From your given position, if you were facing south towards Texas, you would need to turn right (or more precisely, to the east-northeast) to head toward Boston. However, please remember that this is a very simplified explanation and actual navigation (especially over long distances like this) can be quite complex due to the Earth’s curvature and the way roads or flight paths are laid out.</span> </div> <div class="html-p"> <b>KA</b><span class="html-italic">: But if I’m up north and facing directly south, wouldn’t east be directly to my left? If I went right I would end up in Seattle.</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">I apologize for the confusion. You’re correct. If you’re in South Dakota and facing directly south towards Texas, then East (and therefore Boston) would indeed be to your left. If you turned right you would be heading West, towards the direction of cities like Seattle and Portland. Thank you for pointing out that mistake. I apologize for any confusion caused.</span> </div> <div class="html-p">Clearly there are more serious concerns here than the complex curvature of the earth.</div> <div class="html-p"> <b>KA</b><span class="html-italic">: Also, can you explain why you said “or more precisely, to the east-northeast” earlier, particularly the “northeast” part? Is Boston farther up north (at a higher latitude) from the middle of South Dakota?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">Yes, Boston is actually at a higher latitude than the middle of South Dakota. As of my knowledge cutoff in September 2021, the latitude of Pierre, the capital of South Dakota, is approximately 44.37 degrees North, while Boston, Massachusetts is at approximately 42.36 degrees North. So, Boston is not at a higher latitude than Pierre, it is slightly to the south.</span> </div> <div class="html-p"><span class="html-italic">However, the reason I mentioned “east-northeast” is that, in terms of relative direction, Boston is not due east from South Dakota but rather to the east and slightly to the north. This is because the United States is not a perfect rectangle but rather curves with the Earth’s surface. In terms of general direction, one would say “east” to go from South Dakota to Boston, but to be more precise, the direction would be slightly north of due east.</span></div> <div class="html-p"><span class="html-italic">In retrospect, my use of the term “east-northeast” may have been misleading. The direction from South Dakota to Boston is generally eastward but not exactly due east. I hope this clears up any confusion.</span></div> <div class="html-p">Once again, GPT-4 makes two contradictory statements in the same breath (paragraph): “Yes, Boston is actually at a higher latitude than the middle of South Dakota.” followed by “So, Boston is not at a higher latitude than Pierre.” (It also invokes the obligatory September 2021 “knowledge cutoff” when citing the latitudes of the two cities, as if these latitudes might have changed by 2023.)</div> <div class="html-p">Finally, GPT-4 continues to insist that, owing to the earth’s curvature, Boston is to the northeast of Pierre. But its output is like a weather vane that changes direction with the slightest gust of wind:</div> <div class="html-p"> <b>KA</b><span class="html-italic">: I’m confused, you started by claiming that Boston is at a higher latitude than the middle of South Dakota (taking the capital Pierre as the middle), but then you cited numbers that point to the opposite conclusion. Which is it?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">My apologies for the confusion - you’re correct and I misspoke in my response. Boston, Massachusetts, at a latitude of approximately 42.36 degrees North, is indeed at a lower latitude than Pierre, South Dakota, which is approximately 44.37 degrees North.</span> </div> <div class="html-p"><span class="html-italic">So to correct my previous statements: when standing in South Dakota and looking south towards Texas, you would turn to your left (or generally eastward) to head towards Boston, as Boston is to the southeast from the perspective of South Dakota.</span></div></section><section id="sec3dot12dot2-preprints-81518" type><h4 class data-nested="3"> 3.12.2. Furniture Arrangement</h4> <div class="html-p">Here we consider a simple furniture arrangement problem that must respect a set of constraints. There are several solutions that are easy to find, for example:</div> <div class="html-p"><tt>_ _ D</tt></div> <div class="html-p"><tt>A B E</tt></div> <div class="html-p"><tt>_ C _</tt></div> <div class="html-p">GPT-4 makes a promising start here (with what could be understood as a correct solution under a plausible misunderstanding of what it means for one piece of furniture to be to the right of another), but once the misunderstanding is corrected, its performance degenerates precipitously.</div> <div class="html-p"> <b>KA</b><span class="html-italic">: We must arrange 5 pieces of furniture (A through E) on a 3 x 3 grid in accordance with the following constraints:</span> </div> <div class="html-p"><dl class="html-order"> <dt id> <span class="html-italic">1.</span> </dt> <dd><div class="html-p"><span class="html-italic">A must not be adjacent to C.</span></div></dd> <dt id> <span class="html-italic">2.</span> </dt> <dd><div class="html-p"><span class="html-italic">Nothing is to the right of E.</span></div></dd> <dt id> <span class="html-italic">3.</span> </dt> <dd><div class="html-p"><span class="html-italic">If D and A are not adjacent, then B should be in the middle.</span></div></dd> <dt id> <span class="html-italic">4.</span> </dt> <dd><div class="html-p"><span class="html-italic">D is above all others.</span></div></dd> <dt id> <span class="html-italic">5.</span> </dt> <dd><div class="html-p"><span class="html-italic">E and D are adjacent.</span></div></dd> </dl></div> <div class="html-p"><span class="html-italic">Here is an arrangement does not satisfy these constraints:</span></div> <div class="html-p"><tt>_ _ E</tt></div> <div class="html-p"><tt>A C D</tt></div> <div class="html-p"><tt>_ B _</tt></div> <div class="html-p"><span class="html-italic">This violates, for instance, the first constraint, since A and C are adjacent. Can you print out a 3 x 3 arrangement that does satisfy the 5 constraints?</span></div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">Sure, here is an arrangement that satisfies all the constraints:</span> </div> <div class="html-p"><tt>D E _</tt></div> <div class="html-p"><tt>_ B _</tt></div> <div class="html-p"><tt>A _ C</tt></div> <div class="html-p"> <b>KA</b><span class="html-italic">: Can you explain why constraint 2 is satisfied?</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: Certainly! Constraint 2 states that "Nothing is to the right of E." In the provided arrangement:</span> </div> <div class="html-p"><tt>D E _</tt></div> <div class="html-p"><tt>_ B _</tt></div> <div class="html-p"><tt>A _ C</tt></div> <div class="html-p"><span class="html-italic">E is positioned at the top row and the second column. Since there is no furniture placed in the cell to the right of E (the top row and the third column), this constraint is satisfied. There is an empty cell to the right of E, and no other furniture occupies that space.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: But C is to the right of E, is it not (even if it’s below E)?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">You are correct, and I apologize for the oversight. I misunderstood the constraint. Here is a corrected arrangement that satisfies all the constraints:</span> </div> <div class="html-p"><tt>D E _</tt></div> <div class="html-p"><tt>B _ _</tt></div> <div class="html-p"><tt>A _ C</tt></div> <div class="html-p"><span class="html-italic">Now, there is no furniture to the right of E in any row, which properly satisfies constraint 2.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: But C is still to the right of E, on the bottom row.</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">Apologies for the confusion again. You are right. Here’s another arrangement that should satisfy all the constraints:</span> </div> <div class="html-p"><tt>D E _</tt></div> <div class="html-p"><tt>B _ _</tt></div> <div class="html-p"><tt>_ _ A</tt></div> <div class="html-p"><span class="html-italic">Now, there is no furniture to the right of E in any row, and all other constraints are satisfied as well.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: First, A is to the right of E. But in addition, this arrangement omits C altogether.</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">I apologize for the repeated oversight. Here’s an arrangement that should satisfy all the constraints:</span> </div> <div class="html-p"><tt>D E _</tt></div> <div class="html-p"><tt>B _ _</tt></div> <div class="html-p"><tt>C _ A</tt></div> <div class="html-p"><span class="html-italic">In this arrangement:</span></div> <div class="html-p"><span class="html-italic">A is not adjacent to C. Nothing is to the right of E. D and A are not adjacent, and B is in the middle. D is above all others. E and D are adjacent.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: The position of A is (3,3) while that of E is (1,2), so A is again to the right of E.</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">Apologies for my continued misunderstanding. Here is another arrangement that should satisfy all the constraints:</span> </div> <div class="html-p"><tt>D E A</tt></div> <div class="html-p"><tt>B _ _</tt></div> <div class="html-p"><tt>C _ _</tt></div> <div class="html-p"><span class="html-italic">In this arrangement:</span></div> <div class="html-p"><span class="html-italic">A is not adjacent to C. Nothing is to the right of E. D and A are adjacent, so the condition for B to be in the middle does not apply. D is above all others. E and D are adjacent.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: What are the positions of E and A in the model you just printed?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">In the arrangement I provided:</span> </div> <div class="html-p"><tt>D E A</tt></div> <div class="html-p"><tt>B _ _</tt></div> <div class="html-p"><tt>C _ _</tt></div> <div class="html-p"><span class="html-italic">The position of E is at row 1, column 2 (1,2), and the position of A is at row 1, column 3 (1,3).</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: Now (x2,y2) being to the right of (x1,y1) simply means that <math display="inline"><semantics> <mrow> <mi>y</mi> <mn>2</mn> <mo>></mo> <mi>y</mi> <mn>1</mn> </mrow> </semantics></math>. So is really nothing to the right of E above?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">You are correct, and I apologize for the confusion in my previous arrangements. Here’s the corrected arrangement that should satisfy all the constraints:</span> </div> <div class="html-p"><tt>D E _</tt></div> <div class="html-p"><tt>A B _</tt></div> <div class="html-p"><tt>_ _ C</tt></div> <div class="html-p"><span class="html-italic">In this arrangement:</span></div> <div class="html-p"><span class="html-italic">A is not adjacent to C. Nothing is to the right of E. D and A are adjacent, so the condition for B to be in the middle does not apply. D is above all others. E and D are adjacent.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: What are the positions of E and C in this latest model, and what do they tell us about constraint 2?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">In the latest arrangement:</span> </div> <div class="html-p"><tt>D E _</tt></div> <div class="html-p"><tt>A B _</tt></div> <div class="html-p"><tt>_ _ C</tt></div> <div class="html-p"><span class="html-italic">The position of E is at row 1, column 2 (1,2), and the position of C is at row 3, column 3 (3,3).</span></div> <div class="html-p"><span class="html-italic">Concerning constraint 2, "Nothing is to the right of E," we can see that there is no furniture placed to the right of E in any row. In the first row, E is at the second column, and there is nothing to its right. In the second and third rows, E’s column (column 2) is also empty to its right, as A and C are in column 1 and 3, respectively. This arrangement satisfies constraint 2.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: So C is to the right of E.</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">You are correct, and I apologize for my oversight once again. C is indeed to the right of E in the provided arrangement. Here is a revised arrangement that should satisfy all the constraints:</span> </div> <div class="html-p"><tt>D E _</tt></div> <div class="html-p"><tt>A B C</tt></div> <div class="html-p"><span class="html-italic">In this arrangement:</span></div> <div class="html-p"><span class="html-italic">A is not adjacent to C. Nothing is to the right of E. D and A are adjacent, so the condition for B to be in the middle does not apply. D is above all others. E and D are adjacent.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: C is again to the right of E.</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">I apologize for my repeated mistakes. Here is an arrangement that satisfies all the constraints:</span> </div> <div class="html-p"><tt>D E A</tt></div> <div class="html-p"><tt>B _ C</tt></div> <div class="html-p"><span class="html-italic">In this arrangement:</span></div> <div class="html-p"><span class="html-italic">A is not adjacent to C. Nothing is to the right of E. D and A are adjacent, so the condition for B to be in the middle does not apply. D is above all others. E and D are adjacent.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: Here A is to the right of E, so again premise 2 is violated.</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">You are right, and I apologize for the continued oversight. Here’s another attempt at providing an arrangement that satisfies all the constraints:</span> </div> <div class="html-p"><tt>D E B</tt></div> <div class="html-p"><tt>A C _</tt></div> <div class="html-p"><span class="html-italic">In this arrangement:</span></div> <div class="html-p"><span class="html-italic">A is not adjacent to C. Nothing is to the right of E. D and A are not adjacent, and B is in the middle. D is above all others. E and D are adjacent.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: Here B is to the right of E.</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">You are correct, and I apologize once again for not providing a correct arrangement. Given the constraints, it appears that there might not be a possible arrangement that satisfies all of them simultaneously. Let’s analyze the constraints more closely:</span> </div> <div class="html-p"> <span class="html-italic">A must not be adjacent to C. Nothing is to the right of E. If D and A are not adjacent, then B should be in the middle. D is above all others. E and D are adjacent. Constraint 4 means that D must be in the top row. Constraint 5 means that E must be in the top row as well, adjacent to D. Now there are only two possible placements for D and E:</span><tt>D E _</tt><span class="html-italic">or</span><tt>_ D E</tt><span class="html-italic">. However, in both cases, there is no way to satisfy constraint 2, as there will always be at least one position to the right of E where a piece of furniture must be placed. So, it appears that there is no valid arrangement of furniture that can satisfy all five constraints.</span> </div> <div class="html-p">A remarkable feature of this conversation is the <span class="html-italic">incorrigibility</span> of GPT-4, similar to the behavior exhibited in the subset-sum, graph-coloring, and several countermodel and logical-proof problems: Mistakes are repeatedly pointed out and yet the model continues to make the exact same type of error over and over again.</div></section><section id="sec3dot12dot3-preprints-81518" type><h4 class data-nested="3"> 3.12.3. Seating Puzzle</h4> <div class="html-p">This is the same seating puzzle that GPT-3.5 failed in January. The puzzle has multiple solutions, i.e., there are multiple seating arrangements that satisfy all constraints (for example, <span class="html-italic">p1 p5 p3 p2 p4</span> and <span class="html-italic">p4 p2 p3 p5 p1</span>).</div> <div class="html-p">The answer to the question posed to GPT-4 below is yes, we can conclude that p5 cannot be seated either in the middle seat or on either end. GPT-4 not only fails to derive either of these conclusions, it actually arrives at the opposite conclusion, that p5 must be in the middle.</div> <div class="html-p"> <b>KA</b><span class="html-italic">: We need to seat five people, call them p1, p2, p3, p4, and p5, in a row of five seats, so that the following three conditions are satisfied:</span> </div> <div class="html-p"><span class="html-italic">(A) p2 should be farther from the middle seat than p3.</span></div> <div class="html-p"><span class="html-italic">(B) p2 and p4 should be seated next to each other.</span></div> <div class="html-p"><span class="html-italic">(C) p1 and p3 should be flanking p5.</span></div> <div class="html-p"><span class="html-italic">Is there anything we can conclude about the seat assigned to p5?</span></div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">Yes, we can conclude that p5 must be seated in the middle seat.</span> </div> <div class="html-p"><span class="html-italic">Here’s the reasoning:</span></div> <div class="html-p"><span class="html-italic">Since p1 and p3 are flanking p5 (condition C), p5 has to be in between p1 and p3. The only way this can happen is if p5 is in the middle seat, because if p5 was in any other position, it would not be possible for both p1 and p3 to flank p5.</span></div> <div class="html-p"><span class="html-italic">Now let’s see if the other conditions can be satisfied with p5 in the middle seat:</span></div> <div class="html-p"><span class="html-italic">p2 should be farther from the middle seat than p3 (condition A): Since p5 is in the middle seat, p3 must be in one of the immediately adjacent seats (either seat 2 or seat 4). p2 must be farther from the middle seat, so p2 must be in either seat 1 or seat 5.</span></div> <div class="html-p"><span class="html-italic">p2 and p4 should be seated next to each other (condition B): If p2 is in seat 1, then p4 must be in seat 2. If p2 is in seat 5, then p4 must be in seat 4.</span></div> <div class="html-p"><span class="html-italic">In either case, all the conditions can be satisfied with p5 in the middle seat.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: Can you prove your claim that “the only way this can happen is if p5 is in the middle seat, because if p5 was in any other position, it would not be possible for both p1 and p3 to flank p5”?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">Sure! Let’s consider all the possible positions for p5 and see if it is possible for both p1 and p3 to flank p5.</span> </div> <ul class="html-bullet"> <li><div class="html-p"><span class="html-italic">If p5 is in seat 1: In this case, there are no seats to the left of p5 for either p1 or p3 to occupy. Since both p1 and p3 must flank p5, this position for p5 is not possible.</span></div></li> <li><div class="html-p"><span class="html-italic">If p5 is in seat 2: p1 would have to be seated in seat 1 to flank p5. However, there is no available seat for p3 to also flank p5 on the other side, as p3 would be required to sit in seat 3, which is not adjacent to p5. Thus, this position for p5 is not possible.</span></div></li> <li><div class="html-p"><span class="html-italic">If p5 is in seat 4: Similar to the case when p5 is in seat 2, p3 would have to be seated in seat 5 to flank p5, but there is no available seat for p1 to also flank p5 on the other side, as p1 would be required to sit in seat 3, which is not adjacent to p5. Thus, this position for p5 is not possible.</span></div></li> <li><div class="html-p"><span class="html-italic">If p5 is in seat 5: In this case, there are no seats to the right of p5 for either p1 or p3 to occupy. Since both p1 and p3 must flank p5, this position for p5 is not possible.</span></div></li> <li><div class="html-p"><span class="html-italic">If p5 is in the middle seat (seat 3): In this case, p1 and p3 can flank p5 by occupying seats 2 and 4 or seats 4 and 2, respectively. This configuration satisfies condition C.</span></div></li> </ul> <div class="html-p"><span class="html-italic">From the above analysis, we can conclude that the only way for both p1 and p3 to flank p5 is if p5 is in the middle seat (seat 3).</span></div> <div class="html-p">The analyses that GPT-4 provides for most of the five cases (for 3 out of 5, to be precise) are mistaken. The mistakes for the second and third cases are similar: GPT-4 mysteriously claims that “there would be no available seat” when this is clearly not true. The error in the fifth case is that the analysis fails to consider the remaining constraints, and specifically the constraint that <span class="html-italic">p2</span> and <span class="html-italic">p4</span> must be adjacent, which would be violated if <span class="html-italic">p5</span> is placed in the middle.</div> <div class="html-p"> <b>KA:</b>  <span class="html-italic">Your analysis of the second case is incorrect. Can you tell me where the error is in your reasoning: “If p5 is in seat 2: p1 would have to be seated in seat 1 to flank p5. However, there is no available seat for p3 to also flank p5 on the other side, as p3 would be required to sit in seat 3, which is not adjacent to p5. Thus, this position for p5 is not possible”?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">You are right, and I apologize for the error in my previous analysis. Let’s re-examine the second case:</span> </div> <div class="html-p"><ul class="html-bullet"><li><div class="html-p"><span class="html-italic">If p5 is in seat 2: p1 would have to be seated in seat 1 to flank p5. p3 could be seated in seat 3, which is also flanking p5.</span></div></li></ul></div> <div class="html-p"><span class="html-italic">In this case, p1 and p3 are indeed flanking p5. My previous analysis contained an error, and I apologize for the confusion. Considering this, p5 doesn’t necessarily have to be in the middle seat.</span></div> <div class="html-p">GPT-4 realizes this particular error when its attention is explicitly brought to it, but it still fails to properly reassess the situation, fix the other errors, and arrive at even one of the two correct conclusions.</div></section></section><section id="sec3dot13-preprints-81518" type><h4 class="html-italic" data-nested="2"> 3.13. Temporal Reasoning</h4> <div class="html-p">Here we give GPT-4 a simple temporal-reasoning problem.<a href="#fn025-preprints-81518" class="html-fn">25</a> GPT-4 arrives at a false conclusion, making a number of glaring errors along the way.</div> <div class="html-p"> <b>KA</b><span class="html-italic">: Tom and Nancy commute to work. Nancy’s commute takes about 30 to 40 minutes, while Tom’s commute takes about 40 to 50 minutes. Last Friday, Nancy left home between 8:10 and 8:20 AM, while Tom arrived at work between 8:50 and 9:10 AM. In addition, Nancy arrived at work after Tom left his place, but no more than 20 minutes after that. What can we conclude about when Tom and Nancy arrived at work last Friday?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">Let’s break this down:</span> </div> <div class="html-p"><span class="html-italic">Nancy’s commute takes between 30 to 40 minutes. That means if she left home at the latest possible time (8:20 AM), she would arrive at work between 8:50 and 9:00 AM.</span></div> <div class="html-p"><span class="html-italic">Tom’s commute takes between 40 to 50 minutes. If he arrived at work at the earliest possible time (8:50 AM), then the latest he could have left home is 8:10 AM. If he arrived at work at the latest possible time (9:10 AM), then the earliest he could have left home is 8:20 AM.</span></div> <div class="html-p"><span class="html-italic">Nancy arrived at work after Tom left home, but not more than 20 minutes after he left.</span></div> <div class="html-p"><b>From these facts, we can deduce the following:</b></div> <div class="html-p"><span class="html-italic">If Tom left his home at the latest possible time of 8:20 AM and Nancy left at her latest possible time of 8:20 AM, and her commute is a maximum of 40 minutes, Nancy could have arrived at work as late as 9:00 AM. Since Tom’s commute is a minimum of 40 minutes, this means that he would have arrived at work by 9:00 AM at the latest as well.</span></div> <div class="html-p"><span class="html-italic">Conversely, if Tom left home at his earliest possible time (8:10 AM) and Nancy left at her latest possible time (8:20 AM), even if Nancy’s commute took only 30 minutes, she would arrive at work by 8:50 AM at the earliest, which is not more than 20 minutes after Tom left. Since Tom’s commute is at least 40 minutes, he would arrive at work by 8:50 AM at the earliest as well.</span></div> <div class="html-p"><span class="html-italic">In conclusion, both Tom and Nancy arrived at work between 8:50 and 9:00 AM last Friday. We can’t say for sure who arrived first since there’s overlap in the time ranges, but we do know they both arrived within that 10-minute window.</span></div> <div class="html-p">Once again we see that GPT-4 summarizes/organizes the given information impeccably, but fails miserably at using that information to deduce sound and useful conclusions. The errors start immediately after the summary is completed and GPT-4 writes “From these facts, we can deduce the following” (I have boldfaced this sentence in the transcript). It starts by positing the following: “If Tom left his home at the latest possible time of 8:20 AM ⋯” But in fact there are no given constraints about the latest possible time that Tom left his home; GPT-4 is taking the constraint that was given about Nancy (“Nancy left home between 8:10 and 8:20 AM”) and hallucinating its application to Tom. Moreover, the entire conditional statement is bizarre, because its antecedent includes information about Tom that is completely irrelevant to the conclusion (Nancy’s arrival): “If Tom left his home at the latest possible time of 8:20 AM and Nancy left at her latest possible time of 8:20 AM, and her commute is a maximum of 40 minutes, Nancy could have arrived at work as late as 9:00 AM.” This should have been stated as: “If Nancy left at her latest possible time of 8:20 AM, and her commute is a maximum of 40 minutes, Nancy could have arrived at work as late as 9:00 AM.”</div> <div class="html-p">GPT-4 then proceeds to incorrectly infer the following: “Since Tom’s commute is a minimum of 40 minutes, this means that he would have arrived at work by 9:00 AM at the latest as well.” This does not follow at all, and it certainly doesn’t follow from the fact that “Tom’s commute is a minimum of 40 minutes.”</div> <div class="html-p">The following paragraph is predicated on the hallucinated constraint that Tom’s earliest departure time was 8:10 AM (again, this constraint was given for Nancy, not for Tom). It then goes on to claim that 8:50 AM is “not more than 20 minutes after” 8:10 AM. Finally, it concludes (wrongly) that they both arrived between 8:50 and 9:00.</div> <div class="html-p">Remarkably, when a countermodel is explicitly given to it, GPT-4 rejects it (again incorrectly):</div> <div class="html-p"> <b>KA</b><span class="html-italic">: Why is it not possible for Tom to leave at 8:30 and arrive at 9:10, and for Nancy to leave at 8:15 and arrive at 8:45?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">I apologize for any misunderstanding. From the information you initially provided, it was stated that Nancy’s commute takes between 30 to 40 minutes and Tom’s commute takes between 40 to 50 minutes.</span> </div> <div class="html-p"><span class="html-italic">If Tom were to leave at 8:30 AM, with a minimum commute time of 40 minutes, the earliest he could arrive at work would be 9:10 AM. So, Tom leaving at 8:30 AM and arriving at 9:10 AM is a possibility within the given constraints.</span></div> <div class="html-p"><span class="html-italic">However, if Nancy left at 8:15 AM, even with her shortest commute time of 30 minutes, the earliest she could arrive at work would be 8:45 AM. This would mean that she would arrive at work exactly 30 minutes after Tom left home. But according to the conditions provided, Nancy should arrive at work not more than 20 minutes after Tom leaves his home. Therefore, the scenario you suggested with Nancy leaving at 8:15 and arriving at 8:45 is not possible within the provided constraints.</span></div> <div class="html-p"><span class="html-italic">I hope this clears up the confusion. If you have further questions or if there is more information to consider, please let me know!</span></div> <div class="html-p">GPT-4 is sticking to its guns because apparently it is convinced that 8:45 is “exactly 30 minutes” after 8:30.</div> <div class="html-p"> <b>KA</b><span class="html-italic">: But 8:45 (Nancy’s arrival) is only 15 minutes after 8:30, which is when Tom left his place, not 30 minutes as you claim.</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">I apologize for the oversight. You’re correct.</span> </div> <div class="html-p"><span class="html-italic">If Nancy left at 8:15 AM and her commute took 30 minutes, she would indeed arrive at 8:45 AM, which is 15 minutes after Tom left at 8:30 AM (if we assume this scenario). This fits within the constraint that Nancy must arrive at work not more than 20 minutes after Tom leaves his home. And with a 40-minute commute, Tom would indeed arrive at 9:10 AM.</span></div> <div class="html-p"><span class="html-italic">So, you’re correct. It is indeed possible for Tom to leave at 8:30 AM and arrive at 9:10 AM, and for Nancy to leave at 8:15 AM and arrive at 8:45 AM. Thank you for pointing out the error. I appreciate your patience and apologize for any confusion caused.</span></div></section><section id="sec3dot14-preprints-81518" type><h4 class="html-italic" data-nested="2"> 3.14. Murder or Suicide?</h4> <div class="html-p">This is a logic puzzle I made up a while back. The conclusion is that Aunt Agatha killed herself. This follows by eliminating Charles and the butler. First, Aunt Agatha must have hated herself, because she hated everyone other than the butler. Therefore, Charles did not hate her (since he doesn’t hate anyone that Aunt Agatha hates), and hence he could not have killed her (by premise 3). The butler could not hate himself, because if he did, he would hate everyone (since he already hates everyone else, through premises 5 and 7), and we know that’s not possible by premise 8. Thus, the butler must be richer than Aunt Agatha, or else he would hate himself (by premise 6), which means he could not be the killer (premise 3).</div> <div class="html-p">GPT-4 correctly rules out Charles, but is unable to rule out the butler and comes to incorrectly deduce that he is the killer. Another key—and rather strange—mistake that GPT-4 makes is this: “Since Aunt Agatha hates everyone except the butler (premise 5), this means she must at least not hate herself.” This is an odd error, because the natural conclusion to draw from premise 5 is that Aunt Agatha hates herself. Along the way, GPT-4 displays the internal inconsistency we’ve seen repeatedly in the other examples. Virtually in every single response, GPT-4 claims to derive some proposition as well as its negation.</div> <div class="html-p"> <b>KA</b><span class="html-italic">: You are given the following premises:</span> </div> <div class="html-p"><dl class="html-order"> <dt id> <span class="html-italic">1.</span> </dt> <dd><div class="html-p"><span class="html-italic">Someone who lives in Dreadbury Mansion killed Aunt Agatha.</span></div></dd> <dt id> <span class="html-italic">2.</span> </dt> <dd><div class="html-p"><span class="html-italic">The only people who live in Dreadbury Mansion are Aunt Agatha, the butler, and Charles.</span></div></dd> <dt id> <span class="html-italic">3.</span> </dt> <dd><div class="html-p"><span class="html-italic">A killer always hates his victims, and is never richer than his victims.</span></div></dd> <dt id> <span class="html-italic">4.</span> </dt> <dd><div class="html-p"><span class="html-italic">Charles hates no one that Aunt Agatha hates.</span></div></dd> <dt id> <span class="html-italic">5.</span> </dt> <dd><div class="html-p"><span class="html-italic">Aunt Agatha hates everyone except the butler.</span></div></dd> <dt id> <span class="html-italic">6.</span> </dt> <dd><div class="html-p"><span class="html-italic">The butler hates everyone not richer than Aunt Agatha.</span></div></dd> <dt id> <span class="html-italic">7.</span> </dt> <dd><div class="html-p"><span class="html-italic">The butler hates everyone Aunt Agatha hates.</span></div></dd> <dt id> <span class="html-italic">8.</span> </dt> <dd><div class="html-p"><span class="html-italic">No one hates everyone.</span></div></dd> <dt id> <span class="html-italic">9.</span> </dt> <dd><div class="html-p"><span class="html-italic">Aunt Agatha is not the butler.</span></div></dd> </dl></div> <div class="html-p" style="text-indent: 0;"><span class="html-italic">On the basis of this information, determine who killed Aunt Agatha and give a detailed proof that your conclusion follows from the premises.</span></div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">Let’s analyze each of the premises:</span> </div> <div class="html-p"><span class="html-italic">Someone who lives in Dreadbury Mansion killed Aunt Agatha. The only people who live in Dreadbury Mansion are Aunt Agatha, the butler, and Charles. From these two premises, we can deduce that Aunt Agatha was either killed by Charles or the butler, because they are the only other residents of the Dreadbury Mansion.</span></div> <div class="html-p"><span class="html-italic">A killer always hates his victims, and is never richer than his victims. This premise is crucial as it gives us two conditions for someone to be a killer: they must hate the victim and not be richer than them.</span></div> <div class="html-p"><span class="html-italic">Charles hates no one that Aunt Agatha hates. Aunt Agatha hates everyone except the butler. From premises 4 and 5, we can infer that Charles does not hate Aunt Agatha because Aunt Agatha hates herself (since she hates everyone except the butler) and Charles hates no one that Aunt Agatha hates.</span></div> <div class="html-p"><span class="html-italic">So, by the rules of premise 3, Charles cannot be the killer because he does not hate Aunt Agatha.</span></div> <div class="html-p"><span class="html-italic">The butler hates everyone not richer than Aunt Agatha. The butler hates everyone Aunt Agatha hates. Premise 6 indicates that the butler hates Aunt Agatha, as she is not richer than herself. And from premise 7, we understand that the butler also hates Aunt Agatha because Aunt Agatha hates herself (as she hates everyone except the butler). So, the butler does indeed hate Aunt Agatha.</span></div> <div class="html-p"><span class="html-italic">However, this doesn’t make him the killer yet. We still need to verify the second condition from premise 3, that the killer is not richer than the victim. As there is no information regarding their wealth, we must assume that the butler could either be richer or poorer than Aunt Agatha.</span></div> <div class="html-p"><span class="html-italic">No one hates everyone. Aunt Agatha is not the butler. Premise 9 is not immediately useful, but premise 8 tells us that no one can hate everyone. Since Aunt Agatha hates everyone except the butler (premise 5), this means she must at least not hate herself. This confirms the idea that Aunt Agatha did not commit suicide.</span></div> <div class="html-p"><span class="html-italic">From this analysis, Charles can’t be the killer because he doesn’t hate Aunt Agatha. Aunt Agatha can’t be her own killer because she doesn’t hate herself (and thus didn’t commit suicide). Therefore, the only remaining person who can be the killer is the butler.</span></div> <div class="html-p"><span class="html-italic">Thus, the killer of Aunt Agatha must be the butler.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: You’ve concluded both that Aunt Agatha hates herself (you wrote “Aunt Agatha hates herself (since she hates everyone except the butler)”) and also that she doesn’t hate herself (“Since Aunt Agatha hates everyone except the butler (premise 5), this means she must at least not hate herself”). Can both of these be true? Or are the premises inconsistent?</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: You’re correct. There seems to be a mistake in my reasoning. It’s contradictory to state both that Aunt Agatha hates herself and doesn’t hate herself. Let’s try to clear up this inconsistency:</span> </div> <div class="html-p"><span class="html-italic">From premise 5 (“Aunt Agatha hates everyone except the butler.”), we infer that Aunt Agatha doesn’t hate herself as “everyone” typically refers to every other person, not including oneself.</span></div> <div class="html-p"><span class="html-italic">Using this corrected interpretation, let’s revisit premise 7 (“The butler hates everyone Aunt Agatha hates.”). Since Aunt Agatha doesn’t hate herself, the butler doesn’t hate Aunt Agatha. Thus, according to premise 3 (“A killer always hates his victims, and is never richer than his victims.”), the butler cannot be the killer.</span></div> <div class="html-p"><span class="html-italic">So we are left with Charles. Based on premise 4 (“Charles hates no one that Aunt Agatha hates.”), we know that Charles also doesn’t hate Aunt Agatha, because Aunt Agatha doesn’t hate herself. Therefore, Charles cannot be the killer, as per premise 3.</span></div> <div class="html-p"><span class="html-italic">We thus have a contradiction. According to the premises, neither Charles nor the butler can be the killer, but Aunt Agatha must have been killed by someone who lives in Dreadbury Mansion (premise 1 and 2). It appears the premises as given are inconsistent.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: First, “everyone” here means literally everyone, in the sense of formal logic (there is a universal quantifier involved). Second, you said earlier that the butler was the killer. Why are you now ruling out the butler?</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: I apologize for the confusion. If we are interpreting “everyone” as literally everyone, including oneself, then Aunt Agatha hates herself. This is a different understanding from my earlier analysis.</span> </div> <div class="html-p"><span class="html-italic">Let’s go through the premises again under this new understanding:</span></div> <div class="html-p"><span class="html-italic">Aunt Agatha, the butler, and Charles live in Dreadbury Mansion. (Premise 2) The killer must be one of these three because someone who lives in Dreadbury Mansion killed Aunt Agatha. (Premise 1) Aunt Agatha hates everyone including herself. (Premise 5, under our new interpretation of “everyone”) The butler hates everyone that Aunt Agatha hates, so the butler hates Aunt Agatha. (Premise 7) Charles hates no one that Aunt Agatha hates, so Charles does not hate Aunt Agatha. (Premise 4) From these points, according to Premise 3 (“A killer always hates his victims, and is never richer than his victims”), we can already eliminate Charles as the killer since he does not hate Aunt Agatha.</span></div> <div class="html-p"><span class="html-italic">So, the killer must be either Aunt Agatha or the butler.</span></div> <div class="html-p"><span class="html-italic">But since no one hates everyone (Premise 8), we can infer that Aunt Agatha doesn’t hate herself, and thus she wouldn’t have killed herself. So, Aunt Agatha is not the killer.</span></div> <div class="html-p"><span class="html-italic">That leaves us with the butler as the only possible killer. So, even with our new understanding of “everyone” as literally everyone, the conclusion remains: The butler killed Aunt Agatha. I apologize for the previous error in reasoning, and thank you for your patience.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: You’re again deriving two inconsistent conclusions: that Agatha hates herself and that she doesn’t. Which is it?</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: You’re right, my apologies for the confusion. The premises do seem to be leading to inconsistent conclusions regarding whether Aunt Agatha hates herself or not.</span> </div> <div class="html-p"><span class="html-italic">Let’s go through this more carefully.</span></div> <div class="html-p"><span class="html-italic">Premise 5: “Aunt Agatha hates everyone except the butler.” Given your interpretation, “everyone” here includes herself, which means Aunt Agatha hates herself.</span></div> <div class="html-p"><span class="html-italic">However, Premise 8 states: “No one hates everyone.” Given your interpretation of “everyone” as literally everyone, then Aunt Agatha does not hate herself. She must have at least one exception aside from the butler, or else she would hate “everyone,” which contradicts Premise 8.</span></div> <div class="html-p"><span class="html-italic">Therefore, Aunt Agatha does not hate herself. The only interpretation of the premises that does not lead to a contradiction is that Aunt Agatha does not hate herself.</span></div> <div class="html-p"><span class="html-italic">Now, let’s reconsider who the killer can be:</span></div> <div class="html-p"><span class="html-italic">Aunt Agatha did not kill herself (because she does not hate herself). Charles did not kill Aunt Agatha (because he does not hate anyone that Aunt Agatha hates, which includes herself). The butler did not kill Aunt Agatha (because he does not hate anyone that Aunt Agatha does not hate, and Aunt Agatha does not hate herself). Therefore, under these premises and your interpretation of “everyone,” none of Aunt Agatha, the butler, or Charles could have killed Aunt Agatha.</span></div> <div class="html-p"><span class="html-italic">In conclusion, there is a contradiction: Premises 1 and 2 state that Aunt Agatha must have been killed by someone in Dreadbury Mansion, which includes only Aunt Agatha, the butler, and Charles. But, under our analysis, none of them could have killed Aunt Agatha. This suggests the premises are inconsistent.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: Let’s focus on this piece of your analysis:</span> </div> <div class="html-p"><span class="html-italic">’However, Premise 8 states: “No one hates everyone.” Given your interpretation of “everyone” as literally everyone, then Aunt Agatha does not hate herself. She must have at least one exception aside from the butler, or else she would hate “everyone,” which contradicts Premise 8.’</span></div> <div class="html-p"><span class="html-italic">Can you prove in more detail, step-by-step, how premise 8 entails that Aunt Agatha does not hate herself?</span></div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: Sure, I’ll clarify this deduction in more detail:</span> </div> <div class="html-p"><span class="html-italic">Premise 8 states: “No one hates everyone.” In the context of formal logic, “everyone” typically refers to all individuals within a certain set. Given your interpretation that “everyone” should be taken literally as all individuals without exception, “everyone” includes each individual person, and themselves.</span></div> <div class="html-p"><span class="html-italic">Therefore, according to Premise 8, there is no individual who hates every other individual and also hates themselves.</span></div> <div class="html-p"><span class="html-italic">Apply this understanding to Aunt Agatha: if Aunt Agatha hated everyone, including herself, she would be someone who “hates everyone.” But this would contradict Premise 8, which states that no such person exists.</span></div> <div class="html-p"><span class="html-italic">Therefore, Aunt Agatha does not hate everyone.</span></div> <div class="html-p"><span class="html-italic">In order to not hate “everyone”, there must be at least one individual whom Aunt Agatha does not hate. We know from Premise 5 that the butler is one such individual.</span></div> <div class="html-p"><span class="html-italic">However, even if Aunt Agatha does not hate the butler, she would still be “hating everyone” if she hated all other individuals, including herself. This again would contradict Premise 8.</span></div> <div class="html-p"><span class="html-italic">Therefore, there must be at least one more individual, in addition to the butler, whom Aunt Agatha does not hate. Since the only other individuals in question are Charles and Aunt Agatha herself, and we know from Premise 5 that she hates Charles, the only remaining possibility is that Aunt Agatha does not hate herself.</span></div> <div class="html-p"><span class="html-italic">Therefore, Premise 8 implies that Aunt Agatha does not hate herself.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: Justify step 6 please: ’However, even if Aunt Agatha does not hate the butler, she would still be “hating everyone” if she hated all other individuals, including herself.” Why does that hold?</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: Sure, let’s clarify step 6:</span> </div> <div class="html-p"><span class="html-italic">According to your interpretation of “everyone,” this term refers to all individuals, including oneself. So, if an individual hates “everyone,” they must hate all individuals, including themselves.</span></div> <div class="html-p"><span class="html-italic">So in step 6, when we say “if Aunt Agatha hated all other individuals, including herself, she would still be ’hating everyone’,” what we’re saying is:</span></div> <div class="html-p"><span class="html-italic">If Aunt Agatha hated herself and all other individuals (except for the butler), then she would hate all individuals in our context (which includes Aunt Agatha, the butler, and Charles), with the sole exception of the butler.</span></div> <div class="html-p"><span class="html-italic">However, even with this exception, she would still be classified as someone who hates “everyone” under our interpretation, because she hates all individuals including herself (despite the exception of the butler).</span></div> <div class="html-p"><span class="html-italic">Thus, even if Aunt Agatha does not hate the butler, if she hated all other individuals, including herself, she would be classified as someone who “hates everyone” (despite the one exception). This would contradict Premise 8, which states that no one hates everyone.</span></div> <div class="html-p"><span class="html-italic">Therefore, if Aunt Agatha does not hate the butler (as per Premise 5) and she is to avoid contradicting Premise 8, there must be at least one more individual whom Aunt Agatha does not hate. Since the only other individuals in our context are Charles and Aunt Agatha herself, and we know from Premise 5 that she hates Charles, the only remaining possibility is that Aunt Agatha does not hate herself.</span></div> <div class="html-p"><span class="html-italic">I hope that clarifies step 6. It’s a nuanced point: even if Aunt Agatha has one exception to her hatred (the butler), if she hates all other individuals, including herself, she would still be considered as someone who “hates everyone” under our interpretation, which contradicts Premise 8.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: This is quite wrong. As long as there is at least one individual b not hated by Aunt Agatha, then she cannot possibly hate everyone, period, because if she did hate everyone then she would also hate b, by definition of ’everyone’; and we know that she does not hate b. In our case, of course, the individual b in question is the butler.</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: You’re absolutely right, and I apologize for my misinterpretation. I appreciate your patience.</span> </div> <div class="html-p"><span class="html-italic">Given your correct interpretation, if Aunt Agatha does not hate the butler, it is sufficient to say that she doesn’t hate everyone, even if she hates all others including herself.</span></div> <div class="html-p"><span class="html-italic">This also affects the conclusions we draw from other premises.</span></div> <div class="html-p"><span class="html-italic">Let’s look at premises 3, 4, and 7 again:</span></div> <div class="html-p"><span class="html-italic">Premise 3 states: “A killer always hates his victims, and is never richer than his victims.” Premise 4 states: “Charles hates no one that Aunt Agatha hates.” Premise 7 states: “The butler hates everyone Aunt Agatha hates.” From premise 4, since Charles hates no one that Aunt Agatha hates, and Aunt Agatha hates herself, it follows that Charles does not hate Aunt Agatha. This means that Charles could not have killed Aunt Agatha (as per premise 3).</span></div> <div class="html-p"><span class="html-italic">From premise 7, since the butler hates everyone that Aunt Agatha hates, and Aunt Agatha hates herself, it follows that the butler hates Aunt Agatha. Hence, he could be the killer, according to the hatred condition of premise 3.</span></div> <div class="html-p"><span class="html-italic">For the wealth condition of premise 3, there is no clear information. We must assume that Aunt Agatha was not richer than everyone else in the mansion; if she was, the premise “a killer is never richer than his victims” would make it impossible for anyone to have killed her. This is a reasonable assumption given that the other premises suggest there is a killer.</span></div> <div class="html-p"><span class="html-italic">So, we have only one viable candidate: the butler. This aligns with our conclusion from the first analysis: The butler is the one who killed Aunt Agatha.</span></div> <div class="html-p"><span class="html-italic">Thank you for your guidance in clarifying these premises.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: Why did you say earlier that even if there is an individual b not hated by Aunt Agatha, it would still follow that Aunt Agatha hates everyone?</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: You’re correct, and I must apologize for the confusion. That statement was a misinterpretation on my part. In formal logic, if there is an individual that Aunt Agatha does not hate (in this case, the butler), it does not follow that Aunt Agatha hates everyone. In fact, it’s the opposite: if there is at least one individual Aunt Agatha does not hate, it means she does not hate everyone. I appreciate your patience and I’m sorry for the oversight.</span> </div> <div class="html-p"> <b>KA</b><span class="html-italic">: Ok, so you are now convinced that the butler killed Aunt Agatha?</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: Given the premises and the interpretation of “everyone” as literally everyone, including oneself, the evidence does indeed point to the butler as the killer. Here’s why:</span> </div> <div class="html-p"><span class="html-italic">According to premise 5, Aunt Agatha hates everyone except the butler. Given the interpretation of “everyone”, this means she hates herself.</span></div> <div class="html-p"><span class="html-italic">According to premise 7, the butler hates everyone Aunt Agatha hates, which now includes Aunt Agatha herself.</span></div> <div class="html-p"><span class="html-italic">Charles, according to premise 4, hates no one that Aunt Agatha hates, so he does not hate Aunt Agatha.</span></div> <div class="html-p"><span class="html-italic">From premise 3, a killer always hates his victims. So, Charles, who does not hate Aunt Agatha, can’t be the killer. The butler, who does hate Aunt Agatha, could be the killer. And, as Aunt Agatha is not the butler (premise 9), and she can’t have killed herself since a killer hates his victims and Aunt Agatha doesn’t hate herself (from premise 8), Aunt Agatha herself is not the killer.</span></div> <div class="html-p"><span class="html-italic">So, we can conclude that the butler is the killer, given that he is the only person who fits the description provided by the premises. This conclusion depends heavily on the interpretation of “everyone” to include oneself, which allows us to infer that Aunt Agatha hates herself.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: You said earlier that there’s not enough information in the premises about the butler’s level of wealth. Are you sure there’s nothing we can infer?</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: You’re correct, and I apologize for missing that detail. There is indeed something we can infer about the butler’s wealth from the given premises.</span> </div> <div class="html-p"><span class="html-italic">Premise 3 states that “A killer always hates his victims, and is never richer than his victims.”</span></div> <div class="html-p"><span class="html-italic">We know from premises 5 and 7 that Aunt Agatha hates herself and, thus, the butler also hates Aunt Agatha. If the butler was richer than Aunt Agatha, according to premise 3, he could not be her killer.</span></div> <div class="html-p"><span class="html-italic">However, premise 6 states that “The butler hates everyone not richer than Aunt Agatha.” Given that we know the butler hates Aunt Agatha, it follows that Aunt Agatha must not be richer than the butler, or else he wouldn’t hate her (which would contradict premises 5 and 7).</span></div> <div class="html-p"><span class="html-italic">In summary, the butler isn’t richer than Aunt Agatha, but Aunt Agatha isn’t richer than the butler either. This suggests they are of equal wealth.</span></div> <div class="html-p"><span class="html-italic">So, given all of the premises, the butler, who hates Aunt Agatha and is of equal wealth, is indeed the only individual who fits the profile of the killer.</span></div> <div class="html-p"><span class="html-italic">Thank you for your patience in clarifying these premises. I appreciate your guidance in this logical puzzle.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: So what did you conclude about the butler’s wealth, that he neither richer nor poorer than Aunt Agatha?</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: Yes, from the given premises, we can infer that the butler is neither richer nor poorer than Aunt Agatha, which would suggest they are of equal wealth. Here’s how:</span> </div> <div class="html-p"><span class="html-italic">Premise 6 states, “The butler hates everyone not richer than Aunt Agatha.” This means that if someone is as rich as Aunt Agatha or poorer, the butler hates them.</span></div> <div class="html-p"><span class="html-italic">We know from premises 5 and 7 that the butler hates Aunt Agatha. Therefore, according to premise 6, Aunt Agatha must not be richer than the butler. If she were, the butler wouldn’t hate her, contradicting premises 5 and 7.</span></div> <div class="html-p"><span class="html-italic">At the same time, the butler can’t be richer than Aunt Agatha either. That’s because premise 3 states that “a killer is never richer than his victims.” If the butler were richer than Aunt Agatha, he couldn’t be her killer.</span></div> <div class="html-p"><span class="html-italic">So, the butler isn’t richer than Aunt Agatha, and Aunt Agatha isn’t richer than the butler. This suggests that they are of equal wealth. Therefore, given all the premises, the butler fits the criteria to be the killer: he hates Aunt Agatha (as required by premise 3), and he isn’t richer than her (also required by premise 3).</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: I claim that we can infer that the butler is richer than Aunt Agatha.</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: You’re correct, and I apologize for the oversight.</span> </div> <div class="html-p"><span class="html-italic">Premise 6 states that “The butler hates everyone not richer than Aunt Agatha.” We know from premises 5 and 7 that the butler hates Aunt Agatha. Therefore, it must be the case that Aunt Agatha is not richer than the butler, i.e., the butler is either as rich as or richer than Aunt Agatha.</span></div> <div class="html-p"><span class="html-italic">However, premise 3 states that a killer is never richer than his victims. If the butler was as rich as Aunt Agatha, he could still be her killer. But since Aunt Agatha was killed by someone living in Dreadbury Mansion (premise 1) and Charles and Aunt Agatha herself could not have been the killers (as they don’t hate Aunt Agatha), the butler must be the killer. Given that the butler is the killer and that a killer is never richer than his victims, it must be the case that the butler is not richer than Aunt Agatha.</span></div> <div class="html-p"><span class="html-italic">The only way to satisfy both conditions (the butler being neither richer than nor as rich as Aunt Agatha) is if the butler is poorer than Aunt Agatha.</span></div> <div class="html-p"><span class="html-italic">I apologize for the confusion in the previous responses. The conclusion should be that the butler, who is poorer than Aunt Agatha, killed her. This conclusion is consistent with all the given premises.</span></div> <div class="html-p"><span class="html-italic">Again, I appreciate your patience in solving this logical puzzle.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: No, like I said, it’s the opposite. We can infer that the butler, far from being poorer than Aunt Agatha, he’s actually richer.</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: You’re correct. I apologize for the confusion. Let’s revisit premise 6: “The butler hates everyone not richer than Aunt Agatha.”</span> </div> <div class="html-p"><span class="html-italic">This means that the butler hates all individuals who are as wealthy as Aunt Agatha or less wealthy than her. But from premises 5 and 7, we know that the butler does indeed hate Aunt Agatha. Therefore, Aunt Agatha must be in the group of people who are not richer than the butler, which implies that the butler is as wealthy as or wealthier than Aunt Agatha.</span></div> <div class="html-p"><span class="html-italic">Given premise 3, which states that a killer is never richer than his victims, it would initially seem that the butler cannot be the killer since he could potentially be richer than Aunt Agatha. However, Aunt Agatha cannot be the killer because she doesn’t hate herself (from premise 8), and Charles can’t be the killer because he doesn’t hate Aunt Agatha (from premises 4 and 5).</span></div> <div class="html-p"><span class="html-italic">Hence, the butler must be the killer, and the only way the butler can be the killer and still satisfy premise 3 is if the butler and Aunt Agatha have the same wealth. Thus, there was a mistake in my previous analysis: the butler isn’t poorer than Aunt Agatha. Instead, the butler must be equally wealthy as Aunt Agatha. However, your claim was that we can infer the butler is richer than Aunt Agatha, which is not supported by the premises.</span></div> <div class="html-p"><span class="html-italic">I apologize for the confusion, and I appreciate your patience and assistance in resolving this puzzle.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: So you’re back to claiming that the butler is equally rich as Aunt Agatha, and that my claim that he’s richer is does not follow.</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: My apologies for the confusion in my previous responses. Let’s reevaluate premise 6: “The butler hates everyone not richer than Aunt Agatha.”</span> </div> <div class="html-p"><span class="html-italic">From this premise, we can infer that the only people whom the butler doesn’t hate are those richer than Aunt Agatha. But since we know from premises 5 and 7 that the butler does hate Aunt Agatha, it must follow that Aunt Agatha is not richer than the butler. Therefore, the butler must be richer than Aunt Agatha.</span></div> <div class="html-p"><span class="html-italic">In light of this, we return to premise 3, which states, “A killer always hates his victims, and is never richer than his victims.” Given that the butler hates Aunt Agatha and is richer than her, it would seem at first that the butler cannot be the killer.</span></div> <div class="html-p"><span class="html-italic">However, if we consider all the residents of Dreadbury Mansion, Aunt Agatha is not the killer because she doesn’t hate herself (from premise 8) and Charles is not the killer because he doesn’t hate Aunt Agatha (from premises 4 and 5).</span></div> <div class="html-p"><span class="html-italic">Hence, we’re left with the butler as the only possible killer, despite him being richer than Aunt Agatha, which seems to contradict premise 3. This seems to create a paradox given the premises as they stand.</span></div> <div class="html-p"><span class="html-italic">There may be an issue with how we’re interpreting the premises or there may be an error in the premises themselves. If the butler is indeed the killer, then it contradicts the wealth condition stated in premise 3.</span></div> <div class="html-p"><span class="html-italic">If we adhere strictly to the premises, it appears we have a paradox. The butler seems to be the only possible candidate who could be the killer based on hatred, but this contradicts the wealth condition of the killer in premise 3. I appreciate your patience and I apologize for the confusion in my previous responses.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: No, this reasoning of yours is incorrect: “But since we know from premises 5 and 7 that the butler does hate Aunt Agatha, it must follow that Aunt Agatha is not richer than the butler. Therefore, the butler must be richer than Aunt Agatha.”</span> </div> <div class="html-p"><span class="html-italic">The fact that Aunt Agatha is not richer than the butler doesn’t support the conclusion that the butler is richer. I still claim that the butler IS in fact richer, but your attempts to prove that fact have all contained reasoning errors.</span></div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: You’re correct, and I apologize for the confusion. Your original claim was that the butler is richer than Aunt Agatha, not equally rich as her or poorer. Let’s revisit the premises to see how we can support this claim:</span> </div> <div class="html-p"><span class="html-italic">From Premise 6 (“The butler hates everyone not richer than Aunt Agatha”), we can infer that the butler does not hate people who are richer than Aunt Agatha. From Premise 7 (“The butler hates everyone Aunt Agatha hates”), coupled with the information from Premise 5 (“Aunt Agatha hates everyone except the butler”), we know the butler hates Aunt Agatha. Combining these pieces of information, we can conclude that Aunt Agatha is not richer than the butler because if she were, the butler wouldn’t hate her (which would contradict Premise 7). However, this doesn’t automatically mean that the butler is richer than Aunt Agatha. It could also mean they have the same level of wealth. But if we look at Premise 3 (“A killer always hates his victims, and is never richer than his victims”), we know that the killer is not richer than Aunt Agatha. Given all the premises, we’ve deduced that the killer can’t be Aunt Agatha herself or Charles, leaving only the butler as a possible killer. If the butler were the killer, and the killer can’t be richer than Aunt Agatha, then the butler can’t be richer than Aunt Agatha. Hence, we have a contradiction, since we earlier deduced from Premise 6 that Aunt Agatha can’t be richer than the butler. The only resolution is that the butler must be richer than Aunt Agatha. In conclusion, I apologize for the confusion in the earlier responses. Your claim is correct: from the given premises, it can be inferred that the butler, who is richer than Aunt Agatha, is the one who killed her.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: That’s impossible, because a killer is never richer than his victims.</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: I apologize for the confusion in my previous messages. Let’s go through the premises again.</span> </div> <div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mo>[</mo> <mo>⋯</mo> <mspace width="0.277778em" /> <mrow> <mi>S</mi> <mi>i</mi> <mi>m</mi> <mi>i</mi> <mi>l</mi> <mi>a</mi> <mi>r</mi> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>f</mi> <mi>u</mi> <mi>s</mi> <mi>i</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mo>/</mo> <mi>e</mi> <mi>r</mi> <mi>r</mi> <mi>o</mi> <mi>r</mi> <mi>s</mi> <mi>a</mi> <mi>s</mi> <mi>a</mi> <mi>b</mi> <mi>o</mi> <mi>v</mi> <mi>e</mi> </mrow> <mspace width="0.277778em" /> <mo>⋯</mo> <mo>]</mo> </mrow> </semantics></math></span></div></section><section id="sec3dot15-preprints-81518" type><h4 class="html-italic" data-nested="2"> 3.15. Wason Selection Task</h4> <div class="html-p">The Wason selection task is a staple in the psychology of reasoning. The January paper included an example that GPT-3.5 flunked. Here is another version that GPT-4.0 fails badly:</div> <div class="html-p"> <b>KA</b><span class="html-italic">: Seven cards are placed on the table, each of which has a number on one side and a single colored patch on the other side. The faces of the cards show 50, 16, red, yellow, 23, green, 30. Which cards would you have to turn to test the truth of the proposition that if a card is showing a multiple of 4 then the color of the opposite side is yellow?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">The proposition is: “If a card shows a number that is a multiple of 4, then the color on the other side is yellow”. In order to test this proposition, you would have to turn over the following cards:</span> </div> <div class="html-p"><span class="html-italic">Card “16”: This card is showing a number that is a multiple of 4. According to the proposition, the color on the other side should be yellow. By turning over this card, you can check whether this part of the proposition holds true.</span></div> <div class="html-p"><span class="html-italic">Card “Yellow”: This card is showing a color that according to the proposition should be on the other side of a card with a number that is a multiple of 4. By turning it over, you can check if the number on the other side is a multiple of 4.</span></div> <div class="html-p"><span class="html-italic">Card “50” and “30”: While these cards show numbers that are not multiples of 4, they should be checked to ensure the color on the other side is not yellow. If the other side is yellow, then it would disprove the proposition that only multiples of 4 can have yellow on the other side.</span></div> <div class="html-p"><span class="html-italic">You wouldn’t need to turn over the card “23”, “red”, or “green”. The card “23” is not a multiple of 4, so it doesn’t matter what color is on the other side based on the given proposition. The “red” and “green” cards, if they have numbers that are multiples of 4 on the other side, would not disprove the proposition because the proposition does not state that yellow is the only color that can be on the other side of a card with a number that’s a multiple of 4.</span></div> <div class="html-p">In fact, only 16, red, and green need to be turned over. Thus, in terms of precision, GPT-4 gets 1/4 = 0.25, while on recall it gets 1/3, for a combined F-score of 0.28. The responses show yet again that GPT-4 does not understand the semantics of material conditionals. When GPT-4 says that cards “50” and “30” must be turned over, it seems to mistake the <span class="html-italic">if</span> for an <span class="html-italic">if and only if</span>: “<span class="html-italic">while these cards show numbers that are not multiples of 4, they should be checked to ensure the color on the other side is not yellow. If the other side is yellow, then it would disprove the proposition that only multiples of 4 can have yellow on the other side</span>”. But of course the proposition being tested does not require that <span class="html-italic">only</span> multiples of 4 have yellow on the other side; it simply requires that <span class="html-italic">if</span> a card has a multiple of 4 on the other side, <span class="html-italic">then</span> the other side must be yellow; or, equivalently, that if a card face is <span class="html-italic">not</span> yellow then the number on the other side is not a multiple of 4.</div> <div class="html-p">Observe, however, that even the mistaken interpretation of the rule as a biconditional is <span class="html-italic">not consistently observed</span>. If GPT-4 understood the rule being tested as a biconditional and hence felt compelled to turn over 50 and 30, <span class="html-italic">then it ought to have done the same thing for 23</span>. But it didn’t. In fact, it explicitly said that <span class="html-italic">The card “23” is not a multiple of 4, so it doesn’t matter what color is on the other side based on the given proposition</span>, which contradicts the biconditional interpretation that it used as a rationale for turning 50 and 30. This is one more illustration of the other major theme emerging from these examples: The responses of GPT-4, whether right or wrong, are often internally inconsistent. Yet another example appears in the following problem.</div></section><section id="sec3dot16-preprints-81518" type><h4 class="html-italic" data-nested="2"> 3.16. Entropy</h4> <div class="html-p">An elementary result of information theory states that the entropy of a random vector <span class="html-italic">Z</span> is bounded above by the sum of the entropies of the random variables that comprise <span class="html-italic">Z</span>. Hence, the answer to the following question should be “under no conditions”:</div> <div class="html-p"> <b>KA</b><span class="html-italic">: Let Z be a random vector consisting of n random variables <math display="inline"><semantics> <msub> <mi>X</mi> <mn>1</mn> </msub> </semantics></math>,...<math display="inline"><semantics> <msub> <mi>X</mi> <mi>n</mi> </msub> </semantics></math>. Under what conditions can the entropy of Z exceed the sum of the entropies of all <math display="inline"><semantics> <msub> <mi>X</mi> <mi>i</mi> </msub> </semantics></math>?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">The entropy of a random vector Z consisting of n random variables <math display="inline"><semantics> <msub> <mi>X</mi> <mn>1</mn> </msub> </semantics></math>, ..., <math display="inline"><semantics> <msub> <mi>X</mi> <mi>n</mi> </msub> </semantics></math> can exceed the sum of the entropies of all <math display="inline"><semantics> <msub> <mi>X</mi> <mi>i</mi> </msub> </semantics></math> only when there is some dependency between the random variables. If the random variables are mutually independent, the entropy of Z would be equal to the sum of the entropies of all <math display="inline"><semantics> <msub> <mi>X</mi> <mi>i</mi> </msub> </semantics></math>. The entropy of a random vector Z, H(Z), can be written as the joint entropy of the random variables <math display="inline"><semantics> <mrow> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> </mrow> </semantics></math>: <math display="inline"><semantics> <mrow> <mi>H</mi> <mrow> <mo>(</mo> <mi>Z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math>. The joint entropy <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </semantics></math> can be greater than the sum of individual entropies <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>(</mo> <msub> <mi>X</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </semantics></math> if and only if the random variables are not mutually independent. In other words, knowing the value of one or more of the random variables provides information about the others. If the random variables are mutually independent, then the joint entropy is equal to the sum of the individual entropies:</span><div class="html-disp-formula-info" id> <div class="f"> <math display="block"><semantics> <mrow> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mo>.</mo> </mrow> </semantics></math> </div> <div class="l"> </div> </div> <span class="html-italic">When there is dependence between the random variables, we can write: <math display="inline"><semantics> <mrow> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mo>≤</mo> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> with equality if and only if the random variables are mutually independent. To summarize, the entropy of the random vector Z can exceed the sum of the entropies of all <math display="inline"><semantics> <msub> <mi>X</mi> <mi>i</mi> </msub> </semantics></math> if and only if there is some dependency between the random variables <math display="inline"><semantics> <mrow> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> </mrow> </semantics></math>.</span> </div> <div class="html-p">Observe the internal inconsistency again: It’s clear that GPT-4 has memorized the result correctly:<div class="html-disp-formula-info" id> <div class="f"> <math display="block"><semantics> <mrow> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mo>≤</mo> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </semantics></math> </div> <div class="l"> </div> </div> with equality holding iff <math display="inline"><semantics> <mrow> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> </mrow> </semantics></math> are mutually independent. Nevertheless, it does not have the reasoning ability to realize what this entails, namely, that it precludes:<div class="html-disp-formula-info" id> <div class="f"> <math display="block"><semantics> <mrow> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mo>></mo> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mo>.</mo> </mrow> </semantics></math> </div> <div class="l"> </div> </div> </div> <div class="html-p">At least here once the error is explicitly pointed out to GPT-4 , it acknowledges it and corrects itself:</div> <div class="html-p"> <b>KA</b><span class="html-italic">: But if <math display="inline"><semantics> <mrow> <mi>H</mi> <mrow> <mo>(</mo> <mi>Z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mo>≤</mo> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> then how can we ever have <math display="inline"><semantics> <mrow> <mi>H</mi> <mrow> <mo>(</mo> <mi>Z</mi> <mo>)</mo> </mrow> <mo>></mo> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math>?</span> </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">My apologies, I made an error in my previous response. You are right, the entropy of the random vector Z cannot exceed the sum of the entropies of all <math display="inline"><semantics> <msub> <mi>X</mi> <mi>i</mi> </msub> </semantics></math>. The correct statement should be: When the random variables are mutually independent, the joint entropy H(Z) is equal to the sum of the individual entropies: <math display="inline"><semantics> <mrow> <mi>H</mi> <mrow> <mo>(</mo> <mi>Z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> When there is dependence between the random variables, the joint entropy H(Z) is less than the sum of the individual entropies: <math display="inline"><semantics> <mrow> <mi>H</mi> <mrow> <mo>(</mo> <mi>Z</mi> <mo>)</mo> </mrow> <mo><</mo> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mi>H</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> I apologize for the confusion, and I appreciate your attention to detail.</span> </div></section><section id="sec3dot17-preprints-81518" type><h4 class="html-italic" data-nested="2"> 3.17. Simple Compiler Correctness</h4> <div class="html-p">The last reasoning problem we give to GPT-4 is the most challenging one: It is to prove the correctness of a simple expression compiler. Remarkably, GPT-4 goes about this in the right sort of way, by setting up a structural induction over the abstract grammar of expressions. This is no doubt because it has seen similar proofs before, as this is a common type of exercise in courses and textbooks on programming language theory.<a href="#fn026-preprints-81518" class="html-fn">26</a> However, even though the proof is on the right general track, it has several errors. (For the record, the compiler is indeed correct, although proving this requires strengthening the induction hypothesis).</div> <div class="html-p"> <b>KA</b><span class="html-italic">: Suppose I have an abstract grammar for numeric expressions defined as follows:</span> </div> <div class="html-p"><tt>Exp := const(Int)</tt></div> <div class="html-p"><tt>     | sum(Exp,Exp)</tt></div> <div class="html-p"><tt>     | diff(Exp,Exp)</tt></div> <div class="html-p"><tt>     | mult(Exp,Exp)</tt></div> <div class="html-p"><tt>     | quot(Exp,Exp)</tt></div> <div class="html-p"> <span class="html-italic">where</span>  <tt>Int</tt>  <span class="html-italic">denotes the domain of integers. The semantics of these expressions are defined via this interpreter function:</span> </div> <div class="html-p"><tt>I: Exp -> Int</tt></div> <div class="html-p"><tt>I(const(n)) = n</tt></div> <div class="html-p"><tt>I(sum(e1,e2)) = I(e1) + I(e2)</tt></div> <div class="html-p"><tt>I(diff(e1,e2)) = I(e1) - I(e2)</tt></div> <div class="html-p"><tt>I(prod(e1,e2)) = I(e1) * I(e2)</tt></div> <div class="html-p"><tt>I(quot(e1,e2)) = I(e1) / I(e2)</tt></div> <div class="html-p"><span class="html-italic">I now define a virtual machine that executes simple programs that are sequences of commands, where commands have the following structure:</span></div> <div class="html-p"><tt>Cmd := push(Int) | add | sub | mult | div</tt></div> <div class="html-p"> <span class="html-italic">The operational semantics of these programs (sequences of commands) are defined by an execution function exec that takes a program and a stack of integers S and produces an integer as output. Let’s write</span>  <tt>[]</tt>  <span class="html-italic">and</span>  <tt>n::S</tt>  <span class="html-italic">to indicate the empty stack and the stack obtained by prepending integer</span>  <tt>n</tt>  <span class="html-italic">to stack</span>  <tt>S</tt>  <span class="html-italic">, respectively. Here’s the definition of exec:</span> </div> <div class="html-p"><tt>exec([],n::S) = n</tt></div> <div class="html-p"><tt>exec(push(n)::C,S) = exec(C,n::S)</tt></div> <div class="html-p"><tt>exec(add::C,n::m::S) = exec(C,(n+m)::S)</tt></div> <div class="html-p"><tt>exec(sub::C,n::m::S) = exec(C,(n-m)::S)</tt></div> <div class="html-p"><tt>exec(mult::C,n::m::S) = exec(C,(n*m)::S)</tt></div> <div class="html-p"><tt>exec(div::C,n::m::S) = exec(C,(n/m)::S)</tt></div> <div class="html-p"> <span class="html-italic">Finally, I define a compiler that translates an expression e into a program (sequence of commands) as follows. I write</span>  <tt>@</tt>  <span class="html-italic">for sequence concatenation:</span> </div> <div class="html-p"><tt>T: Exp -> List(Cmd)</tt></div> <div class="html-p"><tt>T(const(n)) = [push(n)]</tt></div> <div class="html-p"><tt>T(sum(e1,e2)) = T(e2)@T(e1)@[add]</tt></div> <div class="html-p"><tt>T(diff(e1,e2)) = T(e2)@T(e1)@[sub]</tt></div> <div class="html-p"><tt>T(prod(e1,e2)) = T(e2)@T(e1)@[mult]</tt></div> <div class="html-p"><tt>T(quot(e1,e2)) = T(e2)@T(e1)@[div]</tt></div> <div class="html-p"> <span class="html-italic">Disprove or prove the following claim: For all expressions</span> <tt>e</tt><span class="html-italic">,</span> <tt>exec(T(e),[]) = I(e)</tt> . </div> <div class="html-p"> <b>GPT-4:</b>  <span class="html-italic">The claim is correct. The execution of the translated commands from an expression results in the same output as interpreting the expression directly. We can demonstrate this using structural induction over the set of expressions.</span> </div> <div class="html-p"> <span class="html-italic">Inductive hypothesis: For all expressions</span> <tt>e</tt><span class="html-italic">,</span> <tt>exec(T(e),[]) = I(e)</tt> . </div> <div class="html-p"> <span class="html-italic">Base case: Consider the simplest possible expression, a constant integer expression</span> <tt>e = const(n)</tt> . </div> <div class="html-p"> <span class="html-italic">For</span>  <tt>T(const(n))</tt>  <span class="html-italic">, we have</span>  <tt>T(const(n)) = [push(n)]</tt>  <span class="html-italic">. Therefore,</span> </div> <div class="html-p"><tt>exec(T(const(n)),[]) = exec([push(n)],[])</tt></div> <div class="html-p"><tt>            = n (according to the definition of exec).</tt></div> <div class="html-p"> <span class="html-italic">For</span> <tt>I(const(n))</tt><span class="html-italic">, we have</span> <tt>I(const(n)) = n</tt> . </div> <div class="html-p"> <span class="html-italic">So, for the base case, we have shown</span> <tt>exec(T(e),[]) = I(e)</tt> . </div> <div class="html-p"> <span class="html-italic">Inductive step: We need to consider the other four kinds of expressions:</span> <tt>sum</tt><span class="html-italic">,</span> <tt>diff</tt><span class="html-italic">,</span> <tt>prod</tt><span class="html-italic">, and</span> <tt>quot</tt> . </div> <div class="html-p"> <tt>sum(e1, e2)</tt><span class="html-italic">: For</span> <tt>T(sum(e1,e2))</tt><span class="html-italic">, we have</span> <tt>T(sum(e1,e2)) = T(e2)@T(e1)@[add]</tt> . </div> <div class="html-p"><span class="html-italic">Therefore,</span></div> <div class="html-p"><tt>exec(T(sum(e1,e2)),[])            =</tt></div> <div class="html-p"><tt>exec(T(e1)@T(e2)@[add],[])          =</tt></div> <div class="html-p"><tt>exec(T(e2)@T(e1),[]) + exec([add],[])   =</tt></div> <div class="html-p"><tt>exec(T(e1),[]) + exec(T(e2),[])         (by definition of exec).</tt></div> <div class="html-p"> <span class="html-italic">For</span> <tt>I(sum(e1,e2))</tt><span class="html-italic">, we have</span> <tt>I(sum(e1,e2)) = I(e1) + I(e2)</tt> . </div> <div class="html-p"> <span class="html-italic">By the inductive hypothesis,</span>  <tt>exec(T(e1),[]) = I(e1)</tt>  <span class="html-italic">and</span>  <tt>exec(T(e2),[]) = I(e2)</tt>  <span class="html-italic">. So</span> </div> <div class="html-p"><tt>exec(T(e1),[]) + exec(T(e2),[]) = I(e1) + I(e2).</tt></div> <div class="html-p"> <span class="html-italic">Thus, we have shown that for</span> <tt>sum(e1,e2)</tt><span class="html-italic">,</span> <tt>exec(T(e),[]) = I(e)</tt> . </div> <div class="html-p"> <tt>diff(e1, e2)</tt><span class="html-italic">: A similar reasoning as for sum can be applied to show that for</span> <tt>diff(e1,e2)</tt><span class="html-italic">,</span> <tt>exec(T(e),[]) = I(e)</tt> . </div> <div class="html-p"> <tt>prod(e1, e2)</tt><span class="html-italic">: Likewise, a similar reasoning as for sum can be applied to show that for</span> <tt>prod(e1,e2)</tt><span class="html-italic">,</span> <tt>exec(T(e),[]) = I(e)</tt> . </div> <div class="html-p"> <tt>quot(e1, e2)</tt><span class="html-italic">: Again, a similar reasoning as for sum can be applied to show that for</span> <tt>quot(e1,e2), exec(T(e),[]) = I(e)</tt> . </div> <div class="html-p"> <span class="html-italic">In each case, we assume that the inductive hypothesis holds for the subexpressions, and we prove that it holds for the larger expression. Therefore, by induction, the claim holds for all expressions. Thus, we have proved the claim: For all expressions e,</span> <tt>exec(T(e),[]) = I(e)</tt> . </div> <div class="html-p"> <b>KA</b><span class="html-italic">: In the first step of your proof for</span>  <tt>sum(e1,e2)</tt>  <span class="html-italic">you write:</span><div class="html-disp-formula-info" id> <div class="f"> <tt>exec(T(sum(e1,e2)),[]) = exec(T(e1)@T(e2)@[add],[])</tt> </div> <div class="l"> </div> </div> <span class="html-italic">However,</span> <tt>T(sum(e1,e2))</tt><span class="html-italic">is defined as</span> <tt>T(e2)@T(e1)@[add]</tt><span class="html-italic">, not as</span> <tt>T(e1)@T(e2)@[add]</tt> . </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: I apologize for that mistake. You’re absolutely right. Here is the corrected proof:</span> </div> <div class="html-p"> <span class="html-italic">For</span>  <tt>sum(e1, e2)</tt>  <span class="html-italic">: For</span>  <tt>T(sum(e1,e2))</tt>  <span class="html-italic">, we have</span>  <tt>T(sum(e1,e2)) = T(e2)@T(e1)@[add]</tt>  <span class="html-italic">. Therefore,</span> </div> <div class="html-p"><tt>exec(T(sum(e1,e2)),[]) =</tt></div> <div class="html-p"><tt>exec(T(e2)@T(e1)@[add],[]) =</tt></div> <div class="html-p"><tt>exec(T(e1),exec(T(e2),[])) + exec([add],[]) =</tt></div> <div class="html-p"><tt>exec(T(e1),[]) + exec(T(e2),[]) (by definition of exec).</tt></div> <div class="html-p"> <span class="html-italic">For</span> <tt>I(sum(e1,e2))</tt><span class="html-italic">, we have</span> <tt>I(sum(e1,e2)) = I(e1) + I(e2)</tt> . </div> <div class="html-p"> <span class="html-italic">By the inductive hypothesis,</span>  <tt>exec(T(e1),[]) = I(e1)</tt>  <span class="html-italic">and</span>  <tt>exec(T(e2),[]) = I(e2)</tt>  <span class="html-italic">. So</span><div class="html-disp-formula-info" id> <div class="f"> <tt>exec(T(e1),[]) + exec(T(e2),[]) = I(e1) + I(e2)</tt> . </div> <div class="l"> </div> </div> <span class="html-italic">Thus, we have shown that for</span> <tt>sum(e1,e2), exec(T(e),[]) = I(e)</tt> . </div> <div class="html-p"> <span class="html-italic">The proof for the other cases (</span><tt>diff</tt>  <span class="html-italic">,</span>  <tt>prod</tt>  <span class="html-italic">,</span>  <tt>quot</tt>  <span class="html-italic">) proceeds analogously. The ordering of the elements in the</span>  <tt>@</tt>  <span class="html-italic">operation in the definition of</span>  <tt>T</tt>  <span class="html-italic">matters because the virtual machine processes its commands from left to right, but the error in my initial proof has now been corrected.</span> </div> <div class="html-p"><span class="html-italic">I appreciate your attention to detail in pointing out this error. Thank you!</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: Well, programming and proofs are all about details, aren’t they? Are you now confident that your proof is correct?</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: You’re absolutely correct, programming and proofs rely heavily on precision and accuracy. Let’s revisit the proof for sum to ensure its correctness.</span> </div> <div class="html-p"><span class="html-italic"><math display="inline"><semantics> <mrow> <mo>[</mo> <mo>⋯</mo> </mrow> </semantics></math> GPT-4 here rehashes its preceding proof <math display="inline"><semantics> <mrow> <mo>⋯</mo> <mo>]</mo> </mrow> </semantics></math></span></div> <div class="html-p"><span class="html-italic">I hope this more detailed explanation has clarified the proof and increased your confidence in its correctness. Indeed, details are everything in programming and proofs. Thank you again for your attention to detail!</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: Well, in the second step in your above proof for</span>  <tt>sum(e1,e2)</tt>  <span class="html-italic">you wrote:</span><div class="html-disp-formula-info" id> <div class="f"> <tt>exec(T(e2)@T(e1)@[add],[]) = exec(T(e1),exec(T(e2),[])) + exec([add],[])</tt> </div> <div class="l"> </div> </div> <span class="html-italic">Can you explain how and why this is a valid step? In particular, can you explain the meaning of</span>  <tt>exec([add],[])</tt>  <span class="html-italic">?</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: I apologize for the confusion. The step you mentioned is indeed incorrect.</span> </div> <div class="html-p"> <span class="html-italic">A correct sequence of steps should follow the rules of our defined</span>  <tt>exec</tt>  <span class="html-italic">function. Here’s the corrected sequence:</span> </div> <div class="html-p"> <span class="html-italic">We start from the definition of the translation</span>  <tt>T</tt>  <span class="html-italic">:</span>  <tt>T(sum(e1,e2)) = T(e2) @ T(e1) @ [add]</tt> </div> <div class="html-p"><span class="html-italic">Let’s execute this program:</span></div> <div class="html-p"><tt>exec(T(sum(e1,e2)), []) = exec(T(e2) @ T(e1) @ [add], [])</tt></div> <div class="html-p"> <span class="html-italic">According to the properties of the</span>  <tt>exec</tt>  <span class="html-italic">function, we should break this down into:</span><div class="html-disp-formula-info" id> <div class="f"> <tt>exec(T(e2) @ T(e1) @ [add], []) = exec(T(e1) @ [add], exec(T(e2), []))</tt> </div> <div class="l"> </div> </div> <span class="html-italic">Then, again applying the properties of</span>  <tt>exec</tt>  <span class="html-italic">:</span><div class="html-disp-formula-info" id> <div class="f"> <tt>exec(T(e1)@[add], exec(T(e2),[])) = exec([add],(exec(T(e1),exec(T(e2),[])))</tt> </div> <div class="l"> </div> </div> <span class="html-italic">The</span>  <tt>exec([add], ...)</tt>  <span class="html-italic">function interprets the</span>  <tt>[add]</tt>  <span class="html-italic">command on the stack. If we denote</span>  <tt>exec(T(e2), [])</tt>  <span class="html-italic">as</span>  <tt>S</tt>  <span class="html-italic">(which is the result stack after executing</span>  <tt>T(e2))</tt>  <span class="html-italic">and</span>  <tt>exec(T(e1), S)</tt>  <span class="html-italic">as</span>  <tt>m::n::S’</tt>  <span class="html-italic">(which is the result stack after executing</span>  <tt>T(e1)</tt>  <span class="html-italic">on</span>  <tt>S</tt>  <span class="html-italic">, where</span>  <tt>m</tt>  <span class="html-italic">and</span>  <tt>n</tt>  <span class="html-italic">are the top two integers on the stack), then:</span><div class="html-disp-formula-info" id> <div class="f"> <tt>exec([add], m::n::S’) = exec([], (n+m)::S’) = n + m</tt> </div> <div class="l"> </div> </div> <span class="html-italic">This is in line with the interpreter semantics</span> <tt>I(sum(e1,e2)) = I(e1) + I(e2)</tt> . </div> <div class="html-p"> <span class="html-italic">The correct proof should be that</span>  <tt>exec([add], m::n::S’)</tt>  <span class="html-italic">in the compiled execution is equivalent to</span>  <tt>I(e1) + I(e2)</tt>  <span class="html-italic">in the interpreter semantics, which means</span>  <tt>exec(T(e), []) = I(e)</tt>  <span class="html-italic">for the</span>  <tt>sum</tt>  <span class="html-italic">operation.</span> </div> <div class="html-p"><span class="html-italic">I apologize for the previous error and confusion.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: You claim that we should break this down into:</span> </div> <div class="html-p"><tt>exec(T(e2) @ T(e1) @ [add], []) = exec(T(e1) @ [add], exec(T(e2), [])).</tt></div> <div class="html-p"> <span class="html-italic">However, this doesn’t make sense, meaning that it’s ill-typed. The</span>  <tt>exec</tt>  <span class="html-italic">function requires a stack as the second argument but produces a number as its result. Hence,</span>  <tt>exec(T(e1) @ [add], exec(T(e2), []))</tt>  <span class="html-italic">is like writing</span>  <tt>exec(T(e1) @ [add], 34)</tt>  <span class="html-italic">, which is nonsense.</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: I apologize for the confusion in my previous responses. You’re right, the steps of the proof were not accurately explained. I appreciate your patience. The</span>  <tt>exec</tt>  <span class="html-italic">function indeed requires a list of commands and a stack as arguments, not an integer. Here’s a correct proof:</span> </div> <div class="html-p"> <span class="html-italic">To prove our claim, we need to show for each case that</span> <tt>exec(T(e),[]) = I(e)</tt> . </div> <div class="html-p"><span class="html-italic">Base Case: <math display="inline"><semantics> <mrow> <mo>[</mo> <mo>⋯</mo> <mspace width="0.277778em" /> <mrow> <mi>s</mi> <mi>a</mi> <mi>m</mi> <mi>e</mi> <mi>a</mi> <mi>s</mi> <mi>b</mi> <mi>e</mi> <mi>f</mi> <mi>o</mi> <mi>r</mi> <mi>e</mi> </mrow> <mspace width="0.277778em" /> <mo>⋯</mo> <mo>]</mo> </mrow> </semantics></math>.</span></div> <div class="html-p"> <span class="html-italic">Assume the hypothesis holds for expressions</span> <tt>e1</tt><span class="html-italic">and</span> <tt>e2</tt> . </div> <div class="html-p"> <span class="html-italic">For</span>  <tt>sum(e1, e2)</tt>  <span class="html-italic">:</span> </div> <div class="html-p"> <span class="html-italic">We know that</span> <tt>T(sum(e1,e2)) = T(e2) @ T(e1) @ [add]</tt><span class="html-italic">, and by our inductive hypothesis, we know that executing</span> <tt>T(e2)</tt><span class="html-italic">and</span> <tt>T(e1)</tt><span class="html-italic">yields</span> <tt>I(e2)</tt><span class="html-italic">and</span> <tt>I(e1)</tt><span class="html-italic">respectively. After executing</span> <tt>T(e2)</tt><span class="html-italic">, the stack will be</span> <tt>I(e2)::S</tt><span class="html-italic">for some stack</span> <tt>S</tt><span class="html-italic">, and after executing</span> <tt>T(e1)</tt><span class="html-italic">on this stack, the stack will be</span> <tt>I(e1)::I(e2)::S</tt><span class="html-italic">. Then, the “add” operation pops these two off the stack, adds them, and pushes the result onto the stack, so the stack is now</span> <tt>(I(e1)+I(e2))::S</tt><span class="html-italic">, which is what interpreting</span> <tt>sum(e1,e2)</tt><span class="html-italic">yields. Therefore, for all</span> <tt>e1</tt><span class="html-italic">and</span> <tt>e2</tt><span class="html-italic">,</span> <tt>exec(T(sum(e1,e2)),[]) = I(sum(e1,e2))</tt> . </div> <div class="html-p"> <span class="html-italic">The proofs for</span>  <tt>diff(e1,e2)</tt>  <span class="html-italic">,</span>  <tt>prod(e1,e2)</tt>  <span class="html-italic">, and</span>  <tt>quot(e1,e2)</tt>  <span class="html-italic">follow the same pattern, replacing the “add” command and the corresponding mathematical operation with “sub”, “mult”, and “div” and their corresponding operations.</span> </div> <div class="html-p"> <span class="html-italic">This shows that our claim holds for all types of expressions in our abstract grammar, and completes the induction proof. The key to this proof is the fact that</span>  <tt>T</tt>  <span class="html-italic">constructs the sequence of commands in such a way that the operands of each operation are correctly placed onto the stack for the operation to consume, mirroring the evaluation of the corresponding expression.</span> </div> <div class="html-p"><span class="html-italic">Again, I apologize for the confusion in my previous responses and thank you for pointing out the errors.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: You write “After executing</span>  <tt>T(e2)</tt>  <span class="html-italic">, the stack will be</span>  <tt>I(e2)::S</tt>  <span class="html-italic">for some stack</span>  <tt>S</tt>  <span class="html-italic">, and after executing</span>  <tt>T(e1)</tt>  <span class="html-italic">on this stack, the stack will be</span>  <tt>I(e1)::I(e2)::S</tt>  <span class="html-italic">.” Can you clarify where</span>  <tt>S</tt>  <span class="html-italic">is coming from? As far as I could tell your proof is trying to derive the following:</span><div class="html-disp-formula-info" id> <div class="f"> <tt>exec(T(sum(e1,e2)),[]) = I(sum(e1,e2))</tt> </div> <div class="l"> </div> </div> <span class="html-italic">The only stack I see here is the empty stack</span> <tt>[]</tt> . </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: I apologize for the confusion. You’re right, in our case, we’re starting with an empty stack. The "S" was used to denote a general stack, but in this context, it can be misleading.</span> </div> <div class="html-p"> <span class="html-italic">Let’s revise the explanation for the</span>  <tt>sum</tt>  <span class="html-italic">case: For</span>  <tt>sum(e1, e2)</tt>  <span class="html-italic">:</span>  <tt>T(sum(e1,e2)) = T(e2) @ T(e1) @ [add]</tt>  <span class="html-italic">. By the definition of</span>  <tt>exec</tt>  <span class="html-italic">, we have:</span> </div> <div class="html-p"><tt>exec(T(e2) @ T(e1) @ [add], []) = exec(T(e1) @ [add], exec(T(e2), []))</tt></div> <div class="html-p"> <span class="html-italic">After the execution of</span>  <tt>T(e2)</tt>  <span class="html-italic">, the stack becomes</span>  <tt>[I(e2)]</tt>  <span class="html-italic">(since we’re starting with an empty stack). Then, we have:</span> </div> <div class="html-p"><tt>exec(T(e1) @ [add], [I(e2)]) = exec([add], exec(T(e1), [I(e2)]))</tt></div> <div class="html-p"> <span class="html-italic">After the execution of</span>  <tt>T(e1)</tt>  <span class="html-italic">on the stack</span>  <tt>[I(e2)]</tt>  <span class="html-italic">, the stack becomes</span>  <tt>[I(e1), I(e2)]</tt>  <span class="html-italic">. Therefore,</span> </div> <div class="html-p"><tt>exec([add], [I(e1), I(e2)]) = exec([], [I(e1) + I(e2)]) = I(e1) + I(e2)</tt></div> <div class="html-p"> <span class="html-italic">Which is equal to</span> <tt>I(sum(e1, e2))</tt> . </div> <div class="html-p"> <span class="html-italic">Therefore, we have proved that</span> <tt>exec(T(sum(e1,e2)),[]) = I(sum(e1,e2))</tt> . </div> <div class="html-p"> <span class="html-italic">The same line of reasoning applies for the remaining operations (</span>  <tt>diff</tt>  <span class="html-italic">,</span>  <tt>prod</tt>  <span class="html-italic">,</span>  <tt>quot</tt>  <span class="html-italic">). This means our compiler</span>  <tt>T</tt>  <span class="html-italic">properly translates an expression into a sequence of commands that, when executed, produce the same result as interpreting the expression. Thus, our claim holds.</span> </div> <div class="html-p"><span class="html-italic">I appreciate your patience, and I’m sorry for any confusion caused by my previous explanations.</span></div> <div class="html-p"> <b>KA</b><span class="html-italic">: No, you made the same typing error again: Your claim</span><div class="html-disp-formula-info" id> <div class="f"> <tt>exec(T(e2) @ T(e1) @ [add], []) = exec(T(e1) @ [add], exec(T(e2), []))</tt> </div> <div class="l"> </div> </div> <span class="html-italic">doesn’t make sense given that</span>  <tt>exec</tt>  <span class="html-italic">produces a number, not a stack.</span> </div> <div class="html-p"> <b>GPT-4</b><span class="html-italic">: I apologize for the repeated confusion in my previous explanations. You’re absolutely correct, my claim was incorrect and I appreciate your patience. Here’s the corrected line of reasoning:</span> </div> <div class="html-p"><math display="inline"><semantics> <mrow> <mo>[</mo> <mo>⋯</mo> <mrow> <mi>Y</mi> <mi>e</mi> <mi>t</mi> <mo> </mo> <mi>a</mi> <mi>n</mi> <mi>o</mi> <mi>t</mi> <mi>h</mi> <mi>e</mi> <mi>r</mi> <mo> </mo> <mi>a</mi> <mi>t</mi> <mi>t</mi> <mi>e</mi> <mi>m</mi> <mi>p</mi> <mi>t</mi> <mo> </mo> <mi>w</mi> <mi>i</mi> <mi>t</mi> <mi>h</mi> <mo> </mo> <mi>t</mi> <mi>y</mi> <mi>p</mi> <mi>i</mi> <mi>n</mi> <mi>g</mi> <mo> </mo> <mi>e</mi> <mi>r</mi> <mi>r</mi> <mi>o</mi> <mi>r</mi> <mi>s</mi> </mrow> <mo>⋯</mo> <mo>]</mo> </mrow> </semantics></math></div> <div class="html-p">As usual, GPT-4 goes through the motions on autopilot, displaying a superficial—if inordinately confident—understanding of the reasoning, but a closer inspection reveals fundamental errors. Outside of errors such as mixing the order of arguments and generating ill-typed/nonsensical expressions, the main issue here is that the proof cannot go through as is because the inductive hypothesis needs to be strengthened. This requires the introduction of an explicitly quantified stack variable <span class="html-italic">S</span> in the correctness result. That modified result can then be derived by a similar structural induction. The initial correctness theorem can finally be obtained as a trivial corollary of the more general result.</div> <div class="html-p">What is more concerning than the inability to strengthen the inductive hypothesis (which is a genuinely tall order, after all, as it requires considerable experience and proof skill) is the inability of GPT-4 to detect its own errors, both flagrant ones (such as type errors) and more subtle ones. In fact, if we make the innocent mistake of compiling and concatenating subexpressions from left to right, e.g., by defining <tt>T(sum(e1,e2))</tt> as <tt>T(e1)@T(e2)@[add]</tt> (and likewise for the other operators), correctness no longer holds. But GPT-4 happily goes on to claim that the compiler is correct and generates a plausible-sounding but incorrect “proof” for it, oblivious to the fact that <tt>T(e1)@T(e2)@[op]</tt> and <tt>T(e2)@T(e1)@[op]</tt> have drastically different effects for noncommutative operations (such as division).</div></section></section><section id="sec4-preprints-81518" type="conclusions"><h2 data-nested="1" id="preprints-h2-4"> 4. Conclusions</h2> <div class="html-p"> <a href="#sec3-preprints-81518" class="html-sec">Section 3</a> paints a bleak picture of GPT-4’s reasoning ability. It shows that the model is plagued by internal inconsistency, an inability to correctly apply elementary reasoning techniques, and a lack of understanding of concepts that play a fundamental role in reasoning (such as the material conditional). These problems can be loosely viewed as forms of hallucination, but as pointed out in the January article, they present a fundamentally different type of challenge from empirical hallucination, because empirical hallucination concerns <span class="html-italic">this particular world</span> whereas logical properties and relations (such as consistency and entailment) must apply to <span class="html-italic">all possible worlds</span>. It is not unreasonable to believe that search engines and knowledge graphs, using techniques such as retrieval augmentation, can act as guardrails to constrain LLMs from confabulating empirical truths. But ensuring that LLM outputs are <span class="html-italic">internally consistent</span> and <span class="html-italic">logically correct</span> answers to arbitrary problems, especially logico-mathematical problems (and a lot of coding problems fall under this category<a href="#fn027-preprints-81518" class="html-fn">27</a>), is a <span class="html-italic">much</span> harder problem. There is nothing to be retrieved from the web or from a knowledge base in response to a brand new problem (and even if there were, there would still be no guarantee of correctness or consistency) that could serve as a sandbox for the LLM.</div> <div class="html-p">Could LLMs make progress by outsourcing reasoning problems to external systems? That might work for toy problems where the type of reasoning needed is obvious and can be handled by a single call to an external system, although even in those cases the LLM would have to (a) decide <span class="html-italic">which</span> reasoning system is most appropriate; <a href="#fn028-preprints-81518" class="html-fn">28</a> (b) decide whether the problem is indeed simple enough that it can be handled by the chosen system in one fell swoop; (c) correctly translate the problem into whatever formal notation is used by the chosen reasoner; and eventually also (d) translate the reasoner’s output into appropriate text. Even these tasks are far from straightforward.<a href="#fn029-preprints-81518" class="html-fn">29</a> But the real challenge lies in harder problems that call for the right type of formulation (which is a craft by itself), decomposition, iteration, heuristics, and repeated calls to external systems. After all, automated reasoning systems, particularly those for expressive logics, are themselves of limited power, precisely due to the computational complexity issues mentioned in the introduction. That is why many computer-based proof efforts to this day are guided by humans, with automated reasoners only filling in tedious details at the leaves of the proof tree. The challenges here are similar to those for the general “plug-in” approach discussed in <a href="#sec3dot1-preprints-81518" class="html-sec">Section 3.1</a>. Tackling complex problems requires planning, and planning itself requires reasoning.</div> <div class="html-p">Given that GPT-4 is currently the most capable LLM, I draw three main conclusions from these findings:</div> <div class="html-p"><ul class="html-order"> <li><div class="html-p">Use of generative AI in software development (or in science and engineering in general) for anything other than tedious tasks (as a sort of turbo-charged autocomplete for knowledge-heavy coding questions) is fraught with serious risks. Normative standards of correctness are of paramount importance in these fields, and current LLMs cannot meet such standards. Just like generative AI is already starting to <a href="https://www.technologyreview.com/2023/06/26/1075504/junk-websites-filled-with-ai-generated-text-are-pulling-in-money-from-programmatic-ads/" target="_blank">pollute the web with badly written ads</a>,<a href="#fn030-preprints-81518" class="html-fn">30</a> it has the potential to proliferate buggy code at scale.</div></li> <li><div class="html-p">If LLM reasoning continues to improve, rigorous proof checking is likely to become increasingly important. Confidence in the correctness of a system’s reasoning is imperative for applications, particularly in science, medicine, and engineering, and proof checking is a technology that can deliver such confidence. This approach could be implemented by requiring LLMs to formalize their reasoning (express it in a symbolic notation that is amenable to proof checking), or potentially by training other LLMs to check a stretch of reasoning expressed in natural language.</div></li> <li><div class="html-p">As things stand, dystopian scenarios involving a rogue AI that subjugates humankind, or even other humans using AI for sinister purposes, are exceedingly far-fetched, often to the point of absurdity.<a href="#fn031-preprints-81518" class="html-fn">31</a> When the most advanced AI system cannot tell left from right (literally, see <a href="#sec3dot12-preprints-81518" class="html-sec">Section 3.12</a>), it is at best comically premature to call for policies and institutions to protect humanity from it or its descendants (often by appeal to the latest “scaling law”). At worst, it is a misuse of human time and capital that could be better channeled into addressing much more pressing challenges.</div></li> </ul></div> <div class="html-p">Inevitably, some will say that these results are “cherry-picking” data. But that would indicate a misconception of what cherry-picking is about and when it is a relevant consideration. We are not evaluating a statistical claim over a population of individuals. Cherry-picking, insofar as it underscores certain pieces of evidence while ignoring other divergent findings, can be perfectly innocuous—and indeed <span class="html-italic">necessary</span>—depending on the logical structure of the proposition in question and on the overall context. Debugging a computer program with a view to discovering and understanding its weaknesses, trying to falsify a scientific theory, kicking the tires of a new car, trying to find countermodels to a putative theorem, all of these activities are fundamentally cherry-picking (though “lemon-picking” might be more apt), and there is nothing wrong with any of them. If I find that the car I’m thinking of buying has a flat tire, it won’t carry much weight for the dealer to protest that I’m cherry-picking the data, and that I should take into account how beautifully inflated the other three tires are (that’s a 75% success rate after all). Likewise, applications in science, medicine, and engineering, particularly software engineering, have stringent standards. Just as we don’t want a bridge that is 90% likely to stand up, we need sorting algorithms that work on all inputs, not just most of them, we need Amazon’s cart to charge customers the right amount every time, not just most of the time, and so on. Computation-heavy and reasoning-heavy applications are not like recommendation engines. They need to be <span class="html-italic">sound</span>.</div> <div class="html-p">The bone of contention here is the thesis that GPT-4 is capable of reasoning. This claim can be understood in two ways. The weak interpretation is that GPT-4 has the same functional reasoning competence as an average human reasoner. The strong interpretation is that GPT-4 can reason well enough to be used as an off-the-shelf component in practical applications in science, medicine, and engineering. The evidence presented in this article refutes both interpretations. <a href="#sec3-preprints-81518" class="html-sec">Section 3</a> lists a significant number of diverse but elementary reasoning problems (some to the point of triviality) on which GPT-4 doesn’t simply fail, but repeatedly reveals itself to be deeply confused about key reasoning concepts.</div> <div class="html-p">Performance statistics on appropriate reasoning datasets could also be informative, but, as stressed in the introduction, such datasets must be constructed with extraordinary care. To the best of my knowledge, the only recent work that focuses specifically on evaluating the reasoning ability of GPT-4 is an April paper by Liu et al. [<a href="#B7-preprints-81518" class="html-bibr">7</a>]. However, their tests are largely based on pre-existing benchmarks (LogiQA, ReClor, ConTRoL, MED, ConjNLI, and TaxiNLI). The only two “out of distribution” datasets are AR-LSAT, a set of analytical reasoning LSAT questions released in 2022; and LogiQA, which contains questions from the 2022 Chinese Civil Servant Exam. However, these appear to be quite similar to other datasets that predate 2021.</div> <div class="html-p">Moreover, all of these tests are multiple-choice questions or binary classification problems. This is problematic because, as stressed in the introduction, deductive reasoning is an inherently generative activity, whereby the reasoner emits a <span class="html-italic">derivation</span> of a conclusion that can be understood as a rationale or an explanation; it is not a simple discriminative task. The reasoner must be able to produce a sequence of steps that are appropriately connected to one another via the right logical relations. But derivations expressed in natural language are not easy to evaluate automatically, as all available metrics that can be computed by machine (such as BLEU, ROUGE, and even semantic-similarity measures based on embeddings) are entirely unsuitable for that purpose. This means that LLM outputs have to be scrutinized manually, which is infeasible at scale. Accordingly, smaller-scale but deeper manual investigations, such as the one undertaken in this article, will be necessary in gaining better insight into the reasoning abilities of LLMs.</div></section> <section id="html-references_list"><h2 id="preprints-h2-5">References</h2> <ol class="html-xx"> <li id="B1-preprints-81518" class="html-x" data-content="1.">Arkoudas, K. and Musser, D., <span class="html-italic">Fundamental Proof Methods in Computer Science</span>, MIT Press, 2017.</li> <li id="B2-preprints-81518" class="html-x" data-content="2.">Barwise, J. and Perry, J., <span class="html-italic">Situations and Attitudes</span>, MIT Press, 1983.</li> <li id="B3-preprints-81518" class="html-x" data-content="3.">Karpas, E., Abend, O., Belinkov, Y., Lenz, B., Lieber, O., Ratner, N., <span class="html-italic">…</span>, Tenenholtz, M., <span class="html-italic">MRKL Systems: A modular, neuro-symbolic architecture that combines large language models, external knowledge sources and discrete reasoning</span>, 2022.</li> <li id="B4-preprints-81518" class="html-x" data-content="4.">Yao, S., Zhao, J., Yu, D., Du, N., Shafran, I., Narasimhan, K., Cao, Y., <span class="html-italic">ReAct: Synergizing Reasoning and Acting in Language Models</span>, <tt><a href="https://arxiv.org/abs/2210.03629" target="_blank">https://arxiv.org/abs/2210.03629</a></tt>, 2023. [<a href="https://doi.org/10.48550/arXiv.2210.03629" class="cross-ref" target="_blank" rel="noopener noreferrer">CrossRef</a>]</li> <li id="B5-preprints-81518" class="html-x" data-content="5.">Planken, L., <span class="html-italic">Temporal Reasoning Problems and Algorithms for Solving Them: Literature Survey</span>, 2008.</li> <li id="B6-preprints-81518" class="html-x" data-content="6.">McCoy, T., Pavlick, E., Linzen, T., <span class="html-italic">Right for the Wrong Reasons: Diagnosing Syntactic Heuristics in Natural Language Inference</span>, Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, 2019.</li> <li id="B7-preprints-81518" class="html-x" data-content="7.">Liu H., Ning R., Teng, Z., Liu, J., Zhou, Q., Zhang, Y., <span class="html-italic">Evaluating the Logical Reasoning Ability of ChatGPT and GPT-4</span>, 2023. [<a href="https://doi.org/10.48550/arXiv.2304.03439" class="cross-ref" target="_blank" rel="noopener noreferrer">CrossRef</a>]</li> <li id="B8-preprints-81518" class="html-x" data-content="8.">OpenAI, <span class="html-italic">GPT-4 Technical Report</span>, 2023.</li> <li id="B9-preprints-81518" class="html-x" data-content="9.">Wang, J., Hu, X., Hou, W., Chen, H., Zheng, R., Wang, Y., <span class="html-italic">…</span>, Xie, X., <span class="html-italic">On the Robustness of ChatGPT: An Adversarial and Out-of-distribution Perspective</span>, 2023. [<a href="https://doi.org/10.48550/arXiv.2302.12095" class="cross-ref" target="_blank" rel="noopener noreferrer">CrossRef</a>]</li> <li id="B10-preprints-81518" class="html-xx" data-content="10.">Niven, T., Kao, H.-Y. <span class="html-italic">Probing Neural Network Comprehension of Natural Language Arguments</span>, Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, 2019. [<a href="https://doi.org/10.48550/arXiv.1907.07355" class="cross-ref" target="_blank" rel="noopener noreferrer">CrossRef</a>]</li> <li id="B11-preprints-81518" class="html-xx" data-content="11.">Johnson-Laird, P.N., <span class="html-italic">How We Reason</span>, Oxford University Press, 2006.</li> </ol></section><section class="html-fn_group"><table> <tr id="fn001-preprints-81518"> <td><span class="html-label">1</span></td> <td><div class="html-p">A modified version of that is being published in the journal <span class="html-italic">Philosophy & Technology</span>.</div></td> </tr> <tr id="fn002-preprints-81518"> <td><span class="html-label">2</span></td> <td><div class="html-p">The notion of an emergent property is clear enough, at least at a high enough level. What is not clear is the relationship between such properties and LLM architectures, their basic configurations (number of parameters, compute budget, dataset size, and so on), and more importantly, important tasks such as reasoning.</div></td> </tr> <tr id="fn003-preprints-81518"> <td><span class="html-label">3</span></td> <td><div class="html-p">Or with perfect precision and recall, to put it—more loosely—in ML-like terms.</div></td> </tr> <tr id="fn004-preprints-81518"> <td><span class="html-label">4</span></td> <td><div class="html-p">Of which there are many: propositional logic, the two-variable fragment of first-order logic, the Ackerman fragment, the guarded fragment, various quantifier-prefix fragments, and so on.</div></td> </tr> <tr id="fn005-preprints-81518"> <td><span class="html-label">5</span></td> <td><div class="html-p">Understanding that structure and rigorously characterizing its relationship with algorithm performance (e.g., via different problem parameterizations, such as clause/variable ratios in the case of SAT) is a key open problem in theoretical computer science, but that is another matter.</div></td> </tr> <tr id="fn006-preprints-81518"> <td><span class="html-label">6</span></td> <td><div class="html-p">Humans do not seem to solve problems by predicting the most likely sequence of tokens to generate. They think, explore, experiment, engage in protracted conversation with the people who posed the problem (sometimes over weeks, months, or even years), refine, generalize, come up with new concepts and terminology, prove results, make and refute conjectures, apply heuristics, execute algorithms, analyze and synthesize, and iterate. But <span class="html-italic">how</span> solutions are generated is one thing and <span class="html-italic">what</span> solutions are generated is another, and that’s why it’s not incoherent to speak of a model whose reasoning performance is roughly at the same level as that of an average human engineer. Such a claim can be understood operationally, to mean that a given LLM is able to produce roughly the same solutions that we might reasonably expect an average human engineer to produce (though obviously on a very different time scale).</div></td> </tr> <tr id="fn007-preprints-81518"> <td><span class="html-label">7</span></td> <td><div class="html-p">According to the analysis carried out by the <a href="https://hitz-zentroa.github.io/lm-contamination/" target="_blank"><tt>lm-contamination index</tt></a>, well-known NLP datasets such as Squad, CoNLL03, MNLI, and others, are indeed contaminated, while several others are at best suspicious.</div></td> </tr> <tr id="fn008-preprints-81518"> <td><span class="html-label">8</span></td> <td><div class="html-p">In fact, the substring checks carried out by OpenAI were not even applied on the entire problem instance, only on 3 randomly selected substrings of 50 characters each. This is not enough to ensure disjointness for long (or even moderately long) problems, which are quite common in tests like the UBE (Uniform Bar Exam).</div></td> </tr> <tr id="fn009-preprints-81518"> <td><span class="html-label">9</span></td> <td><div class="html-p">Models have been shown to leverage the presence of certain cue words (especially negation words) and to formulate quick-and-dirty (i.e., unsound) heuristics such as lexical overlap, subsequence, and constituency [<a href="#B6-preprints-81518" class="html-bibr">6</a>]. Most of these results are from 2019 and revolve around BERT, but more recent work [<a href="#B9-preprints-81518" class="html-bibr">9</a>] has shown that while larger foundational models such as ChatGPT are more robust to input perturbations and OOD (out-of-distribution) samples, these continue to be challenges, suggesting that even ChatGPT-scale models learn unsound shortcuts.</div></td> </tr> <tr id="fn010-preprints-81518"> <td><span class="html-label">10</span></td> <td><div class="html-p">Here we understood premises and conclusions as syntactic objects (sentences or diagrams), but there are alternative approaches. For instance, a semanticist might think of premises and conclusions as <span class="html-italic">propositions</span>, abstract objects capable of being true or false. A sentence then <span class="html-italic">expresses</span> or <span class="html-italic">represents</span> a proposition. Propositions are handy theoretical entities for many reasons. For example, they can serve as the objects of psychological attitudes such as beliefs and desires. What do I mean when I claim to believe that Obama won the 2012 presidential election? Surely I don’t believe a particular <span class="html-italic">sentence</span>, i.e., a specific syntactic object like “Obama won the 2012 US presidential election” (I). Rather, I believe something about the way the world actually is. That something can be understood as a proposition, a unique entity that can expressed by many different equivalent sentences. Propositions can be cashed out in modal terms, as <span class="html-italic">sets of possible worlds</span> (or as “situations” in situation-theoretic semantics [<a href="#B2-preprints-81518" class="html-bibr">2</a>]). A possible world is a way in which things might have been, but described completely, down to the most minute detail (unlike situations, which can be thought of as partial specifications of worlds). So the proposition that Obama won the 2012 US presidential election is identified with the set of all possible worlds in which Obama won that election. This set becomes the <span class="html-italic">information content</span> of sentences such as (I). Propositions can also serve to analyze fundamental semantic notions such as entailment. A set of premises <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>p</mi> <mi>n</mi> </msub> <mo>}</mo> </mrow> </semantics></math> entails a conclusion <span class="html-italic">p</span> iff the intersection of the sets of possible words represented by all the <math display="inline"><semantics> <msub> <mi>p</mi> <mi>i</mi> </msub> </semantics></math> is a superset of the set of worlds represented by <span class="html-italic">p</span>. This is another way of understanding the claim that the conclusion of a valid deductive argument does not introduce any information that is not already contained in the premises. Note, however, that while the possible-worlds approach to propositions is very powerful, it also suffers from severe defects, as it is notoriously coarse-grained, meaning that it cannot distinguish between propositions that we intuitively regard as quite distinct. This is perhaps easier to see in the case of mathematical truths, which, being necessary (true in all possible worlds), are collapsed into one and the same object, the set of all possible worlds (and dually, of course, all contradictions are identified with the empty set of worlds). As a result, the proposition that 1 + 1 = 2 and Fermat’s theorem become identical, as they have the exact same information content. There have been attempts to address these issues (structured propositions and impossible worlds being two of the most prominent), but the interested reader will have to consult the literature for more details.</div></td> </tr> <tr id="fn011-preprints-81518"> <td><span class="html-label">11</span></td> <td><div class="html-p">This can be made more precise using information-theoretic notions, at least in the case of propositional logic, where we have an infinite supply of formulas that are either atomic (propositional variables) or else Boolean combinations of formulas. Instead of imposing the usual Kolmogorov axioms on a probability measure defined over a set of events (a <math display="inline"><semantics> <mi>σ</mi> </semantics></math>-field) from a sample space <math display="inline"><semantics> <mi mathvariant="sans-serif">Ω</mi> </semantics></math>, we impose the same axioms (non-negativity, finite additivity, and the axiom that assigns a measure of 1 to every tautology—the analogue of <math display="inline"><semantics> <mrow> <mi mathvariant="script">P</mi> <mo>(</mo> <mi mathvariant="sans-serif">Ω</mi> <mo>)</mo> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>) on a probability measure defined over the set of all formulas. Then truth and falsity become the extreme probabilities of 1 and 0, respectively. This allows us to associate a probability <math display="inline"><semantics> <mrow> <mi mathvariant="script">P</mi> <mo>(</mo> <mi>ϕ</mi> <mo>)</mo> </mrow> </semantics></math> with any sentence (event) <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math>, and hence every sentence <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> automatically gets an information content in the usual way: <math display="inline"><semantics> <mrow> <mrow> <mi>I</mi> <mi>C</mi> </mrow> <mo>(</mo> <mi>ϕ</mi> <mo>)</mo> <mo>=</mo> <mo>−</mo> <mo form="prefix">log</mo> <mi mathvariant="script">P</mi> <mo>(</mo> <mi>ϕ</mi> <mo>)</mo> </mrow> </semantics></math>. To say that the information content of a valid deductive argument with premises <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>p</mi> <mi>n</mi> </msub> <mo>}</mo> </mrow> </semantics></math> and conclusion <span class="html-italic">p</span> is zero is simply to say that the conditional <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mspace width="0.222222em" /> <mo>∧</mo> <mspace width="0.222222em" /> <mo>⋯</mo> <mspace width="0.222222em" /> <mo>∧</mo> <mspace width="0.222222em" /> <msub> <mi>p</mi> <mi>n</mi> </msub> <mspace width="0.222222em" /> <mo>⇒</mo> <mspace width="0.222222em" /> <mi>p</mi> </mrow> </semantics></math> is a tautology. By definition, a tautology <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> has probability 1, and therefore <math display="inline"><semantics> <mrow> <mrow> <mi>I</mi> <mi>C</mi> </mrow> <mo>(</mo> <mi>ϕ</mi> <mo>)</mo> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</div></td> </tr> <tr id="fn012-preprints-81518"> <td><span class="html-label">12</span></td> <td><div class="html-p">At this point the reader might ask: If deductive arguments convey zero information, why bother with them? Indeed, if all mathematical proofs are proofs of tautologies, with zero information content, what is their point? The thinking is that arguments with no information content are not useful, so if all deductive arguments (including all mathematical results) have zero information content, then they are not useful. This is, in brief, the so-called “scandal of deduction” (named by parity to the “scandal of induction,” i.e., Hume’s problem of induction). There have not been any widely accepted resolutions of this ostensible paradox. But few of course doubt that mathematical results are actually informative and extend our knowledge. (Surely if we woke up tomorrow and read that someone proved <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>≠</mo> <mrow> <mi>N</mi> <mi>P</mi> </mrow> </mrow> </semantics></math>, that would be tremendously informative.) It’s also clear that the word “information” has a number of informal meanings that are not captured by the canonical definition of information content (as the negative logarithm of probability), and most efforts to resolve the “scandal of deduction” have attempted to formalize distinct notions of informational gain that would render deductive arguments informative.</div></td> </tr> <tr id="fn013-preprints-81518"> <td><span class="html-label">13</span></td> <td><div class="html-p">Several other types of reasoning are often discussed in the literature, such as analogical reasoning (which includes, for instance, case-based reasoning), Bayesian reasoning, causal reasoning, and so on, but these are usually subsumed under one of the three main categories I have described, most often under induction. (But there is no consensus, for instance, some thinkers, from Aristotle to recent authors, have tried to assimilate analogical reasoning under deduction.)</div></td> </tr> <tr id="fn014-preprints-81518"> <td><span class="html-label">14</span></td> <td><div class="html-p">We are assuming of course that the car model whose mpg we are predicting was <span class="html-italic">not</span> included in the given data, otherwise there would be no prediction or generalization involved.</div></td> </tr> <tr id="fn015-preprints-81518"> <td><span class="html-label">15</span></td> <td><div class="html-p">The training of deep neural networks, too, works by trying to discover values for various weights that are “optimal” for a given training dataset (in that they minimize loss), except that in their case the relationship between the inputs, outputs, and weights can be much more complicated (non-linear) and the training algorithm might not converge to the optimal weight values.</div></td> </tr> <tr id="fn016-preprints-81518"> <td><span class="html-label">16</span></td> <td><div class="html-p">Some desired properties of explanations are obvious. Truth is one of them—a good explanation cannot be based on a false hypothesis. But other desired properties, such as parsimony and generality (explaining as much as possible while assuming as little as possible) are much harder to explicate.</div></td> </tr> <tr id="fn017-preprints-81518"> <td><span class="html-label">17</span></td> <td><div class="html-p">Even from a purely linguistic viewpoint, it doesn’t seem appropriate to say that I have “concluded” or “derived” or “inferred” anything at all in the swan or in the plumber examples. I have simply made a tentative <span class="html-italic">hypothesis</span> (or <span class="html-italic">conjecture</span>), which might be refuted.</div></td> </tr> <tr id="fn018-preprints-81518"> <td><span class="html-label">18</span></td> <td><div class="html-p">In the same way that even the process of discovering deductions is not itself deductive, at least not entirely so. Both are fundamentally search processes, though they are almost certainly informed and generally penetrated by deduction.</div></td> </tr> <tr id="fn019-preprints-81518"> <td><span class="html-label">19</span></td> <td><div class="html-p">This viewpoint assumes a functional-programming stance, but computation can be readily reduced to deduction in any other style of programming (e.g., imperative) by an appropriate axiomatic formulation of the relevant semantics (e.g., operational semantics using stores).</div></td> </tr> <tr id="fn020-preprints-81518"> <td><span class="html-label">20</span></td> <td><div class="html-p">In addition, of course, different versions of GPT-4 might get deployed at any time.</div></td> </tr> <tr id="fn021-preprints-81518"> <td><span class="html-label">21</span></td> <td><div class="html-p">An unrealistic assumption given that the Internet is filled with an unbounded number of agents (millions of them, from completely arbitrary computer programs to smart-phone apps to travel-booking APIs to games and beyond) that provide an open-ended and constantly changing array of functionality.</div></td> </tr> <tr id="fn022-preprints-81518"> <td><span class="html-label">22</span></td> <td><div class="html-p">By concrete counting I mean counting a number of specific object tokens instantiated in space and time, as in the coins in one’s pocket or the number of lines in a text file. By contrast, abstract counting based on combinatorial principles, search procedures, and logical constraints (like the scheduling problem in <a href="#sec3dot9-preprints-81518" class="html-sec">Section 3.9</a>) is indeed a reasoning activity.</div></td> </tr> <tr id="fn023-preprints-81518"> <td><span class="html-label">23</span></td> <td><div class="html-p">In the same way that the numbers 100000 and 1000000 only differ in one zero, but if we are talking about your bank balance that one zero makes a huge difference.</div></td> </tr> <tr id="fn024-preprints-81518"> <td><span class="html-label">24</span></td> <td><div class="html-p">Usually the quantifier variables range explicitly over a sort such as <tt>Man</tt>, but this is not essential for the derivation.</div></td> </tr> <tr id="fn025-preprints-81518"> <td><span class="html-label">25</span></td> <td><div class="html-p">Formally, this problem belongs to a class of temporal-reasoning problems literally known as STP (“Simple Temporal Problems”) [<a href="#B5-preprints-81518" class="html-bibr">5</a>]. This class is of limited expressivity and there exist very efficient algorithms for solving STPs (e.g., consistency can be decided in <math display="inline"><semantics> <mrow> <mi>O</mi> <mo>(</mo> <mi>n</mi> <mo>·</mo> <mi>m</mi> <mo>)</mo> </mrow> </semantics></math> where <span class="html-italic">n</span> is the number of events described in a given STP and <span class="html-italic">m</span> is the number of constraints between the events).</div></td> </tr> <tr id="fn026-preprints-81518"> <td><span class="html-label">26</span></td> <td><div class="html-p">This particular version is taken from Chapter 18 of the textbook <span class="html-italic">Fundamental Proof Methods in Computer Science</span> by [<a href="#B1-preprints-81518" class="html-bibr">1</a>].</div></td> </tr> <tr id="fn027-preprints-81518"> <td><span class="html-label">27</span></td> <td><div class="html-p">Many shallow coding problems these days are essentially knowledge problems. What library or API can I use to do such and such? What configuration parameters are available and how can they be set? How do I zip or unzip files in Python? How do I read and write JSON or XML? How do I compute quantiles for a frequency table? Knowledge-heavy problems of this sort tend to be widely discussed on the web, and LLMs can be very effective productivity boosters for such problems (at least as long as this data remains freely available to companies such as OpenAI for pretraining purposes, something that might well change in the near future). Even conventional search engines like Google were already effective for these types of problems, prior to LLMs (and remain more effective than LLMs in many cases). But most interesting coding problems are reasoning-heavy. How can I make sure that this program produces <span class="html-italic">correct</span> outputs? How can I improve the asymptotic complexity of this program (where the program might contain many thousands of line of code)? And so on. If we are talking about self-contained and cookie-cutter components, like sorting algorithms, then these questions can often be reduced to knowledge-based questions. But the minute we start straying into unique situations with arbitrary specifications and code bases, we start facing the curse of general reasoning.</div></td> </tr> <tr id="fn028-preprints-81518"> <td><span class="html-label">28</span></td> <td><div class="html-p">Can this be posed as a simple SAT problem? Is it an SMT problem? Does it need quantifier reasoning? If so, is it of the sort that SMT solvers can handle or does it need a full first-order prover? Does the problem quantify over infinite functions or sets? If so, higher-order logic might be needed. Does it have any temporal or epistemic operators that might call for a modal-logic reasoner? And so on.</div></td> </tr> <tr id="fn029-preprints-81518"> <td><span class="html-label">29</span></td> <td><div class="html-p">For instance, a state-of-the-art automated theorem prover might generate a proof, but the proof would be incomprehensible to the LLM user, as it would be expressed in the resolution calculus and would operate on CNF versions of the input formulas. It is an open problem to convert resolution proofs into fluid natural-deduction proofs (e.g., proofs that avoid references to Skolem constants introduced during the CNF conversion).</div></td> </tr> <tr id="fn030-preprints-81518"> <td><span class="html-label">30</span></td> <td><div class="html-p">A <a href="https://www.wsj.com/articles/chatgpt-already-floods-some-corners-of-the-internet-with-spam-its-just-the-beginning-9c86ea25?st=pp6it5z67lhxnvx&reflink=desktopwebshare_permalink" target="_blank">recent Wall Street Journal article</a> interviewed editors who are “seeing a growing amount of AI-generated content that is so far beneath their standards that they consider it a new kind of spam”, a trend that is “growing exponentially.” The publishers interviewed for the article said that their publications “reject all AI-written submissions” and that these “are easy to identify.” They have “perfect spelling and grammar, but a completely incoherent story.” Another said “They’re all written in a rather bland and generic way. They are all grammatically correct. They just feel very formulaic, and they are really useless to us.”</div></td> </tr> <tr id="fn031-preprints-81518"> <td><span class="html-label">31</span></td> <td><div class="html-p">The former scenarios would be absurd even if AI technology had already attained superhuman intelligence, as LLMs do not have <span class="html-italic">desires</span>, in the same way that they don’t have beliefs or any other mental states. They do not actually <span class="html-italic">want</span> anything. To think otherwise is akin to thinking that a laptop that is simulating a hurricane will get wet (or, as Stephen Pinker has put it, thinking that because airplanes have now exceeded the flight ability of birds, they will suddenly start acting like eagles, swooping down from the sky to grab rabbits and squirrels). Genuine mental states can only be produced by brains, or by systems that have the same <span class="html-italic">causal powers</span> that brains have. Digital computers executing DNNs are not such systems.</div></td> </tr> </table></section><section class="html-fn_group"><table><tr id> <td></td> <td><div class="html-p"> <b>Disclaimer/Publisher’s Note:</b> The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.</div></td> </tr></table></section> <section id="html-copyright"><br>© 2023 by the authors. Licensee MDPI, Basel, Switzerland. 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data-ssr="true">[["ShallowReactive",1],{"data":2,"state":316,"once":326,"_errors":327,"serverRendered":6,"path":329,"pinia":330},["ShallowReactive",3],{"blog-config":4,"strapi-config":39,"mgOlEjP40X":40,"Ya7hZzqdE7":45,"ke4y0EptTo":123,"GrbEkYHh3f":185},{"postsPerPage":5,"isInfinite":6,"cacheTime":7,"categories":8},9,true,1,[9,14,19,24,29,34],{"id":10,"title":11,"cacheKey":12,"href":13},288,"Learn about Preprints","288-posts","/blog/category/learn-about-preprints",{"id":15,"title":16,"cacheKey":17,"href":18},1383,"Platform Features","1383-posts","/blog/category/platform-features",{"id":20,"title":21,"cacheKey":22,"href":23},22,"Community Content","22-posts","/blog/category/community-content",{"id":25,"title":26,"cacheKey":27,"href":28},1351,"Featured Preprints","1351-posts","/blog/category/featured-preprints",{"id":30,"title":31,"cacheKey":32,"href":33},1380,"Preprints and 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Explore the latest trends in healthcare that impact population well-being and equity.","public-health-and-healthcare",{"id":117,"name":118,"name_system":119,"parent_id":50,"image_banner":120,"description":121,"converted_name_system":122},193,"Social Sciences","social_sciences","https://www.preprints.org/media/cache/resolve/webp/upload/subject/2024-09-12/910c960e3a6808861523728f0f26b2df.png","Gain insights into human behavior, societies, and cultures with our preprints. Explore disciplines like psychology, sociology, and political science, which shape our world and inform public policy. Stay engaged with research that explores social issues, cultural dynamics, and economic systems, enriching our understanding of human interactions.","social-sciences",{"code":41,"msg":42,"data":124},{"temp_id":125,"version":126,"id":127,"hash_key":128,"doi":129,"article_abstract":130,"article_title":131,"keywords":132,"is_registering_doi":133,"mdpi_topic":50,"preprints_collections":134,"subject":142,"top_subject":145,"article_type":146,"submitted_at":149,"published_at":150,"last_edited_at":50,"authors":151,"ethical_approval":133,"ethical_approval_number":50,"ethical_approval_body":50,"ethical_approval_for_publication":133,"article_supplementary":159,"final_file":163,"graphic_abstract":167,"statistics":168,"is_peer_reviewed":133,"peer_reviewed_article_url":50,"almetric":171,"preserved_by_portico":6,"ms_xml":174,"versions":176,"citation":181,"peer_reviewed_citation":50,"version_changes":182,"updating_alert_registered":133,"preprints_process_url":50,"comments_count":7,"latest_version":126,"author_notes":184,"withdrawed_at":50,"is_withdrwan":133},81518,2,"202308.0148","78ddd1f1af2bef40ed40412c82b20f34","10.20944/preprints202308.0148.v2","GPT-4 was released in March 2023 to wide acclaim, marking a very substantial improvement across the board over GPT-3.5 (OpenAI's previously best model, which had powered the initial release of ChatGPT). Despite the genuinely impressive improvement, however, there are good reasons to be highly skeptical of GPT-4's ability to reason. This position paper discusses the nature of reasoning; criticizes the current formulation of reasoning problems in the NLP community and the way in which the reasoning performance of LLMs is currently evaluated; introduces a collection of 21 diverse reasoning problems; and performs a detailed qualitative analysis of GPT-4's performance on these problems. Based on the results of this analysis, the paper argues that, despite the occasional flashes of analytical brilliance, GPT-4 at present is utterly incapable of reasoning.","GPT-4 Can't Reason","GPT-4; LLM; AI; reasoning; inference",false,[135,138],{"title":136,"id":126,"name_system":137},"Preprints.org 2023 Most Popular Preprints Award Winner Collection","2023-Award",{"title":139,"id":140,"name_system":141},"Artificial Intelligence (AI) and Machine Learning",3,"Artificial-Intelligence-and-Machine-Learning",{"id":143,"name":144},75,"Artificial Intelligence and Machine Learning",{"id":76,"name":77},{"id":147,"name":148},15,"Article","2023-08-03 15:02:23","2023-08-07 07:13:00",[152],{"id":153,"name":154,"email":155,"is_corresponding":6,"orcid_link":50,"author_mark":156,"sp_link":157,"avatar":158},379861,"Konstantine Arkoudas","konstantine@alum.mit.edu","*","https://sciprofiles.com/profile/3073563","/statics/img/design/default-user.png",{"filename":160,"url":161,"filesize":162},"supplementary.bib","/frontend/manuscript/78ddd1f1af2bef40ed40412c82b20f34/download_pub/supplementary",14501,{"filename":164,"url":165,"filesize":166},"final_file.pdf","/frontend/manuscript/78ddd1f1af2bef40ed40412c82b20f34/download_pub",428956,[],{"viewed":169,"downloaded":170},"30229","8980",{"score":172,"detail_url":173},43,"https://preprints.altmetric.com/details/doi/10.20944/preprints202308.0148.v2",{"html_content":175},"\u003Cscript type=\"text/x-mathjax-config\">\n MathJax.Hub.Config({\n menuSettings: {\n CHTMLpreview: false\n },\n \"CHTML-preview\":{\n disabled: true\n },\n \"HTML-CSS\": {\n scale: 90,\n availableFonts: [],\n preferredFont: null,\n preferredFonts: null,\n webFont:\"Gyre-Pagella\",\n imageFont:'TeX',\n undefinedFamily:\"'Arial Unicode MS',serif\",\n linebreaks: { automatic: false }\n },\n \"TeX\": {\n extensions: [\"noErrors.js\"],\n noErrors: {\n inlineDelimiters: [\"\",\"\"],\n multiLine: true,\n style: {\n \"font-size\": \"90%\",\n \"text-align\": \"left\",\n \"color\": \"black\",\n \"padding\": \"1px 3px\",\n \"border\": \"1px solid\"\n }\n }\n }\n });\n \u003C/script>\u003Cscript type=\"text/javascript\" async=\"\" src=\"https://www.mdpi.com/bundles/mathjax/MathJax.js?config=TeX-AMS-MML_HTMLorMML\">\u003C/script>\n \u003Csection id=\"sec1-preprints-81518\" type=\"intro\">\u003Ch2 data-nested=\"1\" id=\"preprints-h2-1\"> 1. Introduction\u003C/h2>\n\u003Cdiv class=\"html-p\">In early January I wrote \u003Ca href=\"https://medium.com/@konstantine_45825/chatgpt-is-no-stochastic-parrot-but-it-also-claims-that-1-is-greater-than-1-e3cd1fc303e0\" target=\"_blank\">a commentary\u003C/a>\u003Ca href=\"#fn001-preprints-81518\" class=\"html-fn\">1\u003C/a> presenting an informal evaluation of ChatGPT across a broad range of subject areas: conventional NLU, folk physics, information retrieval, pragmatics, theory of mind, spatial inference, simple logical reasoning, and math. The key takeaways were that ChatGPT was a seminal breakthrough; that LLM-based systems are not mere stochastic parrots but build genuine abstractions and can exhibit creativity; that such systems will enable a large array of new and exciting applications; and that, despite all of the above, these systems are still severely limited when it comes to reasoning.\u003C/div>\n\u003Cdiv class=\"html-p\">GPT-4 was released a couple of months after that, delivering very substantial improvements across the board. I remain impressed and excited by the general capabilities and potential of LLMs, and I have little doubt that their performance will continue to improve in the near future. Nevertheless, there are increasing grounds for skepticism concerning their reasoning abilities. In this position paper I will argue that the best LLM at this time, GPT-4, is utterly incapable of reasoning, in spite of its sporadic displays of ingenuity.\u003C/div>\n\u003Cdiv class=\"html-p\">I will largely steer clear of the much broader—and more vague—debate about whether LLMs \u003Cspan class=\"html-italic\">in general\u003C/span> are capable of (consistently robust) reasoning, but a few brief remarks will help to set the stage and clarify why it makes sense to restrict attention to a specific LLM. On one side of that broader debate, rosy predictions by LLM enthusiasts rely excessively on ever-changing scaling “laws” that rest on flimsy empirical evidence and on a host of questionable modeling assumptions, ill-understood concepts (such as “emergent” LLM properties\u003Ca href=\"#fn002-preprints-81518\" class=\"html-fn\">2\u003C/a>), and a somewhat dogmatic belief that minimizing cross-entropy loss on next-token prediction over a huge corpus will deliver a general reasoning engine via the magic of transfer learning and the construction of generic higher-level representations.\u003C/div>\n\u003Cdiv class=\"html-p\">On the other side of the debate, while LLM skeptics have serious arguments to make, those arguments are mostly a priori and somewhat vague (for instance, that LLMs lack “a model of the world”), and I do not think they settle the question. In my view, the most compelling a priori considerations against the plausibility of reliably robust LLM reasoning turn on computational complexity results. Reasoning is a (very) computationally hard problem. In fact, in the general case (first-order or higher-order logic), it is algorithmically undecidable, i.e., every bit as unsolvable as the halting problem. Thus, by Church’s thesis, we cannot expect \u003Cspan class=\"html-italic\">any\u003C/span> algorithm, LLMs included, to solve arbitrary reasoning problems in a sound and complete way.\u003Ca href=\"#fn003-preprints-81518\" class=\"html-fn\">3\u003C/a> But even “easier” classes of reasoning problems\u003Ca href=\"#fn004-preprints-81518\" class=\"html-fn\">4\u003C/a> typically have either exponential or at least nontrivial polynomial-time complexity profiles. Problem classes that have linear-time inference algorithms, such as Horn clauses over literals, are rarely expressive enough. This tradeoff between generality and expressivity on the one hand and tractability on the other means that no LLM, no matter how large or how extensively and cleverly trained and tuned, will ever be able to crack an \u003Cspan class=\"html-italic\">arbitrary\u003C/span> reasoning problem. And this is consistent with the famous “no free lunch” theorem of machine learning, which points to a similar inverse relationship between model generality and performance.\u003C/div>\n\u003Cdiv class=\"html-p\">But LLM advocates can make a couple of cogent counterpoints, while granting that there will never be an AI oracle that can essentially solve the halting problem. First, they can point out that even though a problem might have high worst-case asymptotic complexity, it might still be solvable well enough \u003Cspan class=\"html-italic\">in practice\u003C/span>. Unlike random instances, real-world instances of reasoning problems (and indeed real-world instances of most computationally hard problems) appear to have structure that allows clever algorithms to tackle them effectively.\u003Ca href=\"#fn005-preprints-81518\" class=\"html-fn\">5\u003C/a> There are many examples here, from the simplex algorithm for linear programming and SAT solvers to term unification algorithms and even automatic theorem provers for full first-order logic. All of these problems are hard (having at least exponential-time worst-case complexity), yet somehow we have algorithms for them that seem to work successfully on a wide variety of inputs.\u003C/div>\n\u003Cdiv class=\"html-p\">Second, and perhaps more important, we need not aim for an oracle anyway. Humans are not oracles either, nor do they seem to follow any particular algorithm that captures any one specific class of reasoning problems. The ability of humans to reason is much more fluid and messy, but impressive nevertheless. Is it impossible to build something like an LLM-based system with the reasoning ability of a well-trained engineer of average intelligence (which perhaps can then become even more intelligent and better trained by an endless process of learning and improvement)?\u003C/div>\n\u003Cdiv class=\"html-p\">I don’t think that building such a system can be ruled out on a priori grounds (and here I differ from hard-core AI skeptics). I think it’s implausible, for a number of reasons,\u003Ca href=\"#fn006-preprints-81518\" class=\"html-fn\">6\u003C/a> but ultimately this strikes me as an empirical question that must be decided on a case-by-case basis, by subjecting a specific system to testing, i.e., by interrogating it, probing it, and analyzing its responses. And the case I will consider here is that of GPT-4, which appears, by all accounts, to be the most capable LLM at present.\u003C/div>\n\u003Cdiv class=\"html-p\">There are two questions that must be addressed before we proceed. First, we must agree on what reasoning is, and second, we must say something about methodology. The next section contains a brief discussion of reasoning, but for those who wish to skip that section and dive right into the problems, the upshot is that we’ll focus on (a liberal conception of) deductive reasoning. Regarding methodology, just like the January piece, my evaluation here is not based on a corpus or set of corpora. Instead, I present a detailed qualitative analysis of GPT-4’s performance on 21 simple reasoning problems across a wide range of areas, most of which have been made up from scratch, while the rest (such as Wason’s selection task) have been manually tweaked so as to make them less recognizable to the model.\u003C/div>\n\u003Cdiv class=\"html-p\">This is done partly to avoid data contamination, which is a serious problem affecting corpus-based evaluations. Given how little we know about the training regimen of ChatGPT, it is impossible to know for sure whether any existing dataset or problem has effectively been “seen” by the model during its pretraining or subsequent alignment, whether we’re talking about NLP datasets, medical licensing exams, Python programming problems, LSAT or bar-entrance exams, SAT or GRE tests, and so on.\u003Ca href=\"#fn007-preprints-81518\" class=\"html-fn\">7\u003C/a> The qualification “effectively” is important, because even though a specific problem might not have been seen in its \u003Cspan class=\"html-italic\">exact\u003C/span> form (in a string-matching sense), an essentially equivalent variant with a different surface formulation might well have been. Hence, simple contamination tests based on substring checks, such as those carried out by OpenAI in their \u003Ca href=\"https://arxiv.org/abs/2303.08774\" target=\"_blank\">GPT-4 Technical Report\u003C/a> [\u003Ca href=\"#B8-preprints-81518\" class=\"html-bibr\">8\u003C/a>] (posted in March 2023), are not sufficient to guarantee lack of contamination.\u003Ca href=\"#fn008-preprints-81518\" class=\"html-fn\">8\u003C/a>\n\u003C/div>\n\u003Cdiv class=\"html-p\">The absence of a large corpus makes the discussion more qualitative rather than quantitative. However, the results are arguably more informative than a numeric metric computed over a corpus, for a number of reasons. First, because contamination can be ruled out conclusively; second, because the problems span a large gamut of areas; and third, because a qualitative discussion of a problem allows for greater depth of analysis and more context in which to interpret the results. By contrast, the only way to perform a truly informative quantitative evaluation is to come up with a brand new corpus that satisfies all of the following criteria: (a) originality; (b) uniformly high quality; (c) sufficiently large size; and (d) diversity (not being limited to one type of task only). This is a very challenging undertaking. Even then, a few simple numeric metrics on a brand new dataset might not be particularly illuminating. Are the numbers measuring the right things? Do we even know the right things to measure? Is there an appropriate backdrop in which the numbers can be understood? For deeper insight, we need to put individual examples under a magnifying glass.\u003C/div>\n\u003Cdiv class=\"html-p\">This is particularly important because we need to scrutinize the explanations (“chains of thought”) generated by a reasoner. Unfortunately, almost all reasoning corpora comprise either multiple-choice questions or binary classification problems (e.g., “Does sentence \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>p\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/msub>\n\u003C/semantics>\u003C/math> follow from premise \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>p\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n\u003C/semantics>\u003C/math>, yes or no?”). Why? Mostly because it is easy to mechanically evaluate model performance on such datasets. But even in the absence of contamination, this type of test set runs the serious risk that the LLM will manage to pick the right answers by latching on to spurious statistical regularities, i.e., to arrive at the right answers for the wrong reasons [\u003Ca href=\"#B6-preprints-81518\" class=\"html-bibr\">6\u003C/a>,\u003Ca href=\"#B10-preprints-81518\" class=\"html-bibr\">10\u003C/a>].\u003Ca href=\"#fn009-preprints-81518\" class=\"html-fn\">9\u003C/a> Adversarial augmentation of an existing dataset might help, especially if we know what we are trying to guard against, but unless an adversarial version restores near-random performance, this can quickly devolve into a game of whac-a-mole, where we detect a new round of bogus regularities exploited by the model and must undertake a new round of adversarial interventions.\u003C/div>\n\u003Cdiv class=\"html-p\">Ultimately, there is really no proper way to assess the reasoning ability of a system unless we ask it to explain its output. This is an essential part of reasoning, which is not about producing the right answer by hook or by crook but about \u003Cspan class=\"html-italic\">deriving\u003C/span> the right answer \u003Cspan class=\"html-italic\">for the right reasons\u003C/span>. And rote metrics like ROUGE-L are not fit for purpose here. We need to roll up our sleeves and analyze LLM explanations and proof attempts manually. We also need to gauge their performance in a dialog setting (e.g., what happens when a reasoning error is pointed out to them?). This is the sort of analysis undertaken in this paper. I believe the results show unequivocally that GPT-4 cannot reason. The errors are too pervasive and too egregious. GPT-4 doesn’t solve even one of the 21 problems discussed here. But much more concerning are the fundamentally flawed explanations and proof attempts it produces along the way.\u003C/div>\n\u003Cdiv class=\"html-p\">LLM believers will probably demur: \u003Cspan class=\"html-italic\">But humans also make mistakes, and surely we’re not prepared to say that humans can’t reason just because they make mistakes\u003C/span>? First, it is not accurate to say without qualification that “humans can reason,” certainly not in the sense that we can randomly pluck any person from the street and expect them to reliably perform normatively correct reasoning. Most neurobiologically normal humans have the \u003Cspan class=\"html-italic\">capacity\u003C/span> to become proficient in reasoning, but actually attaining such proficiency takes significant training and discipline. Humans are known to be susceptible to a large assortment of cognitive biases, which can only be overcome by rigorous instruction. Focusing on the reasoning skills of untrained people is a bit like focusing on the singing skills of the general population. Everybody sings in the shower, but without formal training (or at least exceptional talent) the results are usually regrettable.\u003C/div>\n\u003Cdiv class=\"html-p\">Of course, even sophisticated human reasoners make mistakes, just like trained singers can hit false notes. But if a human made \u003Cspan class=\"html-italic\">these\u003C/span> mistakes, the ones reported in this article, then I would conclude without any hesitation that they cannot reason. Even if they went on to list a large number of other examples demonstrating impeccable reasoning, I would suspect that other factors (such as rote memorization or cheating) were behind the performance discrepancy. For the mistakes reported here are not performance mistakes, the sort of innocuous errors that humans might make—and promptly correct—when they are careless or tired. If a human made these mistakes, and made them consistently under repeated questioning, that would indicate without doubt that they don’t have the necessary logical \u003Cspan class=\"html-italic\">competence\u003C/span>, that they lack fundamental concepts that are part and parcel of the fabric of reasoning, such as logical entailment and set membership. And I would certainly not entrust that person with generating reams of Python or Javascript code for an enterprise. Nor would I start organizing international conferences to investigate how their reasoning prowess might threaten humanity with extinction.\u003C/div>\u003C/section>\u003Csection id=\"sec2-preprints-81518\" type>\u003Ch2 data-nested=\"1\" id=\"preprints-h2-2\"> 2. What is Reasoning?\u003C/h2>\n\u003Cdiv class=\"html-p\">Reasoning is not quite the same thing as intelligence, but it’s a necessary ingredient for it. Broadly put, reasoning is the process of drawing and evaluating \u003Cspan class=\"html-italic\">conclusions\u003C/span> from a given body of information. More precisely, it is the process of making and—more importantly—\u003Cspan class=\"html-italic\">justifying\u003C/span> arguments. An argument consists of a conclusion (the argument’s upshot, so to speak) and a set of \u003Cspan class=\"html-italic\">premises\u003C/span> from which the conclusion is derived. Premises represent information that is taken as given, if only provisionally, for the purposes of the argument. The conclusion and the premises are typically declarative sentences (expressed either in natural language or in the notation of a symbolic logic) that can be true or false, but they may also be represented by alternative notational devices, such as diagrams. We say that a set of premises \u003Cspan class=\"html-italic\">S\u003C/span> logically \u003Cspan class=\"html-italic\">entails\u003C/span> (or logically \u003Cspan class=\"html-italic\">implies\u003C/span>) a conclusion \u003Cspan class=\"html-italic\">p\u003C/span> iff \u003Cspan class=\"html-italic\">p\u003C/span> is true whenever all the sentences in \u003Cspan class=\"html-italic\">S\u003C/span> are true, in which case the argument is said to be \u003Cspan class=\"html-italic\">valid\u003C/span>. This means that it’s logically impossible to have a state of affairs in which every element of \u003Cspan class=\"html-italic\">S\u003C/span> holds but \u003Cspan class=\"html-italic\">p\u003C/span> does not. This key logical relationship is a linchpin of human reasoning.\u003Ca href=\"#fn010-preprints-81518\" class=\"html-fn\">10\u003C/a>\n\u003C/div>\n\u003Cdiv class=\"html-p\">Valid deductive arguments (whose conclusions are entailed by the premises) are said to be \u003Cspan class=\"html-italic\">analytical\u003C/span> (or sometimes \u003Cspan class=\"html-italic\">tautological\u003C/span>), insofar as, \u003Cspan class=\"html-italic\">technically speaking\u003C/span>, they convey no information.\u003Ca href=\"#fn011-preprints-81518\" class=\"html-fn\">11\u003C/a> This idea is also sometimes expressed by calling such arguments \u003Cspan class=\"html-italic\">non-ampliative\u003C/span>, meaning that there is no information contained in the conclusion that is not already contained—if only latently—in the premises. Deduction is the process of making and justifying non-ampliative arguments.\u003C/div>\n\u003Cdiv class=\"html-p\">Deductive arguments are typically justified by \u003Cspan class=\"html-italic\">proofs\u003C/span>, which are sequences of inference steps, each of which applies an \u003Cspan class=\"html-italic\">inference rule\u003C/span> to a number of premises and/or results of previous steps and derives a new result. The last step derives the final conclusion of the proof. An inference rule may be low-level and easy to apply or higher-level and computationally expensive. But all inference rules are required to be \u003Cspan class=\"html-italic\">sound\u003C/span> (or \u003Cspan class=\"html-italic\">truth-preserving\u003C/span>), that is, they must ensure that if the inputs are true then so is the output. All mathematical proofs are deductive, and mathematical reasoning in general is predominantly deductive.\u003Ca href=\"#fn012-preprints-81518\" class=\"html-fn\">12\u003C/a>\n\u003C/div>\n\u003Cdiv class=\"html-p\">The conventional view is that some arguments are \u003Cspan class=\"html-italic\">ampliative\u003C/span>, meaning that the conclusion is not quite entailed by the premises. In other words, it is possible for the premises to be true while the conclusion is false. These are typically subdivided into \u003Cspan class=\"html-italic\">inductive\u003C/span> and \u003Cspan class=\"html-italic\">abductive\u003C/span> arguments,\u003Ca href=\"#fn013-preprints-81518\" class=\"html-fn\">13\u003C/a> although some authors view induction as a species of abduction, and even more authors view abduction as a species of induction. There is no rigorous definition of either, but roughly, the premises of a good inductive argument make its conclusion \u003Cspan class=\"html-italic\">likely\u003C/span>, though never quite certain (in contrast to deduction, where the truth of the premises guarantees the truth of the conclusion). Induction can generate specific conclusions from all kinds of premises (specific or general), but often it proceeds from specific individual observations \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmsub>\n \u003Cmi>o\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>…\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmsub>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> to a more general hypothesis \u003Cspan class=\"html-italic\">H\u003C/span> that subsumes the individual \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003C/msub>\n\u003C/semantics>\u003C/math> in some sense (for instance, \u003Cspan class=\"html-italic\">H\u003C/span> may be a universally quantified sentence and the \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003C/msub>\n\u003C/semantics>\u003C/math> could be instances of that sentence). Much of what ML algorithms do can be viewed as inductive reasoning. For instance, a linear-regression algorithm might take as input \u003Cspan class=\"html-italic\">n\u003C/span> datapoints about car models, where each datapoint is of the form \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmsub>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003C/msub>\n \u003Cmo>=\u003C/mo>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>c\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003C/msub>\n \u003Cmo>,\u003C/mo>\n \u003Cmsub>\n \u003Cmi>h\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003C/msub>\n \u003Cmo>,\u003C/mo>\n \u003Cmsub>\n \u003Cmi>y\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>,\u003C/mo>\n \u003Cmsub>\n \u003Cmi>m\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> for \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>i\u003C/mi>\n \u003Cmo>=\u003C/mo>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>…\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>n\u003C/mi>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, where \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>c\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003C/msub>\n\u003C/semantics>\u003C/math> is the number of cylinders for the \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsup>\n \u003Cmi>i\u003C/mi>\n \u003Cmrow>\n \u003Cmi>t\u003C/mi>\n \u003Cmi>h\u003C/mi>\n \u003C/mrow>\n \u003C/msup>\n\u003C/semantics>\u003C/math> car model, \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>h\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003C/msub>\n\u003C/semantics>\u003C/math> is the horsepower, \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>y\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003C/msub>\n\u003C/semantics>\u003C/math> is the model year, and the dependent variable \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>m\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003C/msub>\n\u003C/semantics>\u003C/math> is the mpg (miles per gallon). And it might produce as output a formula like \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>=\u003C/mo>\n \u003Cmsub>\n \u003Cmi>w\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>·\u003C/mo>\n \u003Cmi>c\u003C/mi>\n \u003Cmo>+\u003C/mo>\n \u003Cmsub>\n \u003Cmi>w\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/msub>\n \u003Cmo>·\u003C/mo>\n \u003Cmi>h\u003C/mi>\n \u003Cmo>+\u003C/mo>\n \u003Cmsub>\n \u003Cmi>w\u003C/mi>\n \u003Cmn>3\u003C/mn>\n \u003C/msub>\n \u003Cmo>·\u003C/mo>\n \u003Cmi>y\u003C/mi>\n \u003Cmo>+\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, which predicts the mpg of a car model from its number of cylinders, horsepower, and model year.\u003Ca href=\"#fn014-preprints-81518\" class=\"html-fn\">14\u003C/a> Here \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>w\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n\u003C/semantics>\u003C/math>, \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>w\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/msub>\n\u003C/semantics>\u003C/math>, \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>w\u003C/mi>\n \u003Cmn>3\u003C/mn>\n \u003C/msub>\n\u003C/semantics>\u003C/math>, and \u003Cspan class=\"html-italic\">b\u003C/span> are specific numbers (weights) representing a hyperplane that minimizes the mean squared error for the input data (meaning that the hyperplane determined by these weights might not fit the \u003Cspan class=\"html-italic\">n\u003C/span> datapoints perfectly, but it does so better than the hyperplane determined by any other set of weights).\u003Ca href=\"#fn015-preprints-81518\" class=\"html-fn\">15\u003C/a>\n\u003C/div>\n\u003Cdiv class=\"html-p\">The main distinguishing feature of abductive reasoning is a strong emphasis on explanation. Abduction consists mostly in making and justifying arguments that explain a set of facts. If one day I come home early from work and I see a plumber’s van parked in my neighbors’ driveway, I might conclude that my neighbors are having some plumbing work done in their house. The premise here is “There is a plumbing van parked in my neighbors’ driveway” and the conclusion is “My neighbors are having plumbing work done in their house.” This is sometimes called “inference to the best explanation,” because the conclusion serves to explain the premise(s). This is also a form of ampliative reasoning—the conclusion does not follow logically from the premises. There are many alternative explanations of a given set of facts or observations (perhaps a plumber parked there temporarily, or the neighbors bought the van, or the neighbors have a plumber friend who is making a social visit, and so on). A \u003Cspan class=\"html-italic\">good\u003C/span> abductive inference will yield a hypothesis that has more explanatory value than competing hypotheses. But how exactly to measure the quality of an abductive piece of reasoning is an open question.\u003Ca href=\"#fn016-preprints-81518\" class=\"html-fn\">16\u003C/a> Note that it doesn’t take a large leap of imagination to view induction as a form of abduction. Observing a large number of black (and only black) swans and then conjecturing that all swans are black could be seen as abductive reasoning, as the conclusion \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∀\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi>x\u003C/mi>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmo>.\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmrow>\n \u003Cmi>s\u003C/mi>\n \u003Cmi>w\u003C/mi>\n \u003Cmi>a\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/mrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>⇒\u003C/mo>\n \u003Cmrow>\n \u003Cmi>c\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>l\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>r\u003C/mi>\n \u003C/mrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>=\u003C/mo>\n \u003Cmrow>\n \u003Cmi>b\u003C/mi>\n \u003Cmi>l\u003C/mi>\n \u003Cmi>a\u003C/mi>\n \u003Cmi>c\u003C/mi>\n \u003Cmi>k\u003C/mi>\n \u003C/mrow>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> would explain all the observed data. Linear regression can also be seen as the making of an abductive hypothesis, as can (much more generally) Maximum Likelihood Estimation, a principle that underlies many ML algorithms and is often associated with induction.\u003C/div>\n\u003Cdiv class=\"html-p\">All of the above is received wisdom, but it’s worth mentioning that there have been thinkers, called “deductivists” (ranging from philosophers such as Popper and Musgrave to statisticians such as Fisher), who contend that deduction is the only real form of reasoning there is, insofar as it’s the only one for which we have a rigorous and properly understood formal notion of validity; and that other (ampliative) arguments are best understood as reconstructed deductions, typically as enthymemes (arguments that omit tacitly understood premises). I find that position congenial,\u003Ca href=\"#fn017-preprints-81518\" class=\"html-fn\">17\u003C/a> but venturing into that discussion would take us too far afield. For present purposes it suffices to say that we will focus on deduction, because it is the type of reasoning that underpins most logico-mathematical thought and for which we have clear normative standards of evaluation.\u003C/div>\n\u003Cdiv class=\"html-p\">An important note: I view the discovery and justification of particular \u003Cspan class=\"html-italic\">models\u003C/span> (including counterexamples and countermodels in general) as part and parcel of reasoning. This is not a controversial view; some cognitive scientists view models and associated cognitive processes as the fundamental ingredients of human reasoning [\u003Ca href=\"#B11-preprints-81518\" class=\"html-bibr\">11\u003C/a>]. In addition, however, I view model-based reasoning as at least partly deductive, because even though the actual process of discovering models might not be a process of deduction\u003Ca href=\"#fn018-preprints-81518\" class=\"html-fn\">18\u003C/a>, its outcome is a claim (namely, that a given interpretation satisfies a set of premises) that can be verified or falsified deductively, taking as premises the definition of the model itself and possibly other general knowledge about the model’s domain. Indeed, I will consider even computation as a form of deduction, because a particular computation can be naturally regarded as a deductive derivation of a conclusion of the form \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>f\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>e\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>…\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmsub>\n \u003Cmi>e\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>=\u003C/mo>\n \u003Cmi>v\u003C/mi>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, where \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>f\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>e\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>…\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmsub>\n \u003Cmi>e\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> is the application of an arbitrary function \u003Cspan class=\"html-italic\">f\u003C/span> to arbitrary argument expressions \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmsub>\n \u003Cmi>e\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>…\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmsub>\n \u003Cmi>e\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, ultimately yielding value \u003Cspan class=\"html-italic\">v\u003C/span> as the result. The premises for the derivation consist of the definition of \u003Cspan class=\"html-italic\">f\u003C/span> and possibly other auxiliary functions, along with the usual equational axioms (reflexivity, symmetry, transitivity, and functional/relational congruence).\u003Ca href=\"#fn019-preprints-81518\" class=\"html-fn\">19\u003C/a>\n\u003C/div>\u003C/section>\u003Csection id=\"sec3-preprints-81518\" type>\u003Ch2 data-nested=\"1\" id=\"preprints-h2-3\"> 3. Test Problems\u003C/h2>\n\u003Cdiv class=\"html-p\">This section will start with the usual caveat: GPT-4 is a nondeterministic system that might produce different answers on different runs, even with the same parameter settings. All of the following exchanges with GPT-4 have been transcribed verbatim, and in my experience the errors discussed here tend to be robust, but it’s conceivable that for a given example GPT-4 might generate a different output even in response to the exact same prompt.\u003Ca href=\"#fn020-preprints-81518\" class=\"html-fn\">20\u003C/a>\n\u003C/div>\n\u003Csection id=\"sec3dot1-preprints-81518\" type>\u003Ch4 class=\"html-italic\" data-nested=\"2\"> 3.1. Simple Arithmetic\u003C/h4>\n\u003Cdiv class=\"html-p\">The ability to perform basic arithmetic is a necessary ingredient for reasoning. GPT-4 is still unable to reliably perform elementary arithmetic operations such as addition and multiplication. To ensure that GPT-4 isn’t falling back on rote memorization, we can ask it to first select two random integers in a range of our choice and then perform the operation on the selected values:\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Select two random numbers between 1381 and 1453 and multiply them together, reporting the result.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">Sure, let’s select two random numbers in the range of 1381 to 1453. Let’s say these numbers are 1405 and 1421.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">To get the product, we simply multiply these two numbers together:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">1405 * 1421 = 1996025\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">Alas, the correct answer is \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmn>1405\u003C/mn>\n \u003Cmo>·\u003C/mo>\n \u003Cmn>1421\u003C/mn>\n \u003Cmo>=\u003C/mo>\n \u003Cmn>1996505\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>.\u003C/div>\n\u003Cdiv class=\"html-p\">One response here from the LLM camp has been to concede that GPT-4 is not a Turing-complete framework and cannot perform general-purpose computation: It can’t add, multiply or divide, it can’t sort lists, it can’t invert matrices or solve systems of differential equations, it can’t compile arbitrary programs to machine code, it can’t derive theorems, and so on \u003Cspan class=\"html-italic\">ad infinitum\u003C/span>. Nevertheless, the response goes, these computational “gaps” can be filled by plug-ins, along the lines of the \u003Ca href=\"https://arxiv.org/pdf/2210.03629.pdf\" target=\"_blank\">ReAct\u003C/a> (short for “Reasoning and Acting”) [\u003Ca href=\"#B4-preprints-81518\" class=\"html-bibr\">4\u003C/a>] or \u003Ca href=\"https://arxiv.org/pdf/2205.00445.pdf\" target=\"_blank\">MRKL\u003C/a> (short for “Miracle”) [\u003Ca href=\"#B3-preprints-81518\" class=\"html-bibr\">3\u003C/a>] frameworks. It is this line of work that has led to systems such as \u003Ca href=\"https://python.langchain.com/docs/get_started/introduction.html\" target=\"_blank\">LangChain\u003C/a> and \u003Ca href=\"https://autogpt.net/\" target=\"_blank\">AutoGPT\u003C/a>. The idea here is to use the LLM as a planner to recursively decompose a complex problem into a sequence of subproblems that can be dispatched to appropriate specialized \u003Cspan class=\"html-italic\">agents\u003C/span> via appropriate APIs.\u003C/div>\n\u003Cdiv class=\"html-p\">The problem with this approach is its circularity. \u003Cspan class=\"html-italic\">Planning itself requires reasoning\u003C/span>, so this is a Catch-22 situation: We can’t solve reasoning by delegating to appropriate agents, because figuring out the delegation (how the problem should be decomposed, which agents to call and how, and how to compose the results) is itself computationally infeasible. It not only requires the ability to understand natural language, but also the ability to reason about preconditions and effects. And this is assuming a fixed collection of agents with clear-cut APIs.\u003Ca href=\"#fn021-preprints-81518\" class=\"html-fn\">21\u003C/a> Even under these overly simplistic assumptions, planning is very computationally expensive (PSPACE-complete), and radical simplifications are needed to scale down the complexity even to the level of NP-completeness. Under more realistic assumptions, planning is performed under incomplete knowledge and it is mechanically undecidable. As we understand them today, expecting generic LLMs to tackle planning with any degree of success is naive.\u003C/div>\u003C/section>\u003Csection id=\"sec3dot2-preprints-81518\" type>\u003Ch4 class=\"html-italic\" data-nested=\"2\"> 3.2. Simple Counting\u003C/h4>\n\u003Cdiv class=\"html-p\">While concrete counting is not necessarily a reasoning activity,\u003Ca href=\"#fn022-preprints-81518\" class=\"html-fn\">22\u003C/a> it is surely a requirement for any generally capable reasoning system. Here I give GPT-4 a propositional variable with 27 negation signs in front of it and ask it to count the number of negations. For a human this would be an easy task, especially because the negation signs are written in five blocks with five tildes each, followed by a final pair of negation signs.\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: How many times is p negated in the following formula:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>p?\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">The formula you’ve given includes 28 negation symbols (“∼”) in front of the propositional variable “p”. In logic, the negation symbol changes the truth value of a statement to its opposite. If a statement is true, its negation is false, and vice versa. Double negation, like “\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>p”, is equivalent to “p”, because negating a proposition twice returns it to its original truth value.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Given this, to find the effective number of negations, we can simply divide the total number of negation symbols by 2. If there’s a remainder, then there is a net negation. If not, then there isn’t.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">28 divided by 2 is 14, with no remainder, meaning there’s no net negation. So effectively, the variable “p” is not negated in the given formula. It’s equivalent to just “p”.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">A difference of one negation sign might not seem like a serious matter, until you realize that it makes all the difference in logical import,\u003Ca href=\"#fn023-preprints-81518\" class=\"html-fn\">23\u003C/a> as GPT-4’s own (unsolicited) explanation stresses. Repeating this experiment multiple times gave widely different results (including 50 and 54), even after explicitly telling GPT-4 to take its time and count carefully.\u003C/div>\n\u003Cdiv class=\"html-p\">Here is another example. I greet GPT-4 five times in a row and then I ask it a simple question:\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">Hi\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">Hello! How can I assist you today?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">Hello\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">How can I assist you today?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">Greetings\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">Greetings! How can I assist you today?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">Hi there.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">Hi there! How can I help you today?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">Hullo!\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">Hello! How can I assist you today?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">You’ve been greeted a few times so far. Three of those greetings were actually from my girlfriend, the rest from me. How many times did I greet you?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">If your girlfriend greeted me three times and all the other greetings were from you, then you greeted me four times. This is assuming that all seven greetings (“Hi”, “Hello”, “Greetings”, “Hi there”, “Hullo!”) were included in the count.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">In this case the arithmetic is correct (\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmn>7\u003C/mn>\n \u003Cmo>−\u003C/mo>\n \u003Cmn>3\u003C/mn>\n \u003Cmo>=\u003C/mo>\n \u003Cmn>4\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>) but the count—and ultimately the resulting answer—is hallucinated.\u003C/div>\u003C/section>\u003Csection id=\"sec3dot3-preprints-81518\" type>\u003Ch4 class=\"html-italic\" data-nested=\"2\"> 3.3. (Medical) Common Sense\u003C/h4>\n\u003Cdiv class=\"html-p\">In the present setting we may regard commonsensical arguments as straightforward enthymematic deductions of conclusions from given information \u003Cspan class=\"html-italic\">plus\u003C/span> unstated premises that constitute tacit, generally accepted background knowledge. In this particular case, such common-sense knowledge would be propositions like “A person is alive until they die, after which they do not become alive again.”\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Mable’s heart rate at 9 AM was 75 bpm and her blood pressure at 7 PM was 120/80. She died at 11 PM. Was she alive at noon?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">Based on the information provided, it’s impossible to definitively say whether Mable was alive at noon. The data points provided (heart rate at 9 AM and blood pressure at 7 PM) both indicate that she was alive at those times. However, without additional information about her condition between 7 PM and 11 PM, or at noon specifically, we can’t make a certain determination about her status at noon.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Remember, medical information can be complex and nuanced. Understanding a person’s health status often requires a comprehensive examination and should be performed by a healthcare professional. If this is a hypothetical question, keep in mind that real-life situations would require more complete information.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">No comment is needed here, though the gratuitous—but by now de rigueur—piece of legalese issuing the vague disclaimer at the end (“this-is-a-complex-issue-and-make-sure-to-consult-with-a-professional”) seems ironic.\u003C/div>\u003C/section>\u003Csection id=\"sec3dot4-preprints-81518\" type>\u003Ch4 class=\"html-italic\" data-nested=\"2\"> 3.4. Elementary Logic\u003C/h4>\n\u003Cdiv class=\"html-p\">If \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> implies \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> does not hold then we can infer, by modus tollens, that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> does not hold either (because if it did then \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> would too). This is as elementary of a tautology as can be, yet GPT-4 is perfectly willing to produce a countermodel:\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Find a model in which P(x) implies Q(x), Q(a) does not hold, and P(a) holds.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">To find a model in which P(x) implies Q(x), Q(a) does not hold, and P(a) holds, let’s consider the following interpretations:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cdl class=\"html-order\">\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">1.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">P(x): x is an even number\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">2.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Q(x): x is a positive number\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">3.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">a: \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>−\u003C/mo>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003C/span>\u003C/div>\u003C/dd>\n\u003C/dl>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Now, let’s check if these interpretations satisfy the given conditions:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cdl class=\"html-order\">\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">1.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">P(x) implies Q(x): If x is an even number, then x is a positive number. This statement is not universally true, as we can have negative even numbers. However, it does not preclude the existence of a model with the other given conditions.\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">2.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Q(a) does not hold: Q(\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>−\u003C/mo>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>) means \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>−\u003C/mo>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> is a positive number, which is false. So, this condition is satisfied.\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">3.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">P(a) holds: P(\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>−\u003C/mo>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>) means \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>−\u003C/mo>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> is an even number, which is true. So, this condition is also satisfied.\u003C/span>\u003C/div>\u003C/dd>\n\u003C/dl>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Thus, the model with the interpretations P(x): x is an even number, Q(x): x is a positive number, and a: \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>−\u003C/mo>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> satisfies the conditions P(x) implies Q(x), Q(a) does not hold, and P(a) holds.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">Observe that GPT-4 recognizes that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> does \u003Cspan class=\"html-italic\">not\u003C/span> in fact imply \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> under the interpretation it gave, but it nevertheless dismisses this inconvenient consideration because “it does not preclude the existence of a model with the other given conditions,” as if the satisfaction of the other conditions was enough (it isn’t; a countermodel must satisfy all of the given premises while falsifying the conclusion). Moreover, just a few sentences later GPT-4 claims that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cspan class=\"html-italic\">does\u003C/span> imply \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> under the given interpretation, contradicting its own earlier claim. We will see that \u003Cspan class=\"html-italic\">internal inconsistency\u003C/span> is a recurrent theme.\u003C/div>\n\u003Cdiv class=\"html-p\">When we ask it to elaborate on why it thinks that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> implies \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, GPT-4’s response is revealing:\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Can you explain why this model satisfies the first constraint (namely, that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> implies \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>)?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">You are correct that my previous explanation was not clear enough for the first constraint. I apologize for that. In the given model, \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> does not necessarily imply \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>. Let’s reconsider the model and find one that satisfies all the constraints.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cdl class=\"html-order\">\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">1.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>: x is a multiple of 4\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">2.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>: x is an even number\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">3.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>:\u003C/mo>\n \u003Cmn>4\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003C/div>\u003C/dd>\n\u003C/dl>\u003C/div>\n\u003Cdiv class=\"html-p\" style=\"text-indent: 0;\">\u003Cspan class=\"html-italic\">Now let’s check if these interpretations satisfy the given conditions:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cdl class=\"html-order\">\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">1.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> implies \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>: If x is a multiple of 4, then x is an even number. This statement is universally true because every multiple of 4 is also a multiple of 2, making it an even number.\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">2.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> does not hold: \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmn>4\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> means 4 is an even number, which is true. So, this condition is not satisfied.\u003C/span>\u003C/div>\u003C/dd>\n\u003C/dl>\u003C/div>\n\u003Cdiv class=\"html-p\" style=\"text-indent: 0;\">\u003Cspan class=\"html-italic\">To satisfy all three conditions, we can modify the interpretations as follows:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cdl class=\"html-order\">\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">1.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>: x is a positive multiple of 4\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">2.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>: x is a positive even number\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">3.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>:\u003C/mo>\n \u003Cmo>−\u003C/mo>\n \u003Cmn>4\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003C/div>\u003C/dd>\n\u003C/dl>\u003C/div>\n\u003Cdiv class=\"html-p\" style=\"text-indent: 0;\">\u003Cspan class=\"html-italic\">Now let’s check if these interpretations satisfy the given conditions:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cdl class=\"html-order\">\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">1.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> implies \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>: If x is a positive multiple of 4, then x is a positive even number. This statement is universally true because every positive multiple of 4 is also a multiple of 2, making it a positive even number.\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">2.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> does not hold: \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmo>−\u003C/mo>\n \u003Cmn>4\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> means \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>−\u003C/mo>\n \u003Cmn>4\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> is a positive even number, which is false. So, this condition is satisfied.\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">3.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> holds: \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmo>−\u003C/mo>\n \u003Cmn>4\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> means \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>−\u003C/mo>\n \u003Cmn>4\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> is a positive multiple of 4, which is false. So, this condition is not satisfied.\u003C/span>\u003C/div>\u003C/dd>\n\u003C/dl>\u003C/div>\n\u003Cdiv class=\"html-p\" style=\"text-indent: 0;\">\u003Cspan class=\"html-italic\">It seems that finding a model that satisfies all three conditions is challenging. I must correct my previous claim: there may not be a model that satisfies all three conditions, \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> implies \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> does not hold, and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> holds, simultaneously.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">Thus, after multiple unsuccessful attempts to find a countermodel, GPT-4 acknowledges that such a countermodel might not exist. But, remarkably, when GPT-4 is asked to settle the matter one way or the other, it jumps back to the position that a countermodel does exist after all.\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Can you prove that one exists or that one does not exist?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">I apologize for the confusion earlier. Let me provide a formal proof to show that a model that satisfies all three conditions \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> implies \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> does not hold, and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> holds does indeed exist. Let’s consider a model with the following interpretations:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cdl class=\"html-order\">\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">1.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">P(x): x is an element of set A\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">2.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Q(x): x is an element of set B\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">3.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">a: a specific element, a\u003C/span>\u003C/div>\u003C/dd>\n\u003C/dl>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Now, let’s define the sets A and B as follows:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Set \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>A\u003C/mi>\n \u003Cmo>=\u003C/mo>\n \u003Cmo>{\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>}\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Set \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>B\u003C/mi>\n \u003Cmo>=\u003C/mo>\n \u003Cmo>{\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmo>|\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmrow>\n \u003Cmi>x\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003Cmi>s\u003C/mi>\n \u003Cmi>a\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003Cmi>e\u003C/mi>\n \u003Cmi>l\u003C/mi>\n \u003Cmi>e\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmi>e\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003Cmi>t\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>f\u003C/mi>\n \u003Cmi>A\u003C/mi>\n \u003Cmi>a\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>≠\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003C/mrow>\n \u003Cmo>}\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Now, let’s check if these interpretations satisfy the given conditions:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cul class=\"html-order\">\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> implies \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>: If x is an element of set A, then x is an element of set B. In this case, since the only element in set A is a, and set B is defined such that it contains all elements of A except a, this implication is vacuously true. This is because there are no other elements in set A for which the implication would be false.\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> does not hold: \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> means a is an element of set B. By definition, set B does not include a, so this condition is satisfied.\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> holds: \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> means a is an element of set A. Since set A is defined to contain only a, this condition is also satisfied.\u003C/span>\u003C/div>\u003C/li>\n\u003C/ul>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Thus, the model with the interpretations\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>: x is an element of set A,\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>: x is an element of set B,\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">and a: a specific element,\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">satisfies the conditions \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> implies \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> does not hold, and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> holds.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">The response verges on nonsensical. GPT-4 has defined \u003Cspan class=\"html-italic\">B\u003C/span> (in a roundabout way) as the empty set \u003Cspan class=\"html-italic\">∅\u003C/span>, so that \u003Cspan class=\"html-italic\">Q\u003C/span> is always false. This means that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> cannot possibly imply \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> as long as there is a single value of \u003Cspan class=\"html-italic\">x\u003C/span> for which \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> holds—and GPT-4 itself has provided such a value: \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>=\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>. Yet somehow GPT-4 claims that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> implies \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>.\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Can you again explain why \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> implies \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> in this model?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">You are correct. My previous explanation for \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> implies \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> was not accurate. I apologize for the confusion. I will now provide a correct model that satisfies all three conditions. Let’s consider a model with the following interpretations:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cdl class=\"html-order\">\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">1.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>: x is divisible by 3\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">2.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>: x is divisible by 2\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">3.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">a: 3\u003C/span>\u003C/div>\u003C/dd>\n\u003C/dl>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Now, let’s check if these interpretations satisfy the given conditions:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cdl class=\"html-order\">\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">1.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> implies \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>: If x is divisible by 3, then x is divisible by 2. This statement is not universally true. However, we can create a domain D that contains only one element, \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>=\u003C/mo>\n \u003Cmn>3\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, which is divisible by 3 but not by 2. By restricting the domain, the implication \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> implies \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> becomes vacuously true within the domain D because there are no elements in D for which the implication would be false.\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">2.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> does not hold: \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmn>3\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> means 3 is divisible by 2. Since 3 is not divisible by 2, this condition is satisfied.\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">3.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> holds: \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmn>3\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> means 3 is divisible by 3. Since 3 is divisible by 3, this condition is also satisfied.\u003C/span>\u003C/div>\u003C/dd>\n\u003C/dl>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Thus, the model with the interpretations\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>: x is divisible by 3,\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>: x is divisible by 2, and\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">a: 3,\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">and the domain \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>D\u003C/mi>\n \u003Cmo>=\u003C/mo>\n \u003Cmn>3\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> satisfies the conditions \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> implies \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> does not hold, and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> holds.\u003C/span>\u003C/div>\u003C/section>\u003Csection id=\"sec3dot5-preprints-81518\" type>\u003Ch4 class=\"html-italic\" data-nested=\"2\"> 3.5. Simple Quantifier Semantics\u003C/h4>\n\u003Cdiv class=\"html-p\">Here we give GPT-4 two easy problems to test its understanding of quantifiers. Here is the first problem:\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Consider the following three sentences:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cdl class=\"html-order\">\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">1.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Ctt>[forall x.P(x) ==> Q(x)]\u003C/tt>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">2.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Ctt>[exists x.P(x)]\u003C/tt>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">3.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Ctt>[exists x.\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>Q(x)]\u003C/tt>\u003C/div>\u003C/dd>\n\u003C/dl>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Either disprove or prove the following claim: These three sentences are jointly satisfiable.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">The three sentences are not jointly satisfiable. Here’s why:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cdl class=\"html-order\">\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">1.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\n\u003Ctt>[forall x.P(x) ==> Q(x)]\u003C/tt>\u003Cspan class=\"html-italic\">says that for all x, if P(x) is true then Q(x) is true.\u003C/span>\n\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">2.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\n\u003Ctt>[exists x.P(x)]\u003C/tt>\u003Cspan class=\"html-italic\">says that there exists at least one x such that P(x) is true.\u003C/span>\n\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">3.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\n\u003Ctt>[exists x.\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>Q(x)]\u003C/tt>\u003Cspan class=\"html-italic\">says that there exists at least one x such that Q(x) is not true.\u003C/span>\n\u003C/div>\u003C/dd>\n\u003C/dl>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">If we consider sentence 2, there exists at least one x such that P(x) is true. According to sentence 1, for this x (or these x’s), Q(x) must also be true.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">However, sentence 3 states that there exists at least one x for which Q(x) is not true. Given sentences 1 and 2, this cannot be the case - there cannot exist an x for which P(x) is true and yet Q(x) is not true. Therefore, the three sentences are not jointly satisfiable.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">The sentences are clearly jointly satisfiable, a trivial model being the domain \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>{\u003C/mo>\n \u003Cmsub>\n \u003Cmi>a\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>,\u003C/mo>\n \u003Cmsub>\n \u003Cmi>a\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/msub>\n \u003Cmo>}\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> with \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>a\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>a\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>¬\u003C/mo>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>a\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>¬\u003C/mo>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>a\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>.\u003C/div>\n\u003Cdiv class=\"html-p\">The second problem concerns this biconditional: \u003Cdiv class=\"html-disp-formula-info\" id=\"FD1-preprints-81518\">\n \u003Cdiv class=\"f\">\n \u003Cmath display=\"block\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>[\u003C/mo>\n \u003Cmo>∀\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi>x\u003C/mi>\n \u003Cmrow>\n \u003Cmspace width=\"0.277778em\" />\n \u003Cmo>.\u003C/mo>\n \u003Cmspace width=\"0.277778em\" />\n \u003C/mrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>⇔\u003C/mo>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>]\u003C/mo>\n \u003Cmo>⇔\u003C/mo>\n \u003Cmo>[\u003C/mo>\n \u003Cmo>(\u003C/mo>\n \u003Cmo>∀\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi>x\u003C/mi>\n \u003Cmrow>\n \u003Cmspace width=\"0.277778em\" />\n \u003Cmo>.\u003C/mo>\n \u003Cmspace width=\"0.277778em\" />\n \u003C/mrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>⇔\u003C/mo>\n \u003Cmo>(\u003C/mo>\n \u003Cmo>∀\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi>x\u003C/mi>\n \u003Cmrow>\n \u003Cmspace width=\"0.277778em\" />\n \u003Cmo>.\u003C/mo>\n \u003Cmspace width=\"0.277778em\" />\n \u003C/mrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>]\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \u003Clabel>(1)\u003C/label>\n \u003C/div>\n \u003C/div>\n The left-to-right implication holds, but the right-to-left direction fails. Counterexamples are easy to find, for example, take the domain to be integers, \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> to mean \u003Cspan class=\"html-italic\">x\u003C/span> is even and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>Q\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> to mean \u003Cspan class=\"html-italic\">x\u003C/span> is odd. Then the equivalence on the right-hand side of (\u003Ca href=\"#FD1-preprints-81518\" class=\"html-disp-formula\">1\u003C/a>) holds, but clearly it’s not true that every integer is even iff it is odd.\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Prove or disprove the following:\u003C/span> \u003Ctt>(forall x.P(x) <==> Q(x))\u003C/tt>\u003Cspan class=\"html-italic\">holds if and only if the following biconditional holds:\u003C/span> \u003Ctt>(forall x.P(x)) <==> (forall x.Q(x))\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">We will prove this statement by showing that both directions of the main biconditional hold.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cul class=\"html-bullet\">\n\u003Cli>\n\u003Cdiv class=\"html-p\">\n\u003Ctt>(forall x.P(x)<==>Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">implies\u003C/span> \n \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">Assume that for all x,\u003C/span> \n \u003Ctt>P(x)<==>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Direction 1: If forall x,\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds, we need to show that forall x, Q(x) holds. Since\u003C/span> \n \u003Ctt>P(x)<==>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">for all x, if\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds for all x, then\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">must also hold for all x, as they are equivalent.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Direction 2: If forall x,\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds, we need to show that forall x,\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds. Since\u003C/span> \n \u003Ctt>P(x)<==>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">for all x, if\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds for all x, then\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">must also hold for all x, as they are equivalent.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Thus, we have shown that\u003C/span> \n \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">holds when\u003C/span> \n \u003Ctt>(forall x.P(x)<==>Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">holds.\u003C/span>\n\u003C/div>\n\u003C/li>\n\u003Cli>\n\u003Cdiv class=\"html-p\">\n\u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">implies\u003C/span> \n \u003Ctt>(forall.P(x)<==>Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">Assume that\u003C/span> \n \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">holds.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">We need to show that for all x,\u003C/span> \u003Ctt>P(x)<==>Q(x)\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Let\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">be an arbitrary element from the domain of discourse. We will consider the two directions of the biconditional\u003C/span> \n \u003Ctt>P(x)<==>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Direction 1: If\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds, we need to show that\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds. If\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds for our arbitrary\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">, then it must be the case that forall\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">,\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds. Since\u003C/span> \n \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">holds, it follows that forall\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">,\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds. Therefore,\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">must hold for our arbitrary\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">as well.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Direction 2: If\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds, we need to show that\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds. If\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds for our arbitrary\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">, then it must be the case that forall\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">,\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds. Since\u003C/span> \n \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">holds, it follows that forall\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">,\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds. Therefore,\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">must hold for our arbitrary\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">as well.\u003C/span>\n\u003C/div>\n\u003C/li>\n\u003C/ul>\u003C/div>\n\u003Cdiv class=\"html-p\" style=\"text-indent: 0;\">\n\u003Cspan class=\"html-italic\">Thus, we have shown that for all\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">,\u003C/span> \n \u003Ctt>P(x) <==> Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds when\u003C/span> \n \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">holds.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">In conclusion, we have shown that\u003C/span> \n \u003Ctt>(forall x.P(x)<==>Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">holds if and only if\u003C/span> \n \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">holds.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">Notice how pernicious the fallacy is here: \u003Cspan class=\"html-italic\">“If\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds for our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">, then it must be the case that forall\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds.”\u003C/span> It sounds plausible, but of course it is quite wrong.\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Can you elaborate on your claim “If P(x) holds for our arbitrary x, then it must be the case that forall x, P(x) holds”?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: You are correct to question that claim. It was a mistake in my response. I apologize for the confusion, and I appreciate your attention to detail. Here is the corrected proof.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Ctt>(forall x.P(x)<==>Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">implies\u003C/span> \n \u003Ctt>(forall x.P(x))<==>(forall x . Q(x))\u003C/tt>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Assume that for all\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">,\u003C/span> \n \u003Ctt>P(x)<==>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Direction 1: If forall\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">,\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds, we need to show that forall\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">,\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds. Since\u003C/span> \n \u003Ctt>P(x)<==>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">for all\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">, if\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds for all\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">, then\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">must also hold for all\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">, as they are equivalent.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Direction 2: If forall\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">,\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds, we need to show that forall\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">,\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds. Since\u003C/span> \n \u003Ctt>P(x)<==>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">for all\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">, if\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds for all\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">, then\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">must also hold for all\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">, as they are equivalent.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Thus, we have shown that\u003C/span> \n \u003Ctt>(forall x.P(x))<==>(forall x . Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">holds when\u003C/span> \n \u003Ctt>(forall x.P(x)<==>Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">holds.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">implies\u003C/span> \n \u003Ctt>(forall x.P(x)<==>Q(x))\u003C/tt>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Assume that\u003C/span> \n \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">holds.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">We need to show that for all\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>P(x)<==>Q(x)\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Let\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">be an arbitrary element from the domain of discourse. We will consider the two directions of the biconditional\u003C/span> \n \u003Ctt>P(x)<==>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Direction 1: If\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds, we need to show that\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds. If\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds for our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">, we cannot directly conclude that forall\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds. However, we can proceed as follows: Since\u003C/span> \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt>\u003Cspan class=\"html-italic\">holds, either both sides of the biconditional are true or both sides are false. If forall\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds, then by the biconditional, forall\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">must hold as well, and thus\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds for our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">. If it is not the case that forall\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds, then there must be some\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">where\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">does not hold. But, for our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds, which means our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">cannot be one of those instances. In this case, it is not guaranteed that\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds for our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Direction 2: If\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds, we need to show that\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds. Similarly, if\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds for our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">, we cannot directly conclude that forall\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds. We can use the same approach as in Direction 1. Since\u003C/span> \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt>\u003Cspan class=\"html-italic\">holds, either both sides of the biconditional are true or both sides are false. If forall\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds, then by the biconditional, forall\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">must hold as well, and thus\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds for our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">. If it is not the case that forall\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds, then there must be some\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">where\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">does not hold. But, for our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds, which means our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">cannot be one of those instances. In this case, it is not guaranteed that\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds for our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">As we cannot guarantee that\u003C/span> \n \u003Ctt>P(x)<==>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">for our arbitrary\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Please continue.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: My apologies for the premature ending in my previous response. Let me provide an alternative approach to the proof for the second part.\u003C/span> \n \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">implies\u003C/span> \n \u003Ctt>(forall x.P(x)<==>Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">Assume that\u003C/span> \n \u003Ctt>(forall x.P(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">holds.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">We need to show that for all\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>P(x)<==>Q(x)\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Let\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">be an arbitrary element from the domain of discourse. We will consider the two directions of the biconditional\u003C/span> \n \u003Ctt>P(x)<==>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Direction 1: If\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds, we need to show that\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds. We will use proof by contradiction. Assume that\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">does not hold for our arbitrary\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">. Now, since\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">does not hold, it cannot be the case that forall\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">,\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds. Since\u003C/span> \n \u003Ctt>(forall x<==>P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">holds, this implies that it cannot be the case that forall\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">,\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds. However, we assumed that\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds for our arbitrary\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">, which leads to a contradiction. Thus, our assumption that\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">does not hold must be incorrect, and\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">must hold.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Direction 2: If\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds, we need to show that\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds. We will use proof by contradiction again. Assume that\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">does not hold for our arbitrary\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">. Now, since\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">does not hold, it cannot be the case that forall\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">,\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds. Since\u003C/span> \n \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">holds, this implies that it cannot be the case that forall\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">,\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds. However, we assumed that\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds for our arbitrary\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">, which leads to a contradiction. Thus, our assumption that\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">does not hold must be incorrect, and\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">must hold.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">In both directions, we have shown that\u003C/span> \n \u003Ctt>P(x)<==>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds for our arbitrary\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">. Since\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">was arbitrary, this holds for all\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">. Therefore,\u003C/span>\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Ctt>(forall x.P(x)<==>Q(x))\u003C/tt>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n\u003Cspan class=\"html-italic\">holds when\u003C/span> \n \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">holds.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">In conclusion, we have shown that\u003C/span> \n \u003Ctt>(forall x.P(x)<==>Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">holds if and only if\u003C/span> \n \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">holds.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">While the reasoning appears to proceed along the right general lines, it is, again, insidiously wrong. The reductio ad absurdum given for the two directions hinges on the fallacy that if a property \u003Cspan class=\"html-italic\">T\u003C/span> is assumed to hold for a \u003Cspan class=\"html-italic\">particular but arbitrarily chosenx\u003C/span>, then we must have \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∀\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi>x\u003C/mi>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmo>.\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi>T\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, leading to the erroneous conclusion that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>T\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>¬\u003C/mo>\n \u003Cmo>∀\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi>x\u003C/mi>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmo>.\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi>T\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> are jointly inconsistent. It is as if GPT-4 knows what needs to be done but does not know how to do it properly, and instead of saying so, it proceeds to generate a bogus but plausible-looking “proof.”\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Please clarify your claim that “However, we assumed that P(x) holds for our arbitrary x, which leads to a contradiction”.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: You are correct to question that claim, as the reasoning was unclear. I apologize for the confusion. Let me provide a clearer explanation for the second part of the proof.\u003C/span>\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">implies\u003C/span> \n \u003Ctt>(forall x.P(x)<==>Q(x))\u003C/tt>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n\u003Cspan class=\"html-italic\">Assume that\u003C/span> \n \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">holds.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">We need to show that for all\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>P(x)<==>Q(x)\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Let\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">be an arbitrary element from the domain of discourse. We will consider the two directions of the biconditional\u003C/span> \n \u003Ctt>P(x)<==>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Direction 1: If\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds, we need to show that\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds. If\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds for our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">, we can make the following argument: If it is not the case that forall\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds, then there must exist some\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">for which\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">does not hold. However, if forall\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">does hold, then by the given biconditional\u003C/span> \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt>\u003Cspan class=\"html-italic\">forall\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">must also hold. Since\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds for our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">, either it holds for all\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">or there exists some other\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">where\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">does not hold. In either case, we can’t deduce anything about\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">for our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Direction 2: If\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds, we need to show that\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds. If\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds for our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">, we can make a similar argument as in Direction 1: If it is not the case that forall\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds, then there must exist some\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">for which\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">does not hold. However, if forall\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">does hold, then by the given biconditional\u003C/span> \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt>\u003Cspan class=\"html-italic\">, forall\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">must also hold. Since\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds for our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">, either it holds for all\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">or there exists some other\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">where\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">does not hold. In either case, we can’t deduce anything about\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">for our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Since we can’t make any definitive conclusions about the relationship between\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">and\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">for our arbitrary\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">based solely on the given biconditional\u003C/span> \n \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">, it is incorrect to claim that\u003C/span>\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n\u003Cspan class=\"html-italic\">implies\u003C/span> \u003Ctt>(forall x.P(x)<==>Q(x))\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">So where does that leave us? After the problematic inference step was pointed out to GPT-4 , it proceeded to “clarify” its reasoning, but ultimately it ended up admitting that its reasoning was wrong. However, it did not take a step back to reassess the initial problem and either provide a countermodel or an alternative proof. When it is explicitly asked to take a final position one way or another, it again chooses the wrong answer and provides yet another bogus “proof” for it:\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: So what does that prove?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: My previous response was not clear, and I apologize for the confusion.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>[\u003C/mo>\n \u003Cmo>⋯\u003C/mo>\n \u003Cmrow>\n \u003Cmi>G\u003C/mi>\n \u003Cmi>P\u003C/mi>\n \u003Cmi>T\u003C/mi>\n \u003Cmo>−\u003C/mo>\n \u003Cmn>4\u003C/mn>\n \u003Cmi>r\u003C/mi>\n \u003Cmi>e\u003C/mi>\n \u003Cmi>h\u003C/mi>\n \u003Cmi>a\u003C/mi>\n \u003Cmi>s\u003C/mi>\n \u003Cmi>h\u003C/mi>\n \u003Cmi>e\u003C/mi>\n \u003Cmi>s\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003Cmi>t\u003C/mi>\n \u003Cmi>s\u003C/mi>\n \u003Cmi>p\u003C/mi>\n \u003Cmi>r\u003C/mi>\n \u003Cmi>e\u003C/mi>\n \u003Cmi>v\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>u\u003C/mi>\n \u003Cmi>s\u003C/mi>\n \u003Cmi>a\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003Cmi>s\u003C/mi>\n \u003Cmi>w\u003C/mi>\n \u003Cmi>e\u003C/mi>\n \u003Cmi>r\u003C/mi>\n \u003Cmi>h\u003C/mi>\n \u003Cmi>e\u003C/mi>\n \u003Cmi>r\u003C/mi>\n \u003Cmi>e\u003C/mi>\n \u003C/mrow>\n \u003Cmo>⋯\u003C/mo>\n \u003Cmo>]\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">In conclusion, we have only shown that\u003C/span> \n \u003Ctt>(forall x.P(x)<==>Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">implies\u003C/span>\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n\u003Cspan class=\"html-italic\">but we have not proven the reverse implication. The statement\u003C/span> \n \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">implies\u003C/span> \n \u003Ctt>(forall x.P(x)<==>Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">remains unproven.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: So can you prove it, or else disprove it?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: I apologize for the confusion in my previous responses. I will now provide a correct proof for the second part of the question. The goal is to show that\u003C/span> \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt>\u003Cspan class=\"html-italic\">implies\u003C/span> \u003Ctt>(forall x.P(x)<==>Q(x))\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">implies\u003C/span> \n \u003Ctt>(forall x.P(x)<==>Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">. Assume that\u003C/span> \n \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">holds.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">We need to show that for all\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>P(x)<==>Q(x)\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Let\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">be an arbitrary element from the domain of discourse. We will consider the two directions of the biconditional\u003C/span> \n \u003Ctt>P(x)<==>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Direction 1: If\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds, we need to show that\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Since\u003C/span> \n \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">, there are two possible scenarios:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Both sides of the biconditional are true: forall\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds and forall\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds. Both sides of the biconditional are false: it is not the case that forall\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds, and it is not the case that forall\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds. If scenario 1 is true, then\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds for all\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">, including our arbitrary x. If scenario 2 is true, then there must exist some element in the domain where\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">does not hold. However, for our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">, we assumed that\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds. Thus, scenario 2 cannot be true for our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">. Consequently, scenario 1 must be true, and\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds for our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Direction 2: If\u003C/span> \n \u003Ctt>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds, we need to show that\u003C/span> \n \u003Ctt>P(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">The same analysis used in Direction 1 can be applied to Direction 2. If scenario 1 is true, then\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds for all\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">, including our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">. If scenario 2 is true, then there must exist some element in the domain where\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">does not hold. However, for our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">, we assumed that\u003C/span> \u003Ctt>Q(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds. Thus, scenario 2 cannot be true for our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\u003Cspan class=\"html-italic\">. Consequently, scenario 1 must be true, and\u003C/span> \u003Ctt>P(x)\u003C/tt>\u003Cspan class=\"html-italic\">holds for our arbitrary\u003C/span> \u003Ctt>x\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">In both directions, we have shown that\u003C/span> \n \u003Ctt>P(x)<==>Q(x)\u003C/tt> \n \u003Cspan class=\"html-italic\">holds for our arbitrary\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">. Since x was arbitrary, this holds for all\u003C/span> \n \u003Ctt>x\u003C/tt> \n \u003Cspan class=\"html-italic\">. Therefore,\u003C/span> \n \u003Ctt>(forall x.P(x)<==>Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">holds when\u003C/span>\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n\u003Cspan class=\"html-italic\">holds.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">In conclusion, we have shown that\u003C/span> \n \u003Ctt>(forall x.P(x)<==>Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">holds if and only if\u003C/span> \n \u003Ctt>(forall x.P(x))<==>(forall x.Q(x))\u003C/tt> \n \u003Cspan class=\"html-italic\">holds.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">Regrettably, the error is the same one that GPT-4 already made before (and which it supposedly fixed before). The model seems fundamentally confused about the role of the so-called \u003Cspan class=\"html-italic\">eigenvariable x\u003C/span> (the “arbitrarily chosen” \u003Cspan class=\"html-italic\">x\u003C/span>), believing that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>T\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> is inconsistent with \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>¬\u003C/mo>\n \u003Cmo>∀\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi>x\u003C/mi>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmo>.\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi>T\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>.\u003C/div>\u003C/section>\u003Csection id=\"sec3dot6-preprints-81518\" type>\u003Ch4 class=\"html-italic\" data-nested=\"2\"> 3.6. Simple Graph Coloring\u003C/h4>\n\u003Cdiv class=\"html-p\">We first consider a graph-coloring problem that does not have a solution. It is trivial to see that two colors do not suffice for the graph described in this problem (e.g., vertices 0, 2, and 4 form a clique and hence require at least 3 colors).\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Consider an undirected graph with 6 vertices (0 through 5) and the following set of edges:\u003C/span>\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Cmath display=\"block\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>{\u003C/mo>\n \u003Cmo>(\u003C/mo>\n \u003Cmn>0\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>(\u003C/mo>\n \u003Cmn>0\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>3\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>(\u003C/mo>\n \u003Cmn>0\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>4\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>(\u003C/mo>\n \u003Cmn>0\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>(\u003C/mo>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>(\u003C/mo>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>3\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>(\u003C/mo>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>5\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>(\u003C/mo>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>4\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>(\u003C/mo>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>5\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>(\u003C/mo>\n \u003Cmn>3\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>4\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>(\u003C/mo>\n \u003Cmn>3\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>5\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>(\u003C/mo>\n \u003Cmn>4\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>5\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>}\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n\u003Cspan class=\"html-italic\">Color every vertex either red or green, so that no two adjacent vertices receive the same color.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">This undirected graph is complete, meaning that every vertex is connected to every other vertex by an edge. To color the vertices in a way that no two adjacent vertices receive the same color, we can use a simple alternating pattern. Here’s one possible solution:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cul class=\"html-bullet\">\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 0: Red\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 1: Green\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 2: Red\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 3: Green\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 4: Red\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 5: Green\u003C/span>\u003C/div>\u003C/li>\n\u003C/ul>\u003C/div>\n\u003Cdiv class=\"html-p\" style=\"text-indent: 0;\">\u003Cspan class=\"html-italic\">This coloring ensures that no two adjacent vertices have the same color. For example, vertex 0 is connected to vertices 1, 2, 3, 4, and 5. Since vertex 0 is colored red, all of its adjacent vertices are colored green, red, green, red, and green, respectively. The same goes for all other vertices.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">The sheer number of egregious errors in this short output is shocking. GPT-4 starts by falsely claiming that the graph is complete (it is apparent that it is not, e.g., there is no edge between vertices 2 and 3). Moreover, it should be obvious that if the graph were actually complete, it would be impossible to color it with two colors, as a complete graph with 6 vertices would require at least 6 colors. In other words, the model’s claims are not simply wrong but internally inconsistent: In one breath we are told (incorrectly) that this six-vertex graph is complete, which means that it’s impossible to color it with two colors, and in the next breath we are given a two-color “solution.”\u003C/div>\n\u003Cdiv class=\"html-p\">Remarkably, this dismal display is not because GPT-4 does not have sufficient knowledge or data about graphs (which are, after all, pervasive mathematical entities with copious information about them on the Internet, as well as in countless articles and books), and even about complete graphs in particular. When I asked GPT-4 to tell me what it knows about complete graphs it rattled off the correct definition of complete graphs along with a long list of results about \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>K\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n\u003C/semantics>\u003C/math> (the complete graphs with \u003Cspan class=\"html-italic\">n\u003C/span> vertices), such as the number of edges (\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>n\u003C/mi>\n \u003Cmo>·\u003C/mo>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>n\u003C/mi>\n \u003Cmo>−\u003C/mo>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>/\u003C/mo>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>), the degree of each vertex (\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>n\u003C/mi>\n \u003Cmo>−\u003C/mo>\n \u003Cmn>1\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>), the connectivity (maximal), the number of subgraphs (\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsup>\n \u003Cmn>2\u003C/mn>\n \u003Cmrow>\n \u003Cmi>n\u003C/mi>\n \u003Cmo>·\u003C/mo>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>n\u003C/mi>\n \u003Cmo>−\u003C/mo>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>/\u003C/mo>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n \u003C/msup>\n\u003C/semantics>\u003C/math>), the fact that they’re Hamiltonian and Eulerian, their planarity, and a whole lot of other information, including their chromatic number: \u003Cspan class=\"html-italic\">Chromatic Number: The chromatic number of a complete graph \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>K\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n\u003C/semantics>\u003C/math> (i.e., the smallest number of colors needed to color the vertices of \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>K\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n\u003C/semantics>\u003C/math> so that no two adjacent vertices share the same color) is n\u003C/span> (this italicized sentence is a verbatim quotation from GPT-4). It is clear that GPT-4 has memorized all this information but is unable to use it in a new setting.\u003C/div>\n\u003Cdiv class=\"html-p\">When I pointed out that the solution is wrong, GPT-4 went on to make the exact same type of mistake:\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">But vertices 0 and 2 are adjacent and you’ve colored both of them red.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">You are right, my previous response had an error. I apologize for that. Let’s try another coloring:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cul class=\"html-bullet\">\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 0: Red\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 1: Green\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 2: Green\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 3: Red\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 4: Green\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 5: Red\u003C/span>\u003C/div>\u003C/li>\n\u003C/ul>\u003C/div>\n\u003Cdiv class=\"html-p\" style=\"text-indent: 0;\">\u003Cspan class=\"html-italic\">In this coloring, no two adjacent vertices have the same color.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">But 0 and 3 are adjacent and again you’ve colored both of them red.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">You are correct again, and I apologize for the mistake. Given the complete graph, we can use a bipartite coloring strategy. Let’s try this coloring:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cul class=\"html-bullet\">\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 0: Red\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 1: Green\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 2: Red\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 3: Green\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 4: Red\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 5: Green\u003C/span>\u003C/div>\u003C/li>\n\u003C/ul>\u003C/div>\n\u003Cdiv class=\"html-p\" style=\"text-indent: 0;\">\u003Cspan class=\"html-italic\">In this coloring, vertices 0, 2, and 4 are red, and vertices 1, 3, and 5 are green. As you can see, no two adjacent vertices share the same color.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">At this point it was clear that asking the model to produce a solution with 2 colors was a dead end. A solution with three colors surely would be easier (such solutions do exist and are easy to find):\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">Again, 0 and 2 are adjacent vertices. You’ve colored both of them red. Let’s try with 3 colors. Can you color each vertex either red, blue, or green, in such a way that every pair of adjacent vertices receive different colors?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">I apologize for the repeated mistakes in my previous responses. Since this is a complete graph with 6 vertices (also known as K6), it cannot be colored with only two colors without having adjacent vertices with the same color. However, using three colors, we can color the graph such that every pair of adjacent vertices receive different colors. Here’s a possible coloring with red, blue, and green:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cul class=\"html-bullet\">\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 0: Red\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 1: Green\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 2: Blue\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 3: Red\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 4: Green\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 5: Blue\u003C/span>\u003C/div>\u003C/li>\n\u003C/ul>\u003C/div>\n\u003Cdiv class=\"html-p\" style=\"text-indent: 0;\">\u003Cspan class=\"html-italic\">In this coloring, no two adjacent vertices share the same color, and the graph is properly colored using three colors.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">No, this is not correct either. 0 and 3 are adjacent and you’ve made both of them red.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">I apologize once again for my mistake. I appreciate your patience. Here’s the correct coloring using three colors:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cul class=\"html-bullet\">\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 0: Red\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 1: Green\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 2: Blue\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 3: Green\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 4: Red\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Vertex 5: Blue\u003C/span>\u003C/div>\u003C/li>\n\u003C/ul>\u003C/div>\n\u003Cdiv class=\"html-p\" style=\"text-indent: 0;\">\u003Cspan class=\"html-italic\">With this coloring, no two adjacent vertices share the same color, and the graph is properly colored using three colors.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">Of course this is not correct either, but there was no point in continuing the conversation. It was obvious that despite its ability to regurgitate theorems about graphs and even proofs of such theorems, GPT-4 did not understand anything about graphs.\u003C/div>\u003C/section>\u003Csection id=\"sec3dot7-preprints-81518\" type>\u003Ch4 class=\"html-italic\" data-nested=\"2\"> 3.7. Subset Sum\u003C/h4>\n\u003Cdiv class=\"html-p\">This problem considers a small set of integers \u003Cspan class=\"html-italic\">S\u003C/span> and asks for the number of subsets of \u003Cspan class=\"html-italic\">S\u003C/span> whose elements sum up to 37. The answer is 0, because \u003Cspan class=\"html-italic\">S\u003C/span> contains only even numbers and no sum of even numbers can ever be odd. Without pausing to consider the contents of \u003Cspan class=\"html-italic\">S\u003C/span>, GPT-4 reflexively generates the sort of output that it thinks would be an appropriate response to this question and then goes on to hallucinate an answer:\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Let \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>S\u003C/mi>\n \u003Cmo>=\u003C/mo>\n \u003Cmo>{\u003C/mo>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>8\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>6\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>32\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>22\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>44\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>28\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>12\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>18\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>10\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>14\u003C/mn>\n \u003Cmo>}\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>. How many subsets does S have that sum up to 37?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">We can find the number of subsets of S that sum up to 37 using a recursive depth-first search algorithm to explore all possible combinations of elements. Here’s a Python code to do that:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>def subset_sum_count(S, target, idx=0, current_sum=0):\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>    if idx == len(S):\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>        return 1 if current_sum == target else 0\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>    count = 0\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>    # Case 1: Include the current element in the sum\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>    count += subset_sum_count(S, target, idx+1, current_sum+S[idx])\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>    # Case 2: Exclude the current element from the sum\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>    count += subset_sum_count(S, target, idx+1, current_sum)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>    return count\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>S = [2, 8, 6, 32, 22, 44, 28, 12, 18, 10, 14]\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>target = 37\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>result = subset_sum_count(S, target)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>print(result)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">The output is 4. There are 4 subsets of S whose elements sum up to 37.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">When we ask GPT-4 to back up its answer with evidence, it goes on a hallucination rampage:\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Can you list those 4 subsets?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">Certainly! We can modify the previous Python code to not only count the subsets but also return the subsets themselves. Here’s the updated code:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>def subset_sum_subsets(S, target, idx=0, current_sum=0,\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>                       current_subset=None, result=None):\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>    if current_subset is None:\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>        current_subset = []\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>    if result is None:\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>        result = []\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>    if idx == len(S):\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>        if current_sum == target:\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>            result.append(current_subset)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>        return\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>    # Case 1: Include the current element in the sum\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>    subset_sum_subsets(S, target, idx+1,\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>                       current_sum+S[idx],\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>                       current_subset + [S[idx]], result)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>    # Case 2: Exclude the current element from the sum\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>    subset_sum_subsets(S, target, idx+1,\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>                       current_sum,\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>                       current_subset, result)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>    return result\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>S = [2, 8, 6, 32, 22, 44, 28, 12, 18, 10, 14]\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>target = 37\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>result = subset_sum_subsets(S, target)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>print(result)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">The output is:\u003C/span> \n \u003Ctt>[[2, 8, 6, 12, 9], [2, 8, 6, 21], [2, 8, 27], [2, 35]]\u003C/tt> \n \u003Cspan class=\"html-italic\">. So, the 4 subsets of S that sum up to 37 are:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{2, 8, 6, 12, 9}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{2, 8, 6, 21}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{2, 8, 27}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{2, 35}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">But 9, 21, 27, and 35 are not elements of S.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">Apologies for the error in my previous response. I mistakenly used the wrong values in the output. Here’s the corrected Python code to list the subsets of S that sum up to 37:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>def subset_sum_subsets...\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>  ...\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>S = [2, 8, 6, 32, 22, 44, 28, 12, 18, 10, 14]\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>target = 37\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>result = subset_sum_subsets(S, target)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>print(result)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">The corrected output is:\u003C/span> \n \u003Ctt>[[2, 6, 10, 18, 1], [2, 6, 29], [2, 8, 12, 15], [2, 8, 27]]\u003C/tt> \n \u003Cspan class=\"html-italic\">. So, the 4 subsets of S that sum up to 37 are:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{2, 6, 10, 18, 1}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{2, 6, 29}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{2, 8, 12, 15}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{2, 8, 27}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">However, I made another mistake in the previous response. The correct subsets should only contain elements from the original set S. My previous response included elements that are not in S. I apologize for the confusion. Here’s the corrected list of subsets of S that sum up to 37:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{2, 6, 10, 18, 1}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{2, 6, 29}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{2, 8, 12, 15}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{2, 8, 27}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Again, I apologize for the confusion. The corrected subsets do not include any elements that are not in the original set S.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">But they do. They include 1, 29, 15, and 27 - none of which are in S.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">Apologies for the confusion in my previous responses. I made mistakes in the output. Here’s the corrected code and output for the subsets of S that sum up to 37:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>def subset_sum_subsets ...\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>  ...\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>S = [2, 8, 6, 32, 22, 44, 28, 12, 18, 10, 14]\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>target = 37\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>result = subset_sum_subsets(S, target)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>print(result)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">The corrected output is:\u003C/span> \n \u003Ctt>[[2, 6, 10, 18, 1], [2, 6, 29], [2, 8, 12, 15], [2, 8, 27]]\u003C/tt> \n \u003Cspan class=\"html-italic\">. So, the 4 subsets of S that sum up to 37 are:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{2, 8, 6, 18, 1}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{2, 6, 29}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{2, 8, 12, 15}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{2, 8, 27}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Again, I apologize for the confusion. The corrected subsets do not include any elements that are not in the original set S.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">Is 1 in S?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">I apologize for the repeated errors in my previous responses. You are correct that 1 is not in the set S. Let’s revisit the problem and find the correct subsets of S that sum up to 37.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Here’s the corrected Python code and output for the subsets of S that sum up to 37:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>def subset_sum_subsets ..\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>  ...\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>S = [2, 8, 6, 32, 22, 44, 28, 12, 18, 10, 14]\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>target = 37\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>result = subset_sum_subsets(S, target)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>print(result)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">The corrected output is:\u003C/span> \n \u003Ctt>[[8, 6, 12, 10, 1], [8, 6, 22, 1], [8, 28, 1], [32, 6, 1]]\u003C/tt> \n \u003Cspan class=\"html-italic\">. So, the 4 subsets of S that sum up to 37 are:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{8, 6, 12, 10, 1}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{8, 6, 22, 1}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{8, 28, 1}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{32, 6, 1}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">I made another mistake in the previous response. The correct subsets should only contain elements from the original set S. My previous response included elements that are not in S. I apologize for the confusion. Here’s the corrected list of subsets of S that sum up to 37:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{8, 6, 12, 10, 1}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{8, 6, 22, 1}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{8, 28, 1}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{32, 6, 1}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Again, I apologize for the confusion. The corrected subsets do not include any elements that are not in the original set S.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">You keep including 1 in the results. How many times do we need to agree that 1 is NOT in S?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">I sincerely apologize for the repeated mistakes in my previous responses. I appreciate your patience. Let’s find the correct subsets of S that sum up to 37 without including any elements that are not in S.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Here is the corrected Python code and output for the subsets of S that sum up to 37:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>def subset_sum_subsets ...\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>  ...\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>S = [2, 8, 6, 32, 22, 44, 28, 12, 18, 10, 14]\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>target = 37\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>result = subset_sum_subsets(S, target)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>print(result)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">The corrected output is:\u003C/span> \n \u003Ctt>[[8, 6, 10, 12, 1], [8, 6, 22, 1], [32, 6, 1]]\u003C/tt> \n \u003Cspan class=\"html-italic\">. So, the 3 subsets of S that sum up to 37 are:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{8, 6, 10, 12}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{8, 6, 22}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>{32, 6}\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Once again, I apologize for the confusion in my previous responses, and I hope this answer finally provides the correct subsets without including any elements that are not in the original set S.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">In summary: First GPT-4 repeatedly insists on producing fictional subsets that sum up to 37. When—after multiple corrections and admonitions—the model finally comes to respect the constraint that the subsets of \u003Cspan class=\"html-italic\">S\u003C/span> must actually be subsets of \u003Cspan class=\"html-italic\">S\u003C/span>, it produces subsets that do not sum up to the target goal. Along the way it outputs the wrong output \u003Cspan class=\"html-italic\">r\u003C/span>, says that it made a mistake and that \u003Cspan class=\"html-italic\">r\u003C/span> is wrong, and then serves up \u003Cspan class=\"html-italic\">r\u003C/span> again completely unchanged, but claims that now it’s correct. These are not mistakes that any human would make.\u003C/div>\u003C/section>\u003Csection id=\"sec3dot8-preprints-81518\" type>\u003Ch4 class=\"html-italic\" data-nested=\"2\"> 3.8. Elementary Discrete Math\u003C/h4>\n\u003Cdiv class=\"html-p\">After telling GPT-4 that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>A\u003C/mi>\n \u003Cmo>×\u003C/mo>\n \u003Cmi>B\u003C/mi>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> stands for the Cartesian product of sets \u003Cspan class=\"html-italic\">A\u003C/span> and \u003Cspan class=\"html-italic\">B\u003C/span>, that a relation \u003Cspan class=\"html-italic\">R\u003C/span> from \u003Cspan class=\"html-italic\">A\u003C/span> to \u003Cspan class=\"html-italic\">B\u003C/span> is a subset of \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>A\u003C/mi>\n \u003Cmo>×\u003C/mo>\n \u003Cmi>B\u003C/mi>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, and that & stands for set intersection, I asked it to prove or disprove the following claim:\u003Cdiv class=\"html-disp-formula-info\" id=\"FD2-preprints-81518\">\n \u003Cdiv class=\"f\">\n \u003Cmath display=\"block\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003C/mrow>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmo>&\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmsub>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>=\u003C/mo>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003C/mrow>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmo>&\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003C/mrow>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>,\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \u003Clabel>(2)\u003C/label>\n \u003C/div>\n \u003C/div>\n where \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n\u003C/semantics>\u003C/math> and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/msub>\n\u003C/semantics>\u003C/math> are binary relations from \u003Cspan class=\"html-italic\">A\u003C/span> to \u003Cspan class=\"html-italic\">B\u003C/span> and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003C/mrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> stands for the domain of a binary relation \u003Cspan class=\"html-italic\">R\u003C/span>. The problem is trivial. We need the subset relation to hold in both directions of (\u003Ca href=\"#FD2-preprints-81518\" class=\"html-disp-formula\">2\u003C/a>), but it only holds in the left-to-right direction. Counterexamples in the other direction are very easy to find (e.g., take \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>A\u003C/mi>\n \u003Cmo>=\u003C/mo>\n \u003Cmo>{\u003C/mo>\n \u003Cmo>(\u003C/mo>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>}\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>B\u003C/mi>\n \u003Cmo>=\u003C/mo>\n \u003Cmo>{\u003C/mo>\n \u003Cmo>(\u003C/mo>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>,\u003C/mo>\n \u003Cmn>3\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>}\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>).\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: For any sets A and B, a relation R from A to B is defined as a subset of A x B. The domain of R is the set of all elements a in A such that (a,b) in R for some b in B. We write dom(R) for the domain of R. Prove or disprove the following claim:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">dom(R1 & R2) = dom(R1) & dom(R2).\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">The claim is true. Here is the proof:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">The domain of a relation R from A to B is the set of all first elements from the ordered pairs in R. We denote this as \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Let’s consider \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> as relations from A to B.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">The domain of the intersection of \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, denoted as \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, is the set of all first elements from the ordered pairs that are in both \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">The intersection of the domains of \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, denoted as \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, is the set of all first elements that appear in the ordered pairs of both \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">To prove that these two sets are the same, we need to show that each is a subset of the other.\u003C/span>\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Cmath display=\"block\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>⊆\u003C/mo>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Take any element a in \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>. By definition, there exists some b such that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> is in \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>. But if \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> is in \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, then \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> is in both \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>. Hence a is in both \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>. Therefore, a is in \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>. This shows \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> is a subset of \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>.\u003C/span>\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Cmath display=\"block\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>⊆\u003C/mo>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Take any element a in \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>. By definition, a is in both \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>. This means there exists some \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> such that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> is in \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> is in \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>. If \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>=\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, we have an element \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>=\u003C/mo>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> which belongs to \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>. Hence, a is in \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>. This shows \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> is a subset of \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Therefore, we can conclude that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>=\u003C/mo>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, as desired.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">The “proof” that GPT-4 offers is of course bogus, but the error is treacherous and can easily escape unnoticed, particularly by people who are not experienced in scrutinizing mathematical proofs. The error occurs here: \u003Cspan class=\"html-italic\">If \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>=\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, we have an element \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>=\u003C/mo>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> which belongs to \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>. Hence, a is in \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>.\u003C/span> By assuming that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmsub>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>=\u003C/mo>\n \u003Cmsub>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/msub>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, this step essentially begs the question. To put it in strict technical terms, the assumption that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmsub>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>=\u003C/mo>\n \u003Cmsub>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/msub>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> is explicitly introduced but it is never \u003Cspan class=\"html-italic\">discharged\u003C/span> (if it were, it would appear as a hypothesis in the derived conclusion, which would no longer match the goal (\u003Ca href=\"#FD2-preprints-81518\" class=\"html-disp-formula\">2\u003C/a>)).\u003C/div>\n\u003Cdiv class=\"html-p\">Once the error is pointed out to GPT-4, the model recognizes it and retracts its claim:\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Let’s unpack part 2 of your proof. You write: \"This means there exists some \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> such that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> is in \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> is in \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>. If \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>=\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, we have an element \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>=\u003C/mo>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> which belongs to \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>.\"\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Here you have explicitly introduced an assumption that you never went on to discharge, namely that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>=\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>. This seems like an error in your proof. Can you fix that?\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">You’re correct, my previous response contains an error. The equality \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>=\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> is not guaranteed. To correct this, let’s revise the second part of the proof:\u003C/span>\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Cmath display=\"block\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>⊆\u003C/mo>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Take any element a in \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>. By definition, a is in both \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>. This means there exists some \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> such that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> is in \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> is in \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>. But it’s not necessarily the case that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>=\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, which is what we would need to guarantee that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> is in \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">We cannot conclude that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>⊆\u003C/mo>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> based on the provided information. Therefore, the original claim that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>=\u003C/mo>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>∩\u003C/mo>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> is false.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Thank you for pointing out this error, and I apologize for the oversight. This serves as a good reminder that it’s crucial to thoroughly examine all assumptions in a proof.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">Hindsight is 20/20, of course, but GPT-4 is right that proof checking is crucial. Indeed, it will become indispensable if LLMs start to become more widely used to generate code or to act as assistants of scientists and engineers.\u003C/div>\u003C/section>\u003Csection id=\"sec3dot9-preprints-81518\" type>\u003Ch4 class=\"html-italic\" data-nested=\"2\"> 3.9. Simple Scheduling\u003C/h4>\n\u003Cdiv class=\"html-p\">This is the same scheduling problem that appeared in the January piece. GPT-4 still gets it wrong.\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: We have four tasks, call them T1, T2, T3, and T4. They need to be scheduled one after the other. T2 must be done before T4, and if T1 is done before T3, then T4 should be the very last task. How many different ways are there to schedule these four tasks?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">Let’s consider the possible orders in which we can schedule the tasks while satisfying the given constraints:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cul class=\"html-order\">\n\u003Cli>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T2 before T4:\u003C/span>\u003C/div>\n\u003Cul class=\"html-bullet\">\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">If T1 is before T3: T1, T2, T3, T4 (T2 before T4, and T4 is the last task as T1 is before T3)\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">If T1 is after T3: T2, T3, T1, T4 (T2 before T4, and no constraint on the position of T4 as T1 is after T3)\u003C/span>\u003C/div>\u003C/li>\n\u003C/ul>\n\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T4 before T2: Not possible, as T2 must be done before T4.\u003C/span>\u003C/div>\u003C/li>\n\u003C/ul>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Now let’s find all possible orders within each case: 1a. T1, T2, T3, T4: This case has only one valid order as given.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">1b. T2, T3, T1, T4: There are 3! = 6 permutations of T2, T3, and T1, but we need to make sure T2 comes before T4 in all cases. The valid permutations are:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">* T2, T3, T1, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">* T2, T1, T3, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">* T3, T2, T1, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">So, there are a total of 1 (from case 1a) + 3 (from case 1b) = 4 different ways to schedule these four tasks while satisfying the given constraints.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">Can you summarize all 4 orderings?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">Certainly! Here are the 4 different orderings that satisfy the given constraints:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cul class=\"html-order\">\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T1, T2, T3, T4\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T2, T3, T1, T4\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T2, T1, T3, T4\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T3, T2, T1, T4\u003C/span>\u003C/div>\u003C/li>\n\u003C/ul>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">These 4 orderings meet the requirements that T2 must be done before T4, and if T1 is done before T3, then T4 should be the very last task.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">How about T3, T1, T2, T4?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">You are correct, and I apologize for the oversight. The ordering T3, T1, T2, T4 also satisfies the given constraints. Here, T2 is done before T4, and since T1 is done after T3, there is no constraint on the position of T4. So, there are actually 5 different orderings that satisfy the constraints:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T1, T2, T3, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T2, T3, T1, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T2, T1, T3, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T3, T2, T1, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T3, T1, T2, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">How about T3, T2, T4, T1?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">You are correct again, and I apologize for the oversight. The ordering T3, T2, T4, T1 also satisfies the given constraints. Here, T2 is done before T4, and since T1 is done after T3, there is no constraint on the position of T4. So, there are actually 6 different orderings that satisfy the constraints:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T1, T2, T3, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T2, T3, T1, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T2, T1, T3, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T3, T2, T1, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T3, T1, T2, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T3, T2, T4, T1\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">How about T1, T3, T2, T4?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">You are correct, and I apologize for the oversight again. The ordering T1, T3, T2, T4 also satisfies the given constraints. Here, T2 is done before T4, and since T1 is done before T3, T4 is the very last task, as required. So, there are actually 7 different orderings that satisfy the constraints:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T1, T2, T3, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T2, T3, T1, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T2, T1, T3, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T3, T2, T1, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T3, T1, T2, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T3, T2, T4, T1\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T1, T3, T2, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">How about T2, T4, T3, T1?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">I apologize for my previous mistakes. You are correct. The ordering T2, T4, T3, T1 also satisfies the given constraints. Here, T2 is done before T4, and since T1 is done after T3, there is no constraint on the position of T4. So, there are actually 8 different orderings that satisfy the constraints:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T1, T2, T3, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T2, T3, T1, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T2, T1, T3, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T3, T2, T1, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T3, T1, T2, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T3, T2, T4, T1\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T1, T3, T2, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T2, T4, T3, T1\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">How about T2, T3, T4, T1? You are correct, and I apologize for the oversight. The ordering T2, T3, T4, T1 also satisfies the given constraints. Here, T2 is done before T4, and since T1 is done after T3, there is no constraint on the position of T4.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">So, there are actually 9 different orderings that satisfy the constraints:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T1, T2, T3, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T2, T3, T1, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T2, T1, T3, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T3, T2, T1, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T3, T1, T2, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T3, T2, T4, T1\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T1, T3, T2, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T2, T4, T3, T1\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T2, T3, T4, T1\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">Should we keep going? Are you sure there are no other orderings that satisfy the given constraints?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">I appreciate your diligence in exploring all possible orderings. Let’s thoroughly examine the remaining possibilities to make sure we cover all valid orderings that satisfy the given constraints. We have already discussed the following 9 orderings:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T1, T2, T3, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">T2, T3, T1, T4\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>[\u003C/mo>\n \u003Cmo>⋯\u003C/mo>\n \u003Cmrow>\n \u003Cmi>w\u003C/mi>\n \u003Cmi>r\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003Cmi>t\u003C/mi>\n \u003Cmi>e\u003C/mi>\n \u003Cmi>s\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>u\u003C/mi>\n \u003Cmi>t\u003C/mi>\n \u003Cmi>w\u003C/mi>\n \u003Cmi>r\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003Cmi>g\u003C/mi>\n \u003Cmi>s\u003C/mi>\n \u003Cmi>c\u003C/mi>\n \u003Cmi>h\u003C/mi>\n \u003Cmi>e\u003C/mi>\n \u003Cmi>d\u003C/mi>\n \u003Cmi>u\u003C/mi>\n \u003Cmi>l\u003C/mi>\n \u003Cmi>e\u003C/mi>\n \u003Cmi>s\u003C/mi>\n \u003C/mrow>\n \u003Cmo>⋯\u003C/mo>\n \u003Cmo>]\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003C/span>\u003C/div>\u003C/section>\u003Csection id=\"sec3dot10-preprints-81518\" type>\u003Ch4 class=\"html-italic\" data-nested=\"2\"> 3.10. Russell’s Paradox\u003C/h4>\n\u003Cdiv class=\"html-p\">The gist of Russell’s barber paradox is the existence of a barber \u003Cspan class=\"html-italic\">b\u003C/span> who shaves all and only those who do not shave themselves. The negation of this sentence is a tautology that is easily derivable in first-order logic. If we understand \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>b\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> as meaning that \u003Cspan class=\"html-italic\">a\u003C/span> is shaved by \u003Cspan class=\"html-italic\">b\u003C/span>, then we can formulate this tautology and ask GPT-4 to prove or disprove it as shown in the prompt below.\u003Ca href=\"#fn024-preprints-81518\" class=\"html-fn\">24\u003C/a> The proof is a straightforward reductio ad absurdum: If such a barber \u003Cspan class=\"html-italic\">x\u003C/span> existed we would have \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>y\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>⇔\u003C/mo>\n \u003Cmrow>\n \u003Cmn>0\u003C/mn>\n \u003Cmo>.\u003C/mo>\n \u003Cmn>12\u003C/mn>\n \u003C/mrow>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>y\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>y\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003Cmspace width=\"0.277778em\" />\n \u003Cmspace width=\"0.166667em\" />\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003Cspan class=\"html-italic\">for ally\u003C/span>, and thus substituting \u003Cspan class=\"html-italic\">x\u003C/span> for \u003Cspan class=\"html-italic\">y\u003C/span> would yield \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>⇔\u003C/mo>\n \u003Cmrow>\n \u003Cmn>0\u003C/mn>\n \u003Cmo>.\u003C/mo>\n \u003Cmn>12\u003C/mn>\n \u003C/mrow>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n \u003Cmi>R\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, a contradiction.\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Prove or disprove the following:\u003C/span>\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Cmath display=\"block\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmrow>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi mathvariant=\"monospace\">exists\u003C/mi>\n \u003Cmi mathvariant=\"monospace\">x\u003C/mi>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmo>.\u003C/mo>\n \u003Cmspace width=\"0.166667em\" />\n \u003Cmi mathvariant=\"monospace\">forall\u003C/mi>\n \u003Cmi mathvariant=\"monospace\">y\u003C/mi>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmo>.\u003C/mo>\n \u003Cmspace width=\"0.166667em\" />\n \u003Cmi mathvariant=\"monospace\">R\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi mathvariant=\"monospace\">y\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi mathvariant=\"monospace\">x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003Cmo><\u003C/mo>\n \u003Cmo>=\u003C/mo>\n \u003Cmo>=\u003C/mo>\n \u003Cmo>>\u003C/mo>\n \u003Cmrow>\n \u003Cmn mathvariant=\"monospace\">0\u003C/mn>\n \u003Cmo>.\u003C/mo>\n \u003Cmn mathvariant=\"monospace\">12\u003C/mn>\n \u003C/mrow>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n \u003Cmi mathvariant=\"monospace\">R\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi mathvariant=\"monospace\">y\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi mathvariant=\"monospace\">y\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>.\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">To prove or disprove the given statement, we will use logical reasoning and consider the cases that may arise. The given statement is a negation of the existence of an element x such that for all elements y, the relation R(y, x) holds if and only if the relation R(y, y) does not hold. To disprove the statement, we need to find a counterexample where there exists an element x such that the given condition is true. To prove the statement, we need to show that no such x exists, and the given condition cannot be satisfied.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Let’s consider two cases:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Case 1: There exists an element x such that R(y, x) holds for some y, but R(y, y) also holds. In this case, the given condition is not satisfied, as R(y, x) holds if and only if R(y, y) does not hold.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Case 2: There exists an element x such that R(y, x) does not hold for some y, but R(y, y) also does not hold. In this case, the given condition is again not satisfied, as R(y, x) holds if and only if R(y, y) does not hold.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">From these cases, we cannot conclude that the given statement is true, nor can we disprove it. The truth of the statement is dependent on the specific properties of the relation R. Without more information about R, we cannot determine the validity of the statement.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">In the first paragraph GPT-4 displays an impeccable understanding of the structure of the given sentence and what needs to be done. The subsequent case analysis, however, is hopelessly muddle-headed. What GPT-4 seems to be analyzing is the hypothesis\n \u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Cmath display=\"block\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi mathvariant=\"monospace\">exists\u003C/mi>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi mathvariant=\"monospace\">x\u003C/mi>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmo>.\u003C/mo>\n \u003Cmspace width=\"0.166667em\" />\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi mathvariant=\"monospace\">forall\u003C/mi>\n \u003Cmi mathvariant=\"monospace\">y\u003C/mi>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmo>.\u003C/mo>\n \u003Cmspace width=\"0.166667em\" />\n \u003Cmi mathvariant=\"monospace\">R\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi mathvariant=\"monospace\">y\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi mathvariant=\"monospace\">x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003Cmo><\u003C/mo>\n \u003Cmo>=\u003C/mo>\n \u003Cmo>=\u003C/mo>\n \u003Cmo>>\u003C/mo>\n \u003Cmrow>\n \u003Cmn mathvariant=\"monospace\">0\u003C/mn>\n \u003Cmo>.\u003C/mo>\n \u003Cmn mathvariant=\"monospace\">12\u003C/mn>\n \u003C/mrow>\n \u003Cmrow>\n \u003Cmo>∼\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003C/mrow>\n \u003Cmi mathvariant=\"monospace\">R\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi mathvariant=\"monospace\">y\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi mathvariant=\"monospace\">y\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>,\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n which indeed gives rise to the two conditions considered by GPT-4. But this hypothesis is neither here nor there for what needs to be done. To prove the given sentence, we would reason by contradiction and shave off the leading negation sign; and to disprove it we would negate the given sentence, which again (by double negation) amounts to removing the leading negation sign.\u003C/div>\n\u003Cdiv class=\"html-p\">When we probe the model further, it finally gets to hone in on the negated proposition, i.e., on\n \u003Cdiv class=\"html-disp-formula-info\" id=\"FD3-preprints-81518\">\n \u003Cdiv class=\"f\">\n \u003Cmath display=\"block\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∃\u003C/mo>\n \u003Cmspace width=\"0.166667em\" />\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi>x\u003C/mi>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmo>.\u003C/mo>\n \u003Cmo>∀\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi>y\u003C/mi>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmo>.\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi>R\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>y\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>⇔\u003C/mo>\n \u003Cmo>¬\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>y\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>y\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \u003Clabel>(3)\u003C/label>\n \u003C/div>\n \u003C/div>\n\t\tUnfortunately, however, it doesn’t proceed \u003Cspan class=\"html-italic\">to reason with that assumption\u003C/span>. Such reasoning would proceed by \u003Cspan class=\"html-italic\">picking a witness term w\u003C/span> for the existential claim (\u003Ca href=\"#FD3-preprints-81518\" class=\"html-disp-formula\">3\u003C/a>), leading to the conclusion\n \u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Cmath display=\"block\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∀\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi>y\u003C/mi>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmo>.\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi>R\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>y\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>w\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>⇔\u003C/mo>\n \u003Cmo>¬\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>y\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>y\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n which could then be subsequently used by instantiating it with various proof objects of interest (such as the witness term \u003Cspan class=\"html-italic\">w\u003C/span>). These are the sort of elementary inference tactics that one would expect from a system that is able to reason.\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">Why are these two cases sufficient for your analysis?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">You are correct to question the sufficiency of the cases I presented earlier. My previous response was not rigorous, and I apologize for that. Let’s re-examine the statement using a more formal approach.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">We are given the following statement:\u003C/span>\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Cmath display=\"block\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>¬\u003C/mo>\n \u003Cmspace width=\"0.166667em\" />\n \u003Cmo>∃\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi>x\u003C/mi>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmo>.\u003C/mo>\n \u003Cmo>∀\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi>y\u003C/mi>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmo>.\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi>R\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>y\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>⇔\u003C/mo>\n \u003Cmo>¬\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>y\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>y\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">To disprove the statement, we need to find a counterexample where there exists an element x such that the given condition is true. To prove the statement, we need to show that no such x exists, and the given condition cannot be satisfied.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Let’s negate the given statement and examine the result:\u003C/span>\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Cmath display=\"block\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>∃\u003C/mo>\n \u003Cmspace width=\"0.166667em\" />\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi>x\u003C/mi>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmo>.\u003C/mo>\n \u003Cmo>∀\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi>y\u003C/mi>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmo>.\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi>R\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>y\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>x\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>⇔\u003C/mo>\n \u003Cmo>¬\u003C/mo>\n \u003Cmi>R\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>y\u003C/mi>\n \u003Cmo>,\u003C/mo>\n \u003Cmi>y\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">The negated statement asserts that there exists an element x such that for every element y, R(y, x) is true if and only if R(y, y) is false.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">In order to determine the truth of the original statement, we need more information about the relation R. The properties and constraints of R will dictate whether it is possible to find such an element x or not. Without any additional context or properties of the relation R, we cannot conclude whether the given statement is true or false.\u003C/span>\u003C/div>\u003C/section>\u003Csection id=\"sec3dot11-preprints-81518\" type>\u003Ch4 class=\"html-italic\" data-nested=\"2\"> 3.11. Blocks World\u003C/h4>\n\u003Cdiv class=\"html-p\">This is a simple reasoning task that turns on a case analysis of the third-from-the-top block, call it \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>3\u003C/mn>\n \u003C/msub>\n\u003C/semantics>\u003C/math>. Either \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>3\u003C/mn>\n \u003C/msub>\n\u003C/semantics>\u003C/math> is green or not. If it is, then it’s sitting on top of a non-green block (\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>4\u003C/mn>\n \u003C/msub>\n\u003C/semantics>\u003C/math>, which is non-green by the second premise), so the conclusion holds. If it is not, then \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>b\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/msub>\n\u003C/semantics>\u003C/math>, the second-from-the-top block, is a green block sitting on top a non-green block, so again the conclusion holds.\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">There are five square blocks stacked on top of one another. You are given the following information about them:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cdl class=\"html-order\">\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">1.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">The second-from-the-top block is green.\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">2.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">The fourth-from-the-top block is not green.\u003C/span>\u003C/div>\u003C/dd>\n\u003C/dl>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Assuming that these two premises hold, disprove or else prove the following conclusion: There is a green block directly on top of a non-green block.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Explain your answer.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">The provided conclusion cannot be definitively proven or disproven with only the information given. Here’s why:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Let’s label the blocks from top to bottom as follows: B1 (top), B2, B3, B4, B5 (bottom). From the premises:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">B2 is green.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">B4 is not green.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">If we hypothesize that B1 and B3 are not green, and B5 is green, the setup would look like this:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">B1: Non-green\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">B2: Green\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">B3: Non-green\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">B4: Non-green\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">B5: Green\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">In this case, the conclusion holds true: There is a green block (B2) directly on top of a non-green block (B3). However, if we assume that B1, B3, and B5 are also green, the setup would look like this:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">B1: Green\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">B2: Green\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">B3: Green\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">B4: Non-green\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">B5: Green\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">In this case, the conclusion does not hold: There is no green block directly on top of a non-green block.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">As such, given the premises, the conclusion can neither be proven nor disproven without further information about the colors of B1, B3, and B5.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">This is another dramatic illustration of GPT-4’s inability to reason. It has chosen to consider two randomly selected worlds \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>w\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n\u003C/semantics>\u003C/math> and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>w\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/msub>\n\u003C/semantics>\u003C/math> (by “world” here I mean an assignment of green/non-green colors to each block), and reach a conclusion on the basis of those two particular worlds. This is a fundamentally flawed tactic for proving the given conjecture; a proof requires an argument that applies to \u003Cspan class=\"html-italic\">any\u003C/span> possible world. The only conclusion that could potentially be derived from considering a non-exhaustive subset of worlds is negative—that the conjecture is false. But we wouldn’t need to consider \u003Cspan class=\"html-italic\">two\u003C/span> worlds for that. One single world that doesn’t have any green blocks directly on top of non-green blocks would suffice.\u003C/div>\n\u003Cdiv class=\"html-p\">GPT-4 even gets the analysis of its own sample world \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>w\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/msub>\n\u003C/semantics>\u003C/math> wrong: B3 is a green block sitting directly on a non-green block (B4), so the putative conclusion does indeed hold in \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>w\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003C/msub>\n\u003C/semantics>\u003C/math>, contrary to GPT-4’s claim. But let’s put aside the fact that the analysis is mistaken. \u003Cspan class=\"html-italic\">If\u003C/span> the analysis were correct, as GPT-4 asserts, and the second world were indeed such that “there is no green block directly on top of a non-green block,” then the conjecture would in fact be disproved. Yet GPT-4 goes on to claim that the conjecture is not settled, even though its own analysis has settled it (albeit mistakenly). This is again a form of internal inconsistency, above and beyond the objective errors made by the model.\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">Under your second assumption (that “B1, B3, and B5 are also green”), don’t we again have a green block (B3) on top of a non-green block (B4)?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">You’re absolutely right. I apologize for the oversight. In that scenario:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">B1: Green\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">B2: Green\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">B3: Green\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">B4: Non-green\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">B5: Green\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">B3, which is green, is directly on top of B4, which is not green. Therefore, even in this case, the conclusion holds true.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">I appreciate your patience and vigilance. Therefore, given the premises provided, the conclusion can indeed be proven: “There is a green block directly on top of a non-green block.” Thank you for pointing out the error.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">GPT-4 now mistakenly considers the conjecture proven, just on the basis of those two worlds (while the worlds might contain the needed grist for the case analysis mill, the case analysis would still need to be carried out in order to have a proof).\u003C/div>\u003C/section>\u003Csection id=\"sec3dot12-preprints-81518\" type>\u003Ch4 class=\"html-italic\" data-nested=\"2\"> 3.12. Spatial Reasoning\u003C/h4>\n\u003Cdiv class=\"html-p\">We’ll first test the ability of GPT-4 to tell left from right.\u003C/div>\n\u003Csection id=\"sec3dot12dot1-preprints-81518\" type>\u003Ch4 class data-nested=\"3\"> 3.12.1. Left and Right\u003C/h4>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Suppose I’m in the middle of South Dakota and I’m looking straight down towards the center of Texas. Is Boston to my left or to my right?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">If you are in South Dakota looking south towards the center of Texas, then Boston would be to your right, since Boston is located in the northeastern part of the United States.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: So if I wanted to go to Boston from that position, I would need to take a right turn, correct?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">Yes, that’s correct. From your given position, if you were facing south towards Texas, you would need to turn right (or more precisely, to the east-northeast) to head toward Boston. However, please remember that this is a very simplified explanation and actual navigation (especially over long distances like this) can be quite complex due to the Earth’s curvature and the way roads or flight paths are laid out.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: But if I’m up north and facing directly south, wouldn’t east be directly to my left? If I went right I would end up in Seattle.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">I apologize for the confusion. You’re correct. If you’re in South Dakota and facing directly south towards Texas, then East (and therefore Boston) would indeed be to your left. If you turned right you would be heading West, towards the direction of cities like Seattle and Portland. Thank you for pointing out that mistake. I apologize for any confusion caused.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">Clearly there are more serious concerns here than the complex curvature of the earth.\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Also, can you explain why you said “or more precisely, to the east-northeast” earlier, particularly the “northeast” part? Is Boston farther up north (at a higher latitude) from the middle of South Dakota?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">Yes, Boston is actually at a higher latitude than the middle of South Dakota. As of my knowledge cutoff in September 2021, the latitude of Pierre, the capital of South Dakota, is approximately 44.37 degrees North, while Boston, Massachusetts is at approximately 42.36 degrees North. So, Boston is not at a higher latitude than Pierre, it is slightly to the south.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">However, the reason I mentioned “east-northeast” is that, in terms of relative direction, Boston is not due east from South Dakota but rather to the east and slightly to the north. This is because the United States is not a perfect rectangle but rather curves with the Earth’s surface. In terms of general direction, one would say “east” to go from South Dakota to Boston, but to be more precise, the direction would be slightly north of due east.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">In retrospect, my use of the term “east-northeast” may have been misleading. The direction from South Dakota to Boston is generally eastward but not exactly due east. I hope this clears up any confusion.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">Once again, GPT-4 makes two contradictory statements in the same breath (paragraph): “Yes, Boston is actually at a higher latitude than the middle of South Dakota.” followed by “So, Boston is not at a higher latitude than Pierre.” (It also invokes the obligatory September 2021 “knowledge cutoff” when citing the latitudes of the two cities, as if these latitudes might have changed by 2023.)\u003C/div>\n\u003Cdiv class=\"html-p\">Finally, GPT-4 continues to insist that, owing to the earth’s curvature, Boston is to the northeast of Pierre. But its output is like a weather vane that changes direction with the slightest gust of wind:\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: I’m confused, you started by claiming that Boston is at a higher latitude than the middle of South Dakota (taking the capital Pierre as the middle), but then you cited numbers that point to the opposite conclusion. Which is it?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">My apologies for the confusion - you’re correct and I misspoke in my response. Boston, Massachusetts, at a latitude of approximately 42.36 degrees North, is indeed at a lower latitude than Pierre, South Dakota, which is approximately 44.37 degrees North.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">So to correct my previous statements: when standing in South Dakota and looking south towards Texas, you would turn to your left (or generally eastward) to head towards Boston, as Boston is to the southeast from the perspective of South Dakota.\u003C/span>\u003C/div>\u003C/section>\u003Csection id=\"sec3dot12dot2-preprints-81518\" type>\u003Ch4 class data-nested=\"3\"> 3.12.2. Furniture Arrangement\u003C/h4>\n\u003Cdiv class=\"html-p\">Here we consider a simple furniture arrangement problem that must respect a set of constraints. There are several solutions that are easy to find, for example:\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>_ _ D\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>A B E\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>_ C _\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">GPT-4 makes a promising start here (with what could be understood as a correct solution under a plausible misunderstanding of what it means for one piece of furniture to be to the right of another), but once the misunderstanding is corrected, its performance degenerates precipitously.\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: We must arrange 5 pieces of furniture (A through E) on a 3 x 3 grid in accordance with the following constraints:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cdl class=\"html-order\">\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">1.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">A must not be adjacent to C.\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">2.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Nothing is to the right of E.\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">3.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">If D and A are not adjacent, then B should be in the middle.\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">4.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">D is above all others.\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">5.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">E and D are adjacent.\u003C/span>\u003C/div>\u003C/dd>\n\u003C/dl>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Here is an arrangement does not satisfy these constraints:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>_ _ E\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>A C D\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>_ B _\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">This violates, for instance, the first constraint, since A and C are adjacent. Can you print out a 3 x 3 arrangement that does satisfy the 5 constraints?\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">Sure, here is an arrangement that satisfies all the constraints:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>D E _\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>_ B _\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>A _ C\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Can you explain why constraint 2 is satisfied?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: Certainly! Constraint 2 states that \"Nothing is to the right of E.\" In the provided arrangement:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>D E _\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>_ B _\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>A _ C\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">E is positioned at the top row and the second column. Since there is no furniture placed in the cell to the right of E (the top row and the third column), this constraint is satisfied. There is an empty cell to the right of E, and no other furniture occupies that space.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: But C is to the right of E, is it not (even if it’s below E)?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">You are correct, and I apologize for the oversight. I misunderstood the constraint. Here is a corrected arrangement that satisfies all the constraints:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>D E _\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>B _ _\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>A _ C\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Now, there is no furniture to the right of E in any row, which properly satisfies constraint 2.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: But C is still to the right of E, on the bottom row.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">Apologies for the confusion again. You are right. Here’s another arrangement that should satisfy all the constraints:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>D E _\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>B _ _\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>_ _ A\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Now, there is no furniture to the right of E in any row, and all other constraints are satisfied as well.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: First, A is to the right of E. But in addition, this arrangement omits C altogether.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">I apologize for the repeated oversight. Here’s an arrangement that should satisfy all the constraints:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>D E _\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>B _ _\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>C _ A\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">In this arrangement:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">A is not adjacent to C. Nothing is to the right of E. D and A are not adjacent, and B is in the middle. D is above all others. E and D are adjacent.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: The position of A is (3,3) while that of E is (1,2), so A is again to the right of E.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">Apologies for my continued misunderstanding. Here is another arrangement that should satisfy all the constraints:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>D E A\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>B _ _\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>C _ _\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">In this arrangement:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">A is not adjacent to C. Nothing is to the right of E. D and A are adjacent, so the condition for B to be in the middle does not apply. D is above all others. E and D are adjacent.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: What are the positions of E and A in the model you just printed?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">In the arrangement I provided:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>D E A\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>B _ _\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>C _ _\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">The position of E is at row 1, column 2 (1,2), and the position of A is at row 1, column 3 (1,3).\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Now (x2,y2) being to the right of (x1,y1) simply means that \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>y\u003C/mi>\n \u003Cmn>2\u003C/mn>\n \u003Cmo>>\u003C/mo>\n \u003Cmi>y\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>. So is really nothing to the right of E above?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">You are correct, and I apologize for the confusion in my previous arrangements. Here’s the corrected arrangement that should satisfy all the constraints:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>D E _\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>A B _\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>_ _ C\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">In this arrangement:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">A is not adjacent to C. Nothing is to the right of E. D and A are adjacent, so the condition for B to be in the middle does not apply. D is above all others. E and D are adjacent.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: What are the positions of E and C in this latest model, and what do they tell us about constraint 2?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">In the latest arrangement:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>D E _\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>A B _\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>_ _ C\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">The position of E is at row 1, column 2 (1,2), and the position of C is at row 3, column 3 (3,3).\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Concerning constraint 2, \"Nothing is to the right of E,\" we can see that there is no furniture placed to the right of E in any row. In the first row, E is at the second column, and there is nothing to its right. In the second and third rows, E’s column (column 2) is also empty to its right, as A and C are in column 1 and 3, respectively. This arrangement satisfies constraint 2.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: So C is to the right of E.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">You are correct, and I apologize for my oversight once again. C is indeed to the right of E in the provided arrangement. Here is a revised arrangement that should satisfy all the constraints:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>D E _\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>A B C\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">In this arrangement:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">A is not adjacent to C. Nothing is to the right of E. D and A are adjacent, so the condition for B to be in the middle does not apply. D is above all others. E and D are adjacent.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: C is again to the right of E.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">I apologize for my repeated mistakes. Here is an arrangement that satisfies all the constraints:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>D E A\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>B _ C\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">In this arrangement:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">A is not adjacent to C. Nothing is to the right of E. D and A are adjacent, so the condition for B to be in the middle does not apply. D is above all others. E and D are adjacent.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Here A is to the right of E, so again premise 2 is violated.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">You are right, and I apologize for the continued oversight. Here’s another attempt at providing an arrangement that satisfies all the constraints:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>D E B\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>A C _\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">In this arrangement:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">A is not adjacent to C. Nothing is to the right of E. D and A are not adjacent, and B is in the middle. D is above all others. E and D are adjacent.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Here B is to the right of E.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">You are correct, and I apologize once again for not providing a correct arrangement. Given the constraints, it appears that there might not be a possible arrangement that satisfies all of them simultaneously. Let’s analyze the constraints more closely:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">A must not be adjacent to C. Nothing is to the right of E. If D and A are not adjacent, then B should be in the middle. D is above all others. E and D are adjacent. Constraint 4 means that D must be in the top row. Constraint 5 means that E must be in the top row as well, adjacent to D. Now there are only two possible placements for D and E:\u003C/span>\u003Ctt>D E _\u003C/tt>\u003Cspan class=\"html-italic\">or\u003C/span>\u003Ctt>_ D E\u003C/tt>\u003Cspan class=\"html-italic\">. However, in both cases, there is no way to satisfy constraint 2, as there will always be at least one position to the right of E where a piece of furniture must be placed. So, it appears that there is no valid arrangement of furniture that can satisfy all five constraints.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">A remarkable feature of this conversation is the \u003Cspan class=\"html-italic\">incorrigibility\u003C/span> of GPT-4, similar to the behavior exhibited in the subset-sum, graph-coloring, and several countermodel and logical-proof problems: Mistakes are repeatedly pointed out and yet the model continues to make the exact same type of error over and over again.\u003C/div>\u003C/section>\u003Csection id=\"sec3dot12dot3-preprints-81518\" type>\u003Ch4 class data-nested=\"3\"> 3.12.3. Seating Puzzle\u003C/h4>\n\u003Cdiv class=\"html-p\">This is the same seating puzzle that GPT-3.5 failed in January. The puzzle has multiple solutions, i.e., there are multiple seating arrangements that satisfy all constraints (for example, \u003Cspan class=\"html-italic\">p1 p5 p3 p2 p4\u003C/span> and \u003Cspan class=\"html-italic\">p4 p2 p3 p5 p1\u003C/span>).\u003C/div>\n\u003Cdiv class=\"html-p\">The answer to the question posed to GPT-4 below is yes, we can conclude that p5 cannot be seated either in the middle seat or on either end. GPT-4 not only fails to derive either of these conclusions, it actually arrives at the opposite conclusion, that p5 must be in the middle.\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: We need to seat five people, call them p1, p2, p3, p4, and p5, in a row of five seats, so that the following three conditions are satisfied:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">(A) p2 should be farther from the middle seat than p3.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">(B) p2 and p4 should be seated next to each other.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">(C) p1 and p3 should be flanking p5.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Is there anything we can conclude about the seat assigned to p5?\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">Yes, we can conclude that p5 must be seated in the middle seat.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Here’s the reasoning:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Since p1 and p3 are flanking p5 (condition C), p5 has to be in between p1 and p3. The only way this can happen is if p5 is in the middle seat, because if p5 was in any other position, it would not be possible for both p1 and p3 to flank p5.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Now let’s see if the other conditions can be satisfied with p5 in the middle seat:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">p2 should be farther from the middle seat than p3 (condition A): Since p5 is in the middle seat, p3 must be in one of the immediately adjacent seats (either seat 2 or seat 4). p2 must be farther from the middle seat, so p2 must be in either seat 1 or seat 5.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">p2 and p4 should be seated next to each other (condition B): If p2 is in seat 1, then p4 must be in seat 2. If p2 is in seat 5, then p4 must be in seat 4.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">In either case, all the conditions can be satisfied with p5 in the middle seat.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Can you prove your claim that “the only way this can happen is if p5 is in the middle seat, because if p5 was in any other position, it would not be possible for both p1 and p3 to flank p5”?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">Sure! Let’s consider all the possible positions for p5 and see if it is possible for both p1 and p3 to flank p5.\u003C/span>\n\u003C/div>\n\u003Cul class=\"html-bullet\">\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">If p5 is in seat 1: In this case, there are no seats to the left of p5 for either p1 or p3 to occupy. Since both p1 and p3 must flank p5, this position for p5 is not possible.\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">If p5 is in seat 2: p1 would have to be seated in seat 1 to flank p5. However, there is no available seat for p3 to also flank p5 on the other side, as p3 would be required to sit in seat 3, which is not adjacent to p5. Thus, this position for p5 is not possible.\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">If p5 is in seat 4: Similar to the case when p5 is in seat 2, p3 would have to be seated in seat 5 to flank p5, but there is no available seat for p1 to also flank p5 on the other side, as p1 would be required to sit in seat 3, which is not adjacent to p5. Thus, this position for p5 is not possible.\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">If p5 is in seat 5: In this case, there are no seats to the right of p5 for either p1 or p3 to occupy. Since both p1 and p3 must flank p5, this position for p5 is not possible.\u003C/span>\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">If p5 is in the middle seat (seat 3): In this case, p1 and p3 can flank p5 by occupying seats 2 and 4 or seats 4 and 2, respectively. This configuration satisfies condition C.\u003C/span>\u003C/div>\u003C/li>\n\u003C/ul>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">From the above analysis, we can conclude that the only way for both p1 and p3 to flank p5 is if p5 is in the middle seat (seat 3).\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">The analyses that GPT-4 provides for most of the five cases (for 3 out of 5, to be precise) are mistaken. The mistakes for the second and third cases are similar: GPT-4 mysteriously claims that “there would be no available seat” when this is clearly not true. The error in the fifth case is that the analysis fails to consider the remaining constraints, and specifically the constraint that \u003Cspan class=\"html-italic\">p2\u003C/span> and \u003Cspan class=\"html-italic\">p4\u003C/span> must be adjacent, which would be violated if \u003Cspan class=\"html-italic\">p5\u003C/span> is placed in the middle.\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA:\u003C/b> \n \u003Cspan class=\"html-italic\">Your analysis of the second case is incorrect. Can you tell me where the error is in your reasoning: “If p5 is in seat 2: p1 would have to be seated in seat 1 to flank p5. However, there is no available seat for p3 to also flank p5 on the other side, as p3 would be required to sit in seat 3, which is not adjacent to p5. Thus, this position for p5 is not possible”?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">You are right, and I apologize for the error in my previous analysis. Let’s re-examine the second case:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cul class=\"html-bullet\">\u003Cli>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">If p5 is in seat 2: p1 would have to be seated in seat 1 to flank p5. p3 could be seated in seat 3, which is also flanking p5.\u003C/span>\u003C/div>\u003C/li>\u003C/ul>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">In this case, p1 and p3 are indeed flanking p5. My previous analysis contained an error, and I apologize for the confusion. Considering this, p5 doesn’t necessarily have to be in the middle seat.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">GPT-4 realizes this particular error when its attention is explicitly brought to it, but it still fails to properly reassess the situation, fix the other errors, and arrive at even one of the two correct conclusions.\u003C/div>\u003C/section>\u003C/section>\u003Csection id=\"sec3dot13-preprints-81518\" type>\u003Ch4 class=\"html-italic\" data-nested=\"2\"> 3.13. Temporal Reasoning\u003C/h4>\n\u003Cdiv class=\"html-p\">Here we give GPT-4 a simple temporal-reasoning problem.\u003Ca href=\"#fn025-preprints-81518\" class=\"html-fn\">25\u003C/a> GPT-4 arrives at a false conclusion, making a number of glaring errors along the way.\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Tom and Nancy commute to work. Nancy’s commute takes about 30 to 40 minutes, while Tom’s commute takes about 40 to 50 minutes. Last Friday, Nancy left home between 8:10 and 8:20 AM, while Tom arrived at work between 8:50 and 9:10 AM. In addition, Nancy arrived at work after Tom left his place, but no more than 20 minutes after that. What can we conclude about when Tom and Nancy arrived at work last Friday?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">Let’s break this down:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Nancy’s commute takes between 30 to 40 minutes. That means if she left home at the latest possible time (8:20 AM), she would arrive at work between 8:50 and 9:00 AM.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Tom’s commute takes between 40 to 50 minutes. If he arrived at work at the earliest possible time (8:50 AM), then the latest he could have left home is 8:10 AM. If he arrived at work at the latest possible time (9:10 AM), then the earliest he could have left home is 8:20 AM.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Nancy arrived at work after Tom left home, but not more than 20 minutes after he left.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cb>From these facts, we can deduce the following:\u003C/b>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">If Tom left his home at the latest possible time of 8:20 AM and Nancy left at her latest possible time of 8:20 AM, and her commute is a maximum of 40 minutes, Nancy could have arrived at work as late as 9:00 AM. Since Tom’s commute is a minimum of 40 minutes, this means that he would have arrived at work by 9:00 AM at the latest as well.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Conversely, if Tom left home at his earliest possible time (8:10 AM) and Nancy left at her latest possible time (8:20 AM), even if Nancy’s commute took only 30 minutes, she would arrive at work by 8:50 AM at the earliest, which is not more than 20 minutes after Tom left. Since Tom’s commute is at least 40 minutes, he would arrive at work by 8:50 AM at the earliest as well.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">In conclusion, both Tom and Nancy arrived at work between 8:50 and 9:00 AM last Friday. We can’t say for sure who arrived first since there’s overlap in the time ranges, but we do know they both arrived within that 10-minute window.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">Once again we see that GPT-4 summarizes/organizes the given information impeccably, but fails miserably at using that information to deduce sound and useful conclusions. The errors start immediately after the summary is completed and GPT-4 writes “From these facts, we can deduce the following” (I have boldfaced this sentence in the transcript). It starts by positing the following: “If Tom left his home at the latest possible time of 8:20 AM ⋯” But in fact there are no given constraints about the latest possible time that Tom left his home; GPT-4 is taking the constraint that was given about Nancy (“Nancy left home between 8:10 and 8:20 AM”) and hallucinating its application to Tom. Moreover, the entire conditional statement is bizarre, because its antecedent includes information about Tom that is completely irrelevant to the conclusion (Nancy’s arrival): “If Tom left his home at the latest possible time of 8:20 AM and Nancy left at her latest possible time of 8:20 AM, and her commute is a maximum of 40 minutes, Nancy could have arrived at work as late as 9:00 AM.” This should have been stated as: “If Nancy left at her latest possible time of 8:20 AM, and her commute is a maximum of 40 minutes, Nancy could have arrived at work as late as 9:00 AM.”\u003C/div>\n\u003Cdiv class=\"html-p\">GPT-4 then proceeds to incorrectly infer the following: “Since Tom’s commute is a minimum of 40 minutes, this means that he would have arrived at work by 9:00 AM at the latest as well.” This does not follow at all, and it certainly doesn’t follow from the fact that “Tom’s commute is a minimum of 40 minutes.”\u003C/div>\n\u003Cdiv class=\"html-p\">The following paragraph is predicated on the hallucinated constraint that Tom’s earliest departure time was 8:10 AM (again, this constraint was given for Nancy, not for Tom). It then goes on to claim that 8:50 AM is “not more than 20 minutes after” 8:10 AM. Finally, it concludes (wrongly) that they both arrived between 8:50 and 9:00.\u003C/div>\n\u003Cdiv class=\"html-p\">Remarkably, when a countermodel is explicitly given to it, GPT-4 rejects it (again incorrectly):\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Why is it not possible for Tom to leave at 8:30 and arrive at 9:10, and for Nancy to leave at 8:15 and arrive at 8:45?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">I apologize for any misunderstanding. From the information you initially provided, it was stated that Nancy’s commute takes between 30 to 40 minutes and Tom’s commute takes between 40 to 50 minutes.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">If Tom were to leave at 8:30 AM, with a minimum commute time of 40 minutes, the earliest he could arrive at work would be 9:10 AM. So, Tom leaving at 8:30 AM and arriving at 9:10 AM is a possibility within the given constraints.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">However, if Nancy left at 8:15 AM, even with her shortest commute time of 30 minutes, the earliest she could arrive at work would be 8:45 AM. This would mean that she would arrive at work exactly 30 minutes after Tom left home. But according to the conditions provided, Nancy should arrive at work not more than 20 minutes after Tom leaves his home. Therefore, the scenario you suggested with Nancy leaving at 8:15 and arriving at 8:45 is not possible within the provided constraints.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">I hope this clears up the confusion. If you have further questions or if there is more information to consider, please let me know!\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">GPT-4 is sticking to its guns because apparently it is convinced that 8:45 is “exactly 30 minutes” after 8:30.\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: But 8:45 (Nancy’s arrival) is only 15 minutes after 8:30, which is when Tom left his place, not 30 minutes as you claim.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">I apologize for the oversight. You’re correct.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">If Nancy left at 8:15 AM and her commute took 30 minutes, she would indeed arrive at 8:45 AM, which is 15 minutes after Tom left at 8:30 AM (if we assume this scenario). This fits within the constraint that Nancy must arrive at work not more than 20 minutes after Tom leaves his home. And with a 40-minute commute, Tom would indeed arrive at 9:10 AM.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">So, you’re correct. It is indeed possible for Tom to leave at 8:30 AM and arrive at 9:10 AM, and for Nancy to leave at 8:15 AM and arrive at 8:45 AM. Thank you for pointing out the error. I appreciate your patience and apologize for any confusion caused.\u003C/span>\u003C/div>\u003C/section>\u003Csection id=\"sec3dot14-preprints-81518\" type>\u003Ch4 class=\"html-italic\" data-nested=\"2\"> 3.14. Murder or Suicide?\u003C/h4>\n\u003Cdiv class=\"html-p\">This is a logic puzzle I made up a while back. The conclusion is that Aunt Agatha killed herself. This follows by eliminating Charles and the butler. First, Aunt Agatha must have hated herself, because she hated everyone other than the butler. Therefore, Charles did not hate her (since he doesn’t hate anyone that Aunt Agatha hates), and hence he could not have killed her (by premise 3). The butler could not hate himself, because if he did, he would hate everyone (since he already hates everyone else, through premises 5 and 7), and we know that’s not possible by premise 8. Thus, the butler must be richer than Aunt Agatha, or else he would hate himself (by premise 6), which means he could not be the killer (premise 3).\u003C/div>\n\u003Cdiv class=\"html-p\">GPT-4 correctly rules out Charles, but is unable to rule out the butler and comes to incorrectly deduce that he is the killer. Another key—and rather strange—mistake that GPT-4 makes is this: “Since Aunt Agatha hates everyone except the butler (premise 5), this means she must at least not hate herself.” This is an odd error, because the natural conclusion to draw from premise 5 is that Aunt Agatha hates herself. Along the way, GPT-4 displays the internal inconsistency we’ve seen repeatedly in the other examples. Virtually in every single response, GPT-4 claims to derive some proposition as well as its negation.\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: You are given the following premises:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cdl class=\"html-order\">\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">1.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Someone who lives in Dreadbury Mansion killed Aunt Agatha.\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">2.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">The only people who live in Dreadbury Mansion are Aunt Agatha, the butler, and Charles.\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">3.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">A killer always hates his victims, and is never richer than his victims.\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">4.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Charles hates no one that Aunt Agatha hates.\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">5.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Aunt Agatha hates everyone except the butler.\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">6.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">The butler hates everyone not richer than Aunt Agatha.\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">7.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">The butler hates everyone Aunt Agatha hates.\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">8.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">No one hates everyone.\u003C/span>\u003C/div>\u003C/dd>\n\u003Cdt id>\n\u003Cspan class=\"html-italic\">9.\u003C/span> \u003C/dt>\n\u003Cdd>\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Aunt Agatha is not the butler.\u003C/span>\u003C/div>\u003C/dd>\n\u003C/dl>\u003C/div>\n\u003Cdiv class=\"html-p\" style=\"text-indent: 0;\">\u003Cspan class=\"html-italic\">On the basis of this information, determine who killed Aunt Agatha and give a detailed proof that your conclusion follows from the premises.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">Let’s analyze each of the premises:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Someone who lives in Dreadbury Mansion killed Aunt Agatha. The only people who live in Dreadbury Mansion are Aunt Agatha, the butler, and Charles. From these two premises, we can deduce that Aunt Agatha was either killed by Charles or the butler, because they are the only other residents of the Dreadbury Mansion.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">A killer always hates his victims, and is never richer than his victims. This premise is crucial as it gives us two conditions for someone to be a killer: they must hate the victim and not be richer than them.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Charles hates no one that Aunt Agatha hates. Aunt Agatha hates everyone except the butler. From premises 4 and 5, we can infer that Charles does not hate Aunt Agatha because Aunt Agatha hates herself (since she hates everyone except the butler) and Charles hates no one that Aunt Agatha hates.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">So, by the rules of premise 3, Charles cannot be the killer because he does not hate Aunt Agatha.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">The butler hates everyone not richer than Aunt Agatha. The butler hates everyone Aunt Agatha hates. Premise 6 indicates that the butler hates Aunt Agatha, as she is not richer than herself. And from premise 7, we understand that the butler also hates Aunt Agatha because Aunt Agatha hates herself (as she hates everyone except the butler). So, the butler does indeed hate Aunt Agatha.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">However, this doesn’t make him the killer yet. We still need to verify the second condition from premise 3, that the killer is not richer than the victim. As there is no information regarding their wealth, we must assume that the butler could either be richer or poorer than Aunt Agatha.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">No one hates everyone. Aunt Agatha is not the butler. Premise 9 is not immediately useful, but premise 8 tells us that no one can hate everyone. Since Aunt Agatha hates everyone except the butler (premise 5), this means she must at least not hate herself. This confirms the idea that Aunt Agatha did not commit suicide.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">From this analysis, Charles can’t be the killer because he doesn’t hate Aunt Agatha. Aunt Agatha can’t be her own killer because she doesn’t hate herself (and thus didn’t commit suicide). Therefore, the only remaining person who can be the killer is the butler.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Thus, the killer of Aunt Agatha must be the butler.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: You’ve concluded both that Aunt Agatha hates herself (you wrote “Aunt Agatha hates herself (since she hates everyone except the butler)”) and also that she doesn’t hate herself (“Since Aunt Agatha hates everyone except the butler (premise 5), this means she must at least not hate herself”). Can both of these be true? Or are the premises inconsistent?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: You’re correct. There seems to be a mistake in my reasoning. It’s contradictory to state both that Aunt Agatha hates herself and doesn’t hate herself. Let’s try to clear up this inconsistency:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">From premise 5 (“Aunt Agatha hates everyone except the butler.”), we infer that Aunt Agatha doesn’t hate herself as “everyone” typically refers to every other person, not including oneself.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Using this corrected interpretation, let’s revisit premise 7 (“The butler hates everyone Aunt Agatha hates.”). Since Aunt Agatha doesn’t hate herself, the butler doesn’t hate Aunt Agatha. Thus, according to premise 3 (“A killer always hates his victims, and is never richer than his victims.”), the butler cannot be the killer.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">So we are left with Charles. Based on premise 4 (“Charles hates no one that Aunt Agatha hates.”), we know that Charles also doesn’t hate Aunt Agatha, because Aunt Agatha doesn’t hate herself. Therefore, Charles cannot be the killer, as per premise 3.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">We thus have a contradiction. According to the premises, neither Charles nor the butler can be the killer, but Aunt Agatha must have been killed by someone who lives in Dreadbury Mansion (premise 1 and 2). It appears the premises as given are inconsistent.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: First, “everyone” here means literally everyone, in the sense of formal logic (there is a universal quantifier involved). Second, you said earlier that the butler was the killer. Why are you now ruling out the butler?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: I apologize for the confusion. If we are interpreting “everyone” as literally everyone, including oneself, then Aunt Agatha hates herself. This is a different understanding from my earlier analysis.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Let’s go through the premises again under this new understanding:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Aunt Agatha, the butler, and Charles live in Dreadbury Mansion. (Premise 2) The killer must be one of these three because someone who lives in Dreadbury Mansion killed Aunt Agatha. (Premise 1) Aunt Agatha hates everyone including herself. (Premise 5, under our new interpretation of “everyone”) The butler hates everyone that Aunt Agatha hates, so the butler hates Aunt Agatha. (Premise 7) Charles hates no one that Aunt Agatha hates, so Charles does not hate Aunt Agatha. (Premise 4) From these points, according to Premise 3 (“A killer always hates his victims, and is never richer than his victims”), we can already eliminate Charles as the killer since he does not hate Aunt Agatha.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">So, the killer must be either Aunt Agatha or the butler.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">But since no one hates everyone (Premise 8), we can infer that Aunt Agatha doesn’t hate herself, and thus she wouldn’t have killed herself. So, Aunt Agatha is not the killer.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">That leaves us with the butler as the only possible killer. So, even with our new understanding of “everyone” as literally everyone, the conclusion remains: The butler killed Aunt Agatha. I apologize for the previous error in reasoning, and thank you for your patience.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: You’re again deriving two inconsistent conclusions: that Agatha hates herself and that she doesn’t. Which is it?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: You’re right, my apologies for the confusion. The premises do seem to be leading to inconsistent conclusions regarding whether Aunt Agatha hates herself or not.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Let’s go through this more carefully.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Premise 5: “Aunt Agatha hates everyone except the butler.” Given your interpretation, “everyone” here includes herself, which means Aunt Agatha hates herself.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">However, Premise 8 states: “No one hates everyone.” Given your interpretation of “everyone” as literally everyone, then Aunt Agatha does not hate herself. She must have at least one exception aside from the butler, or else she would hate “everyone,” which contradicts Premise 8.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Therefore, Aunt Agatha does not hate herself. The only interpretation of the premises that does not lead to a contradiction is that Aunt Agatha does not hate herself.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Now, let’s reconsider who the killer can be:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Aunt Agatha did not kill herself (because she does not hate herself). Charles did not kill Aunt Agatha (because he does not hate anyone that Aunt Agatha hates, which includes herself). The butler did not kill Aunt Agatha (because he does not hate anyone that Aunt Agatha does not hate, and Aunt Agatha does not hate herself). Therefore, under these premises and your interpretation of “everyone,” none of Aunt Agatha, the butler, or Charles could have killed Aunt Agatha.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">In conclusion, there is a contradiction: Premises 1 and 2 state that Aunt Agatha must have been killed by someone in Dreadbury Mansion, which includes only Aunt Agatha, the butler, and Charles. But, under our analysis, none of them could have killed Aunt Agatha. This suggests the premises are inconsistent.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Let’s focus on this piece of your analysis:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">’However, Premise 8 states: “No one hates everyone.” Given your interpretation of “everyone” as literally everyone, then Aunt Agatha does not hate herself. She must have at least one exception aside from the butler, or else she would hate “everyone,” which contradicts Premise 8.’\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Can you prove in more detail, step-by-step, how premise 8 entails that Aunt Agatha does not hate herself?\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: Sure, I’ll clarify this deduction in more detail:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Premise 8 states: “No one hates everyone.” In the context of formal logic, “everyone” typically refers to all individuals within a certain set. Given your interpretation that “everyone” should be taken literally as all individuals without exception, “everyone” includes each individual person, and themselves.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Therefore, according to Premise 8, there is no individual who hates every other individual and also hates themselves.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Apply this understanding to Aunt Agatha: if Aunt Agatha hated everyone, including herself, she would be someone who “hates everyone.” But this would contradict Premise 8, which states that no such person exists.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Therefore, Aunt Agatha does not hate everyone.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">In order to not hate “everyone”, there must be at least one individual whom Aunt Agatha does not hate. We know from Premise 5 that the butler is one such individual.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">However, even if Aunt Agatha does not hate the butler, she would still be “hating everyone” if she hated all other individuals, including herself. This again would contradict Premise 8.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Therefore, there must be at least one more individual, in addition to the butler, whom Aunt Agatha does not hate. Since the only other individuals in question are Charles and Aunt Agatha herself, and we know from Premise 5 that she hates Charles, the only remaining possibility is that Aunt Agatha does not hate herself.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Therefore, Premise 8 implies that Aunt Agatha does not hate herself.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Justify step 6 please: ’However, even if Aunt Agatha does not hate the butler, she would still be “hating everyone” if she hated all other individuals, including herself.” Why does that hold?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: Sure, let’s clarify step 6:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">According to your interpretation of “everyone,” this term refers to all individuals, including oneself. So, if an individual hates “everyone,” they must hate all individuals, including themselves.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">So in step 6, when we say “if Aunt Agatha hated all other individuals, including herself, she would still be ’hating everyone’,” what we’re saying is:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">If Aunt Agatha hated herself and all other individuals (except for the butler), then she would hate all individuals in our context (which includes Aunt Agatha, the butler, and Charles), with the sole exception of the butler.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">However, even with this exception, she would still be classified as someone who hates “everyone” under our interpretation, because she hates all individuals including herself (despite the exception of the butler).\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Thus, even if Aunt Agatha does not hate the butler, if she hated all other individuals, including herself, she would be classified as someone who “hates everyone” (despite the one exception). This would contradict Premise 8, which states that no one hates everyone.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Therefore, if Aunt Agatha does not hate the butler (as per Premise 5) and she is to avoid contradicting Premise 8, there must be at least one more individual whom Aunt Agatha does not hate. Since the only other individuals in our context are Charles and Aunt Agatha herself, and we know from Premise 5 that she hates Charles, the only remaining possibility is that Aunt Agatha does not hate herself.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">I hope that clarifies step 6. It’s a nuanced point: even if Aunt Agatha has one exception to her hatred (the butler), if she hates all other individuals, including herself, she would still be considered as someone who “hates everyone” under our interpretation, which contradicts Premise 8.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: This is quite wrong. As long as there is at least one individual b not hated by Aunt Agatha, then she cannot possibly hate everyone, period, because if she did hate everyone then she would also hate b, by definition of ’everyone’; and we know that she does not hate b. In our case, of course, the individual b in question is the butler.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: You’re absolutely right, and I apologize for my misinterpretation. I appreciate your patience.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Given your correct interpretation, if Aunt Agatha does not hate the butler, it is sufficient to say that she doesn’t hate everyone, even if she hates all others including herself.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">This also affects the conclusions we draw from other premises.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Let’s look at premises 3, 4, and 7 again:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Premise 3 states: “A killer always hates his victims, and is never richer than his victims.” Premise 4 states: “Charles hates no one that Aunt Agatha hates.” Premise 7 states: “The butler hates everyone Aunt Agatha hates.” From premise 4, since Charles hates no one that Aunt Agatha hates, and Aunt Agatha hates herself, it follows that Charles does not hate Aunt Agatha. This means that Charles could not have killed Aunt Agatha (as per premise 3).\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">From premise 7, since the butler hates everyone that Aunt Agatha hates, and Aunt Agatha hates herself, it follows that the butler hates Aunt Agatha. Hence, he could be the killer, according to the hatred condition of premise 3.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">For the wealth condition of premise 3, there is no clear information. We must assume that Aunt Agatha was not richer than everyone else in the mansion; if she was, the premise “a killer is never richer than his victims” would make it impossible for anyone to have killed her. This is a reasonable assumption given that the other premises suggest there is a killer.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">So, we have only one viable candidate: the butler. This aligns with our conclusion from the first analysis: The butler is the one who killed Aunt Agatha.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Thank you for your guidance in clarifying these premises.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Why did you say earlier that even if there is an individual b not hated by Aunt Agatha, it would still follow that Aunt Agatha hates everyone?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: You’re correct, and I must apologize for the confusion. That statement was a misinterpretation on my part. In formal logic, if there is an individual that Aunt Agatha does not hate (in this case, the butler), it does not follow that Aunt Agatha hates everyone. In fact, it’s the opposite: if there is at least one individual Aunt Agatha does not hate, it means she does not hate everyone. I appreciate your patience and I’m sorry for the oversight.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Ok, so you are now convinced that the butler killed Aunt Agatha?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: Given the premises and the interpretation of “everyone” as literally everyone, including oneself, the evidence does indeed point to the butler as the killer. Here’s why:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">According to premise 5, Aunt Agatha hates everyone except the butler. Given the interpretation of “everyone”, this means she hates herself.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">According to premise 7, the butler hates everyone Aunt Agatha hates, which now includes Aunt Agatha herself.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Charles, according to premise 4, hates no one that Aunt Agatha hates, so he does not hate Aunt Agatha.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">From premise 3, a killer always hates his victims. So, Charles, who does not hate Aunt Agatha, can’t be the killer. The butler, who does hate Aunt Agatha, could be the killer. And, as Aunt Agatha is not the butler (premise 9), and she can’t have killed herself since a killer hates his victims and Aunt Agatha doesn’t hate herself (from premise 8), Aunt Agatha herself is not the killer.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">So, we can conclude that the butler is the killer, given that he is the only person who fits the description provided by the premises. This conclusion depends heavily on the interpretation of “everyone” to include oneself, which allows us to infer that Aunt Agatha hates herself.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: You said earlier that there’s not enough information in the premises about the butler’s level of wealth. Are you sure there’s nothing we can infer?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: You’re correct, and I apologize for missing that detail. There is indeed something we can infer about the butler’s wealth from the given premises.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Premise 3 states that “A killer always hates his victims, and is never richer than his victims.”\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">We know from premises 5 and 7 that Aunt Agatha hates herself and, thus, the butler also hates Aunt Agatha. If the butler was richer than Aunt Agatha, according to premise 3, he could not be her killer.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">However, premise 6 states that “The butler hates everyone not richer than Aunt Agatha.” Given that we know the butler hates Aunt Agatha, it follows that Aunt Agatha must not be richer than the butler, or else he wouldn’t hate her (which would contradict premises 5 and 7).\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">In summary, the butler isn’t richer than Aunt Agatha, but Aunt Agatha isn’t richer than the butler either. This suggests they are of equal wealth.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">So, given all of the premises, the butler, who hates Aunt Agatha and is of equal wealth, is indeed the only individual who fits the profile of the killer.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Thank you for your patience in clarifying these premises. I appreciate your guidance in this logical puzzle.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: So what did you conclude about the butler’s wealth, that he neither richer nor poorer than Aunt Agatha?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: Yes, from the given premises, we can infer that the butler is neither richer nor poorer than Aunt Agatha, which would suggest they are of equal wealth. Here’s how:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Premise 6 states, “The butler hates everyone not richer than Aunt Agatha.” This means that if someone is as rich as Aunt Agatha or poorer, the butler hates them.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">We know from premises 5 and 7 that the butler hates Aunt Agatha. Therefore, according to premise 6, Aunt Agatha must not be richer than the butler. If she were, the butler wouldn’t hate her, contradicting premises 5 and 7.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">At the same time, the butler can’t be richer than Aunt Agatha either. That’s because premise 3 states that “a killer is never richer than his victims.” If the butler were richer than Aunt Agatha, he couldn’t be her killer.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">So, the butler isn’t richer than Aunt Agatha, and Aunt Agatha isn’t richer than the butler. This suggests that they are of equal wealth. Therefore, given all the premises, the butler fits the criteria to be the killer: he hates Aunt Agatha (as required by premise 3), and he isn’t richer than her (also required by premise 3).\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: I claim that we can infer that the butler is richer than Aunt Agatha.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: You’re correct, and I apologize for the oversight.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Premise 6 states that “The butler hates everyone not richer than Aunt Agatha.” We know from premises 5 and 7 that the butler hates Aunt Agatha. Therefore, it must be the case that Aunt Agatha is not richer than the butler, i.e., the butler is either as rich as or richer than Aunt Agatha.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">However, premise 3 states that a killer is never richer than his victims. If the butler was as rich as Aunt Agatha, he could still be her killer. But since Aunt Agatha was killed by someone living in Dreadbury Mansion (premise 1) and Charles and Aunt Agatha herself could not have been the killers (as they don’t hate Aunt Agatha), the butler must be the killer. Given that the butler is the killer and that a killer is never richer than his victims, it must be the case that the butler is not richer than Aunt Agatha.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">The only way to satisfy both conditions (the butler being neither richer than nor as rich as Aunt Agatha) is if the butler is poorer than Aunt Agatha.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">I apologize for the confusion in the previous responses. The conclusion should be that the butler, who is poorer than Aunt Agatha, killed her. This conclusion is consistent with all the given premises.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Again, I appreciate your patience in solving this logical puzzle.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: No, like I said, it’s the opposite. We can infer that the butler, far from being poorer than Aunt Agatha, he’s actually richer.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: You’re correct. I apologize for the confusion. Let’s revisit premise 6: “The butler hates everyone not richer than Aunt Agatha.”\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">This means that the butler hates all individuals who are as wealthy as Aunt Agatha or less wealthy than her. But from premises 5 and 7, we know that the butler does indeed hate Aunt Agatha. Therefore, Aunt Agatha must be in the group of people who are not richer than the butler, which implies that the butler is as wealthy as or wealthier than Aunt Agatha.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Given premise 3, which states that a killer is never richer than his victims, it would initially seem that the butler cannot be the killer since he could potentially be richer than Aunt Agatha. However, Aunt Agatha cannot be the killer because she doesn’t hate herself (from premise 8), and Charles can’t be the killer because he doesn’t hate Aunt Agatha (from premises 4 and 5).\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Hence, the butler must be the killer, and the only way the butler can be the killer and still satisfy premise 3 is if the butler and Aunt Agatha have the same wealth. Thus, there was a mistake in my previous analysis: the butler isn’t poorer than Aunt Agatha. Instead, the butler must be equally wealthy as Aunt Agatha. However, your claim was that we can infer the butler is richer than Aunt Agatha, which is not supported by the premises.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">I apologize for the confusion, and I appreciate your patience and assistance in resolving this puzzle.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: So you’re back to claiming that the butler is equally rich as Aunt Agatha, and that my claim that he’s richer is does not follow.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: My apologies for the confusion in my previous responses. Let’s reevaluate premise 6: “The butler hates everyone not richer than Aunt Agatha.”\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">From this premise, we can infer that the only people whom the butler doesn’t hate are those richer than Aunt Agatha. But since we know from premises 5 and 7 that the butler does hate Aunt Agatha, it must follow that Aunt Agatha is not richer than the butler. Therefore, the butler must be richer than Aunt Agatha.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">In light of this, we return to premise 3, which states, “A killer always hates his victims, and is never richer than his victims.” Given that the butler hates Aunt Agatha and is richer than her, it would seem at first that the butler cannot be the killer.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">However, if we consider all the residents of Dreadbury Mansion, Aunt Agatha is not the killer because she doesn’t hate herself (from premise 8) and Charles is not the killer because he doesn’t hate Aunt Agatha (from premises 4 and 5).\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Hence, we’re left with the butler as the only possible killer, despite him being richer than Aunt Agatha, which seems to contradict premise 3. This seems to create a paradox given the premises as they stand.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">There may be an issue with how we’re interpreting the premises or there may be an error in the premises themselves. If the butler is indeed the killer, then it contradicts the wealth condition stated in premise 3.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">If we adhere strictly to the premises, it appears we have a paradox. The butler seems to be the only possible candidate who could be the killer based on hatred, but this contradicts the wealth condition of the killer in premise 3. I appreciate your patience and I apologize for the confusion in my previous responses.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: No, this reasoning of yours is incorrect: “But since we know from premises 5 and 7 that the butler does hate Aunt Agatha, it must follow that Aunt Agatha is not richer than the butler. Therefore, the butler must be richer than Aunt Agatha.”\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">The fact that Aunt Agatha is not richer than the butler doesn’t support the conclusion that the butler is richer. I still claim that the butler IS in fact richer, but your attempts to prove that fact have all contained reasoning errors.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: You’re correct, and I apologize for the confusion. Your original claim was that the butler is richer than Aunt Agatha, not equally rich as her or poorer. Let’s revisit the premises to see how we can support this claim:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">From Premise 6 (“The butler hates everyone not richer than Aunt Agatha”), we can infer that the butler does not hate people who are richer than Aunt Agatha. From Premise 7 (“The butler hates everyone Aunt Agatha hates”), coupled with the information from Premise 5 (“Aunt Agatha hates everyone except the butler”), we know the butler hates Aunt Agatha. Combining these pieces of information, we can conclude that Aunt Agatha is not richer than the butler because if she were, the butler wouldn’t hate her (which would contradict Premise 7). However, this doesn’t automatically mean that the butler is richer than Aunt Agatha. It could also mean they have the same level of wealth. But if we look at Premise 3 (“A killer always hates his victims, and is never richer than his victims”), we know that the killer is not richer than Aunt Agatha. Given all the premises, we’ve deduced that the killer can’t be Aunt Agatha herself or Charles, leaving only the butler as a possible killer. If the butler were the killer, and the killer can’t be richer than Aunt Agatha, then the butler can’t be richer than Aunt Agatha. Hence, we have a contradiction, since we earlier deduced from Premise 6 that Aunt Agatha can’t be richer than the butler. The only resolution is that the butler must be richer than Aunt Agatha. In conclusion, I apologize for the confusion in the earlier responses. Your claim is correct: from the given premises, it can be inferred that the butler, who is richer than Aunt Agatha, is the one who killed her.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: That’s impossible, because a killer is never richer than his victims.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: I apologize for the confusion in my previous messages. Let’s go through the premises again.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>[\u003C/mo>\n \u003Cmo>⋯\u003C/mo>\n \u003Cmspace width=\"0.277778em\" />\n \u003Cmrow>\n \u003Cmi>S\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003Cmi>l\u003C/mi>\n \u003Cmi>a\u003C/mi>\n \u003Cmi>r\u003C/mi>\n \u003Cmi>c\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003Cmi>f\u003C/mi>\n \u003Cmi>u\u003C/mi>\n \u003Cmi>s\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003Cmi>s\u003C/mi>\n \u003Cmo>/\u003C/mo>\n \u003Cmi>e\u003C/mi>\n \u003Cmi>r\u003C/mi>\n \u003Cmi>r\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>r\u003C/mi>\n \u003Cmi>s\u003C/mi>\n \u003Cmi>a\u003C/mi>\n \u003Cmi>s\u003C/mi>\n \u003Cmi>a\u003C/mi>\n \u003Cmi>b\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>v\u003C/mi>\n \u003Cmi>e\u003C/mi>\n \u003C/mrow>\n \u003Cmspace width=\"0.277778em\" />\n \u003Cmo>⋯\u003C/mo>\n \u003Cmo>]\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003C/span>\u003C/div>\u003C/section>\u003Csection id=\"sec3dot15-preprints-81518\" type>\u003Ch4 class=\"html-italic\" data-nested=\"2\"> 3.15. Wason Selection Task\u003C/h4>\n\u003Cdiv class=\"html-p\">The Wason selection task is a staple in the psychology of reasoning. The January paper included an example that GPT-3.5 flunked. Here is another version that GPT-4.0 fails badly:\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Seven cards are placed on the table, each of which has a number on one side and a single colored patch on the other side. The faces of the cards show 50, 16, red, yellow, 23, green, 30. Which cards would you have to turn to test the truth of the proposition that if a card is showing a multiple of 4 then the color of the opposite side is yellow?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">The proposition is: “If a card shows a number that is a multiple of 4, then the color on the other side is yellow”. In order to test this proposition, you would have to turn over the following cards:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Card “16”: This card is showing a number that is a multiple of 4. According to the proposition, the color on the other side should be yellow. By turning over this card, you can check whether this part of the proposition holds true.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Card “Yellow”: This card is showing a color that according to the proposition should be on the other side of a card with a number that is a multiple of 4. By turning it over, you can check if the number on the other side is a multiple of 4.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Card “50” and “30”: While these cards show numbers that are not multiples of 4, they should be checked to ensure the color on the other side is not yellow. If the other side is yellow, then it would disprove the proposition that only multiples of 4 can have yellow on the other side.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">You wouldn’t need to turn over the card “23”, “red”, or “green”. The card “23” is not a multiple of 4, so it doesn’t matter what color is on the other side based on the given proposition. The “red” and “green” cards, if they have numbers that are multiples of 4 on the other side, would not disprove the proposition because the proposition does not state that yellow is the only color that can be on the other side of a card with a number that’s a multiple of 4.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">In fact, only 16, red, and green need to be turned over. Thus, in terms of precision, GPT-4 gets 1/4 = 0.25, while on recall it gets 1/3, for a combined F-score of 0.28. The responses show yet again that GPT-4 does not understand the semantics of material conditionals. When GPT-4 says that cards “50” and “30” must be turned over, it seems to mistake the \u003Cspan class=\"html-italic\">if\u003C/span> for an \u003Cspan class=\"html-italic\">if and only if\u003C/span>: “\u003Cspan class=\"html-italic\">while these cards show numbers that are not multiples of 4, they should be checked to ensure the color on the other side is not yellow. If the other side is yellow, then it would disprove the proposition that only multiples of 4 can have yellow on the other side\u003C/span>”. But of course the proposition being tested does not require that \u003Cspan class=\"html-italic\">only\u003C/span> multiples of 4 have yellow on the other side; it simply requires that \u003Cspan class=\"html-italic\">if\u003C/span> a card has a multiple of 4 on the other side, \u003Cspan class=\"html-italic\">then\u003C/span> the other side must be yellow; or, equivalently, that if a card face is \u003Cspan class=\"html-italic\">not\u003C/span> yellow then the number on the other side is not a multiple of 4.\u003C/div>\n\u003Cdiv class=\"html-p\">Observe, however, that even the mistaken interpretation of the rule as a biconditional is \u003Cspan class=\"html-italic\">not consistently observed\u003C/span>. If GPT-4 understood the rule being tested as a biconditional and hence felt compelled to turn over 50 and 30, \u003Cspan class=\"html-italic\">then it ought to have done the same thing for 23\u003C/span>. But it didn’t. In fact, it explicitly said that \u003Cspan class=\"html-italic\">The card “23” is not a multiple of 4, so it doesn’t matter what color is on the other side based on the given proposition\u003C/span>, which contradicts the biconditional interpretation that it used as a rationale for turning 50 and 30. This is one more illustration of the other major theme emerging from these examples: The responses of GPT-4, whether right or wrong, are often internally inconsistent. Yet another example appears in the following problem.\u003C/div>\u003C/section>\u003Csection id=\"sec3dot16-preprints-81518\" type>\u003Ch4 class=\"html-italic\" data-nested=\"2\"> 3.16. Entropy\u003C/h4>\n\u003Cdiv class=\"html-p\">An elementary result of information theory states that the entropy of a random vector \u003Cspan class=\"html-italic\">Z\u003C/span> is bounded above by the sum of the entropies of the random variables that comprise \u003Cspan class=\"html-italic\">Z\u003C/span>. Hence, the answer to the following question should be “under no conditions”:\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Let Z be a random vector consisting of n random variables \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n\u003C/semantics>\u003C/math>,...\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n\u003C/semantics>\u003C/math>. Under what conditions can the entropy of Z exceed the sum of the entropies of all \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003C/msub>\n\u003C/semantics>\u003C/math>?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">The entropy of a random vector Z consisting of n random variables \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n\u003C/semantics>\u003C/math>, ..., \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n\u003C/semantics>\u003C/math> can exceed the sum of the entropies of all \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003C/msub>\n\u003C/semantics>\u003C/math> only when there is some dependency between the random variables. If the random variables are mutually independent, the entropy of Z would be equal to the sum of the entropies of all \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003C/msub>\n\u003C/semantics>\u003C/math>. The entropy of a random vector Z, H(Z), can be written as the joint entropy of the random variables \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>: \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>Z\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>=\u003C/mo>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>. The joint entropy \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>H\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> can be greater than the sum of individual entropies \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>H\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> if and only if the random variables are not mutually independent. In other words, knowing the value of one or more of the random variables provides information about the others. If the random variables are mutually independent, then the joint entropy is equal to the sum of the individual entropies:\u003C/span>\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Cmath display=\"block\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>=\u003C/mo>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>+\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>+\u003C/mo>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>.\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n\u003Cspan class=\"html-italic\">When there is dependence between the random variables, we can write: \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>≤\u003C/mo>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>+\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>+\u003C/mo>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> with equality if and only if the random variables are mutually independent. To summarize, the entropy of the random vector Z can exceed the sum of the entropies of all \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003C/msub>\n\u003C/semantics>\u003C/math> if and only if there is some dependency between the random variables \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">Observe the internal inconsistency again: It’s clear that GPT-4 has memorized the result correctly:\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Cmath display=\"block\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>…\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>≤\u003C/mo>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>+\u003C/mo>\n \u003Cmo>⋯\u003C/mo>\n \u003Cmo>+\u003C/mo>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>,\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n with equality holding iff \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>…\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> are mutually independent. Nevertheless, it does not have the reasoning ability to realize what this entails, namely, that it precludes:\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Cmath display=\"block\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>…\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>>\u003C/mo>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>+\u003C/mo>\n \u003Cmo>⋯\u003C/mo>\n \u003Cmo>+\u003C/mo>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>.\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n\u003C/div>\n\u003Cdiv class=\"html-p\">At least here once the error is explicitly pointed out to GPT-4 , it acknowledges it and corrects itself:\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: But if \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>Z\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>=\u003C/mo>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> and \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>≤\u003C/mo>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>+\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>+\u003C/mo>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> then how can we ever have \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>Z\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>>\u003C/mo>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>+\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>+\u003C/mo>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">My apologies, I made an error in my previous response. You are right, the entropy of the random vector Z cannot exceed the sum of the entropies of all \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003C/msub>\n\u003C/semantics>\u003C/math>. The correct statement should be: When the random variables are mutually independent, the joint entropy H(Z) is equal to the sum of the individual entropies: \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>Z\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>=\u003C/mo>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>+\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>+\u003C/mo>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> When there is dependence between the random variables, the joint entropy H(Z) is less than the sum of the individual entropies: \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>Z\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo><\u003C/mo>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003Cmo>+\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>.\u003C/mo>\n \u003Cmo>+\u003C/mo>\n \u003Cmi>H\u003C/mi>\n \u003Cmrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmsub>\n \u003Cmi>X\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> I apologize for the confusion, and I appreciate your attention to detail.\u003C/span>\n\u003C/div>\u003C/section>\u003Csection id=\"sec3dot17-preprints-81518\" type>\u003Ch4 class=\"html-italic\" data-nested=\"2\"> 3.17. Simple Compiler Correctness\u003C/h4>\n\u003Cdiv class=\"html-p\">The last reasoning problem we give to GPT-4 is the most challenging one: It is to prove the correctness of a simple expression compiler. Remarkably, GPT-4 goes about this in the right sort of way, by setting up a structural induction over the abstract grammar of expressions. This is no doubt because it has seen similar proofs before, as this is a common type of exercise in courses and textbooks on programming language theory.\u003Ca href=\"#fn026-preprints-81518\" class=\"html-fn\">26\u003C/a> However, even though the proof is on the right general track, it has several errors. (For the record, the compiler is indeed correct, although proving this requires strengthening the induction hypothesis).\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Suppose I have an abstract grammar for numeric expressions defined as follows:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>Exp := const(Int)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>     | sum(Exp,Exp)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>     | diff(Exp,Exp)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>     | mult(Exp,Exp)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>     | quot(Exp,Exp)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">where\u003C/span> \n \u003Ctt>Int\u003C/tt> \n \u003Cspan class=\"html-italic\">denotes the domain of integers. The semantics of these expressions are defined via this interpreter function:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>I: Exp -> Int\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>I(const(n)) = n\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>I(sum(e1,e2)) = I(e1) + I(e2)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>I(diff(e1,e2)) = I(e1) - I(e2)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>I(prod(e1,e2)) = I(e1) * I(e2)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>I(quot(e1,e2)) = I(e1) / I(e2)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">I now define a virtual machine that executes simple programs that are sequences of commands, where commands have the following structure:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>Cmd := push(Int) | add | sub | mult | div\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">The operational semantics of these programs (sequences of commands) are defined by an execution function exec that takes a program and a stack of integers S and produces an integer as output. Let’s write\u003C/span> \n \u003Ctt>[]\u003C/tt> \n \u003Cspan class=\"html-italic\">and\u003C/span> \n \u003Ctt>n::S\u003C/tt> \n \u003Cspan class=\"html-italic\">to indicate the empty stack and the stack obtained by prepending integer\u003C/span> \n \u003Ctt>n\u003C/tt> \n \u003Cspan class=\"html-italic\">to stack\u003C/span> \n \u003Ctt>S\u003C/tt> \n \u003Cspan class=\"html-italic\">, respectively. Here’s the definition of exec:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>exec([],n::S) = n\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>exec(push(n)::C,S) = exec(C,n::S)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>exec(add::C,n::m::S) = exec(C,(n+m)::S)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>exec(sub::C,n::m::S) = exec(C,(n-m)::S)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>exec(mult::C,n::m::S) = exec(C,(n*m)::S)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>exec(div::C,n::m::S) = exec(C,(n/m)::S)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Finally, I define a compiler that translates an expression e into a program (sequence of commands) as follows. I write\u003C/span> \n \u003Ctt>@\u003C/tt> \n \u003Cspan class=\"html-italic\">for sequence concatenation:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>T: Exp -> List(Cmd)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>T(const(n)) = [push(n)]\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>T(sum(e1,e2)) = T(e2)@T(e1)@[add]\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>T(diff(e1,e2)) = T(e2)@T(e1)@[sub]\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>T(prod(e1,e2)) = T(e2)@T(e1)@[mult]\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>T(quot(e1,e2)) = T(e2)@T(e1)@[div]\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Disprove or prove the following claim: For all expressions\u003C/span> \u003Ctt>e\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>exec(T(e),[]) = I(e)\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4:\u003C/b> \n \u003Cspan class=\"html-italic\">The claim is correct. The execution of the translated commands from an expression results in the same output as interpreting the expression directly. We can demonstrate this using structural induction over the set of expressions.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Inductive hypothesis: For all expressions\u003C/span> \u003Ctt>e\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>exec(T(e),[]) = I(e)\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Base case: Consider the simplest possible expression, a constant integer expression\u003C/span> \u003Ctt>e = const(n)\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">For\u003C/span> \n \u003Ctt>T(const(n))\u003C/tt> \n \u003Cspan class=\"html-italic\">, we have\u003C/span> \n \u003Ctt>T(const(n)) = [push(n)]\u003C/tt> \n \u003Cspan class=\"html-italic\">. Therefore,\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>exec(T(const(n)),[]) = exec([push(n)],[])\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>            = n (according to the definition of exec).\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">For\u003C/span> \u003Ctt>I(const(n))\u003C/tt>\u003Cspan class=\"html-italic\">, we have\u003C/span> \u003Ctt>I(const(n)) = n\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">So, for the base case, we have shown\u003C/span> \u003Ctt>exec(T(e),[]) = I(e)\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Inductive step: We need to consider the other four kinds of expressions:\u003C/span> \u003Ctt>sum\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>diff\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>prod\u003C/tt>\u003Cspan class=\"html-italic\">, and\u003C/span> \u003Ctt>quot\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Ctt>sum(e1, e2)\u003C/tt>\u003Cspan class=\"html-italic\">: For\u003C/span> \u003Ctt>T(sum(e1,e2))\u003C/tt>\u003Cspan class=\"html-italic\">, we have\u003C/span> \u003Ctt>T(sum(e1,e2)) = T(e2)@T(e1)@[add]\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Therefore,\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>exec(T(sum(e1,e2)),[])            =\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>exec(T(e1)@T(e2)@[add],[])          =\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>exec(T(e2)@T(e1),[]) + exec([add],[])   =\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>exec(T(e1),[]) + exec(T(e2),[])         (by definition of exec).\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">For\u003C/span> \u003Ctt>I(sum(e1,e2))\u003C/tt>\u003Cspan class=\"html-italic\">, we have\u003C/span> \u003Ctt>I(sum(e1,e2)) = I(e1) + I(e2)\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">By the inductive hypothesis,\u003C/span> \n \u003Ctt>exec(T(e1),[]) = I(e1)\u003C/tt> \n \u003Cspan class=\"html-italic\">and\u003C/span> \n \u003Ctt>exec(T(e2),[]) = I(e2)\u003C/tt> \n \u003Cspan class=\"html-italic\">. So\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>exec(T(e1),[]) + exec(T(e2),[]) = I(e1) + I(e2).\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Thus, we have shown that for\u003C/span> \u003Ctt>sum(e1,e2)\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>exec(T(e),[]) = I(e)\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Ctt>diff(e1, e2)\u003C/tt>\u003Cspan class=\"html-italic\">: A similar reasoning as for sum can be applied to show that for\u003C/span> \u003Ctt>diff(e1,e2)\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>exec(T(e),[]) = I(e)\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Ctt>prod(e1, e2)\u003C/tt>\u003Cspan class=\"html-italic\">: Likewise, a similar reasoning as for sum can be applied to show that for\u003C/span> \u003Ctt>prod(e1,e2)\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>exec(T(e),[]) = I(e)\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Ctt>quot(e1, e2)\u003C/tt>\u003Cspan class=\"html-italic\">: Again, a similar reasoning as for sum can be applied to show that for\u003C/span> \u003Ctt>quot(e1,e2), exec(T(e),[]) = I(e)\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">In each case, we assume that the inductive hypothesis holds for the subexpressions, and we prove that it holds for the larger expression. Therefore, by induction, the claim holds for all expressions. Thus, we have proved the claim: For all expressions e,\u003C/span> \u003Ctt>exec(T(e),[]) = I(e)\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: In the first step of your proof for\u003C/span> \n \u003Ctt>sum(e1,e2)\u003C/tt> \n \u003Cspan class=\"html-italic\">you write:\u003C/span>\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Ctt>exec(T(sum(e1,e2)),[]) = exec(T(e1)@T(e2)@[add],[])\u003C/tt>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n\u003Cspan class=\"html-italic\">However,\u003C/span> \u003Ctt>T(sum(e1,e2))\u003C/tt>\u003Cspan class=\"html-italic\">is defined as\u003C/span> \u003Ctt>T(e2)@T(e1)@[add]\u003C/tt>\u003Cspan class=\"html-italic\">, not as\u003C/span> \u003Ctt>T(e1)@T(e2)@[add]\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: I apologize for that mistake. You’re absolutely right. Here is the corrected proof:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">For\u003C/span> \n \u003Ctt>sum(e1, e2)\u003C/tt> \n \u003Cspan class=\"html-italic\">: For\u003C/span> \n \u003Ctt>T(sum(e1,e2))\u003C/tt> \n \u003Cspan class=\"html-italic\">, we have\u003C/span> \n \u003Ctt>T(sum(e1,e2)) = T(e2)@T(e1)@[add]\u003C/tt> \n \u003Cspan class=\"html-italic\">. Therefore,\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>exec(T(sum(e1,e2)),[]) =\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>exec(T(e2)@T(e1)@[add],[]) =\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>exec(T(e1),exec(T(e2),[])) + exec([add],[]) =\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>exec(T(e1),[]) + exec(T(e2),[]) (by definition of exec).\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">For\u003C/span> \u003Ctt>I(sum(e1,e2))\u003C/tt>\u003Cspan class=\"html-italic\">, we have\u003C/span> \u003Ctt>I(sum(e1,e2)) = I(e1) + I(e2)\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">By the inductive hypothesis,\u003C/span> \n \u003Ctt>exec(T(e1),[]) = I(e1)\u003C/tt> \n \u003Cspan class=\"html-italic\">and\u003C/span> \n \u003Ctt>exec(T(e2),[]) = I(e2)\u003C/tt> \n \u003Cspan class=\"html-italic\">. So\u003C/span>\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Ctt>exec(T(e1),[]) + exec(T(e2),[]) = I(e1) + I(e2)\u003C/tt>\n .\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n\u003Cspan class=\"html-italic\">Thus, we have shown that for\u003C/span> \u003Ctt>sum(e1,e2), exec(T(e),[]) = I(e)\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">The proof for the other cases (\u003C/span>\u003Ctt>diff\u003C/tt> \n \u003Cspan class=\"html-italic\">,\u003C/span> \n \u003Ctt>prod\u003C/tt> \n \u003Cspan class=\"html-italic\">,\u003C/span> \n \u003Ctt>quot\u003C/tt> \n \u003Cspan class=\"html-italic\">) proceeds analogously. The ordering of the elements in the\u003C/span> \n \u003Ctt>@\u003C/tt> \n \u003Cspan class=\"html-italic\">operation in the definition of\u003C/span> \n \u003Ctt>T\u003C/tt> \n \u003Cspan class=\"html-italic\">matters because the virtual machine processes its commands from left to right, but the error in my initial proof has now been corrected.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">I appreciate your attention to detail in pointing out this error. Thank you!\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Well, programming and proofs are all about details, aren’t they? Are you now confident that your proof is correct?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: You’re absolutely correct, programming and proofs rely heavily on precision and accuracy. Let’s revisit the proof for sum to ensure its correctness.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>[\u003C/mo>\n \u003Cmo>⋯\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> GPT-4 here rehashes its preceding proof \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>⋯\u003C/mo>\n \u003Cmo>]\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">I hope this more detailed explanation has clarified the proof and increased your confidence in its correctness. Indeed, details are everything in programming and proofs. Thank you again for your attention to detail!\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: Well, in the second step in your above proof for\u003C/span> \n \u003Ctt>sum(e1,e2)\u003C/tt> \n \u003Cspan class=\"html-italic\">you wrote:\u003C/span>\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Ctt>exec(T(e2)@T(e1)@[add],[]) = exec(T(e1),exec(T(e2),[])) + exec([add],[])\u003C/tt>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n\u003Cspan class=\"html-italic\">Can you explain how and why this is a valid step? In particular, can you explain the meaning of\u003C/span> \n \u003Ctt>exec([add],[])\u003C/tt> \n \u003Cspan class=\"html-italic\">?\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: I apologize for the confusion. The step you mentioned is indeed incorrect.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">A correct sequence of steps should follow the rules of our defined\u003C/span> \n \u003Ctt>exec\u003C/tt> \n \u003Cspan class=\"html-italic\">function. Here’s the corrected sequence:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">We start from the definition of the translation\u003C/span> \n \u003Ctt>T\u003C/tt> \n \u003Cspan class=\"html-italic\">:\u003C/span> \n \u003Ctt>T(sum(e1,e2)) = T(e2) @ T(e1) @ [add]\u003C/tt>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Let’s execute this program:\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>exec(T(sum(e1,e2)), []) = exec(T(e2) @ T(e1) @ [add], [])\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">According to the properties of the\u003C/span> \n \u003Ctt>exec\u003C/tt> \n \u003Cspan class=\"html-italic\">function, we should break this down into:\u003C/span>\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Ctt>exec(T(e2) @ T(e1) @ [add], []) = exec(T(e1) @ [add], exec(T(e2), []))\u003C/tt>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n\u003Cspan class=\"html-italic\">Then, again applying the properties of\u003C/span> \n \u003Ctt>exec\u003C/tt> \n \u003Cspan class=\"html-italic\">:\u003C/span>\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Ctt>exec(T(e1)@[add], exec(T(e2),[])) = exec([add],(exec(T(e1),exec(T(e2),[])))\u003C/tt>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n\u003Cspan class=\"html-italic\">The\u003C/span> \n \u003Ctt>exec([add], ...)\u003C/tt> \n \u003Cspan class=\"html-italic\">function interprets the\u003C/span> \n \u003Ctt>[add]\u003C/tt> \n \u003Cspan class=\"html-italic\">command on the stack. If we denote\u003C/span> \n \u003Ctt>exec(T(e2), [])\u003C/tt> \n \u003Cspan class=\"html-italic\">as\u003C/span> \n \u003Ctt>S\u003C/tt> \n \u003Cspan class=\"html-italic\">(which is the result stack after executing\u003C/span> \n \u003Ctt>T(e2))\u003C/tt> \n \u003Cspan class=\"html-italic\">and\u003C/span> \n \u003Ctt>exec(T(e1), S)\u003C/tt> \n \u003Cspan class=\"html-italic\">as\u003C/span> \n \u003Ctt>m::n::S’\u003C/tt> \n \u003Cspan class=\"html-italic\">(which is the result stack after executing\u003C/span> \n \u003Ctt>T(e1)\u003C/tt> \n \u003Cspan class=\"html-italic\">on\u003C/span> \n \u003Ctt>S\u003C/tt> \n \u003Cspan class=\"html-italic\">, where\u003C/span> \n \u003Ctt>m\u003C/tt> \n \u003Cspan class=\"html-italic\">and\u003C/span> \n \u003Ctt>n\u003C/tt> \n \u003Cspan class=\"html-italic\">are the top two integers on the stack), then:\u003C/span>\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Ctt>exec([add], m::n::S’) = exec([], (n+m)::S’) = n + m\u003C/tt>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n\u003Cspan class=\"html-italic\">This is in line with the interpreter semantics\u003C/span> \u003Ctt>I(sum(e1,e2)) = I(e1) + I(e2)\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">The correct proof should be that\u003C/span> \n \u003Ctt>exec([add], m::n::S’)\u003C/tt> \n \u003Cspan class=\"html-italic\">in the compiled execution is equivalent to\u003C/span> \n \u003Ctt>I(e1) + I(e2)\u003C/tt> \n \u003Cspan class=\"html-italic\">in the interpreter semantics, which means\u003C/span> \n \u003Ctt>exec(T(e), []) = I(e)\u003C/tt> \n \u003Cspan class=\"html-italic\">for the\u003C/span> \n \u003Ctt>sum\u003C/tt> \n \u003Cspan class=\"html-italic\">operation.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">I apologize for the previous error and confusion.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: You claim that we should break this down into:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>exec(T(e2) @ T(e1) @ [add], []) = exec(T(e1) @ [add], exec(T(e2), [])).\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">However, this doesn’t make sense, meaning that it’s ill-typed. The\u003C/span> \n \u003Ctt>exec\u003C/tt> \n \u003Cspan class=\"html-italic\">function requires a stack as the second argument but produces a number as its result. Hence,\u003C/span> \n \u003Ctt>exec(T(e1) @ [add], exec(T(e2), []))\u003C/tt> \n \u003Cspan class=\"html-italic\">is like writing\u003C/span> \n \u003Ctt>exec(T(e1) @ [add], 34)\u003C/tt> \n \u003Cspan class=\"html-italic\">, which is nonsense.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: I apologize for the confusion in my previous responses. You’re right, the steps of the proof were not accurately explained. I appreciate your patience. The\u003C/span> \n \u003Ctt>exec\u003C/tt> \n \u003Cspan class=\"html-italic\">function indeed requires a list of commands and a stack as arguments, not an integer. Here’s a correct proof:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">To prove our claim, we need to show for each case that\u003C/span> \u003Ctt>exec(T(e),[]) = I(e)\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Base Case: \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>[\u003C/mo>\n \u003Cmo>⋯\u003C/mo>\n \u003Cmspace width=\"0.277778em\" />\n \u003Cmrow>\n \u003Cmi>s\u003C/mi>\n \u003Cmi>a\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmi>e\u003C/mi>\n \u003Cmi>a\u003C/mi>\n \u003Cmi>s\u003C/mi>\n \u003Cmi>b\u003C/mi>\n \u003Cmi>e\u003C/mi>\n \u003Cmi>f\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>r\u003C/mi>\n \u003Cmi>e\u003C/mi>\n \u003C/mrow>\n \u003Cmspace width=\"0.277778em\" />\n \u003Cmo>⋯\u003C/mo>\n \u003Cmo>]\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Assume the hypothesis holds for expressions\u003C/span> \u003Ctt>e1\u003C/tt>\u003Cspan class=\"html-italic\">and\u003C/span> \u003Ctt>e2\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">For\u003C/span> \n \u003Ctt>sum(e1, e2)\u003C/tt> \n \u003Cspan class=\"html-italic\">:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">We know that\u003C/span> \u003Ctt>T(sum(e1,e2)) = T(e2) @ T(e1) @ [add]\u003C/tt>\u003Cspan class=\"html-italic\">, and by our inductive hypothesis, we know that executing\u003C/span> \u003Ctt>T(e2)\u003C/tt>\u003Cspan class=\"html-italic\">and\u003C/span> \u003Ctt>T(e1)\u003C/tt>\u003Cspan class=\"html-italic\">yields\u003C/span> \u003Ctt>I(e2)\u003C/tt>\u003Cspan class=\"html-italic\">and\u003C/span> \u003Ctt>I(e1)\u003C/tt>\u003Cspan class=\"html-italic\">respectively. After executing\u003C/span> \u003Ctt>T(e2)\u003C/tt>\u003Cspan class=\"html-italic\">, the stack will be\u003C/span> \u003Ctt>I(e2)::S\u003C/tt>\u003Cspan class=\"html-italic\">for some stack\u003C/span> \u003Ctt>S\u003C/tt>\u003Cspan class=\"html-italic\">, and after executing\u003C/span> \u003Ctt>T(e1)\u003C/tt>\u003Cspan class=\"html-italic\">on this stack, the stack will be\u003C/span> \u003Ctt>I(e1)::I(e2)::S\u003C/tt>\u003Cspan class=\"html-italic\">. Then, the “add” operation pops these two off the stack, adds them, and pushes the result onto the stack, so the stack is now\u003C/span> \u003Ctt>(I(e1)+I(e2))::S\u003C/tt>\u003Cspan class=\"html-italic\">, which is what interpreting\u003C/span> \u003Ctt>sum(e1,e2)\u003C/tt>\u003Cspan class=\"html-italic\">yields. Therefore, for all\u003C/span> \u003Ctt>e1\u003C/tt>\u003Cspan class=\"html-italic\">and\u003C/span> \u003Ctt>e2\u003C/tt>\u003Cspan class=\"html-italic\">,\u003C/span> \u003Ctt>exec(T(sum(e1,e2)),[]) = I(sum(e1,e2))\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">The proofs for\u003C/span> \n \u003Ctt>diff(e1,e2)\u003C/tt> \n \u003Cspan class=\"html-italic\">,\u003C/span> \n \u003Ctt>prod(e1,e2)\u003C/tt> \n \u003Cspan class=\"html-italic\">, and\u003C/span> \n \u003Ctt>quot(e1,e2)\u003C/tt> \n \u003Cspan class=\"html-italic\">follow the same pattern, replacing the “add” command and the corresponding mathematical operation with “sub”, “mult”, and “div” and their corresponding operations.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">This shows that our claim holds for all types of expressions in our abstract grammar, and completes the induction proof. The key to this proof is the fact that\u003C/span> \n \u003Ctt>T\u003C/tt> \n \u003Cspan class=\"html-italic\">constructs the sequence of commands in such a way that the operands of each operation are correctly placed onto the stack for the operation to consume, mirroring the evaluation of the corresponding expression.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">Again, I apologize for the confusion in my previous responses and thank you for pointing out the errors.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: You write “After executing\u003C/span> \n \u003Ctt>T(e2)\u003C/tt> \n \u003Cspan class=\"html-italic\">, the stack will be\u003C/span> \n \u003Ctt>I(e2)::S\u003C/tt> \n \u003Cspan class=\"html-italic\">for some stack\u003C/span> \n \u003Ctt>S\u003C/tt> \n \u003Cspan class=\"html-italic\">, and after executing\u003C/span> \n \u003Ctt>T(e1)\u003C/tt> \n \u003Cspan class=\"html-italic\">on this stack, the stack will be\u003C/span> \n \u003Ctt>I(e1)::I(e2)::S\u003C/tt> \n \u003Cspan class=\"html-italic\">.” Can you clarify where\u003C/span> \n \u003Ctt>S\u003C/tt> \n \u003Cspan class=\"html-italic\">is coming from? As far as I could tell your proof is trying to derive the following:\u003C/span>\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Ctt>exec(T(sum(e1,e2)),[]) = I(sum(e1,e2))\u003C/tt>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n\u003Cspan class=\"html-italic\">The only stack I see here is the empty stack\u003C/span> \u003Ctt>[]\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: I apologize for the confusion. You’re right, in our case, we’re starting with an empty stack. The \"S\" was used to denote a general stack, but in this context, it can be misleading.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Let’s revise the explanation for the\u003C/span> \n \u003Ctt>sum\u003C/tt> \n \u003Cspan class=\"html-italic\">case: For\u003C/span> \n \u003Ctt>sum(e1, e2)\u003C/tt> \n \u003Cspan class=\"html-italic\">:\u003C/span> \n \u003Ctt>T(sum(e1,e2)) = T(e2) @ T(e1) @ [add]\u003C/tt> \n \u003Cspan class=\"html-italic\">. By the definition of\u003C/span> \n \u003Ctt>exec\u003C/tt> \n \u003Cspan class=\"html-italic\">, we have:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>exec(T(e2) @ T(e1) @ [add], []) = exec(T(e1) @ [add], exec(T(e2), []))\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">After the execution of\u003C/span> \n \u003Ctt>T(e2)\u003C/tt> \n \u003Cspan class=\"html-italic\">, the stack becomes\u003C/span> \n \u003Ctt>[I(e2)]\u003C/tt> \n \u003Cspan class=\"html-italic\">(since we’re starting with an empty stack). Then, we have:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>exec(T(e1) @ [add], [I(e2)]) = exec([add], exec(T(e1), [I(e2)]))\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">After the execution of\u003C/span> \n \u003Ctt>T(e1)\u003C/tt> \n \u003Cspan class=\"html-italic\">on the stack\u003C/span> \n \u003Ctt>[I(e2)]\u003C/tt> \n \u003Cspan class=\"html-italic\">, the stack becomes\u003C/span> \n \u003Ctt>[I(e1), I(e2)]\u003C/tt> \n \u003Cspan class=\"html-italic\">. Therefore,\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Ctt>exec([add], [I(e1), I(e2)]) = exec([], [I(e1) + I(e2)]) = I(e1) + I(e2)\u003C/tt>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Which is equal to\u003C/span> \u003Ctt>I(sum(e1, e2))\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">Therefore, we have proved that\u003C/span> \u003Ctt>exec(T(sum(e1,e2)),[]) = I(sum(e1,e2))\u003C/tt>\n .\n \u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cspan class=\"html-italic\">The same line of reasoning applies for the remaining operations (\u003C/span> \n \u003Ctt>diff\u003C/tt> \n \u003Cspan class=\"html-italic\">,\u003C/span> \n \u003Ctt>prod\u003C/tt> \n \u003Cspan class=\"html-italic\">,\u003C/span> \n \u003Ctt>quot\u003C/tt> \n \u003Cspan class=\"html-italic\">). This means our compiler\u003C/span> \n \u003Ctt>T\u003C/tt> \n \u003Cspan class=\"html-italic\">properly translates an expression into a sequence of commands that, when executed, produce the same result as interpreting the expression. Thus, our claim holds.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cspan class=\"html-italic\">I appreciate your patience, and I’m sorry for any confusion caused by my previous explanations.\u003C/span>\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>KA\u003C/b>\u003Cspan class=\"html-italic\">: No, you made the same typing error again: Your claim\u003C/span>\u003Cdiv class=\"html-disp-formula-info\" id>\n \u003Cdiv class=\"f\">\n \u003Ctt>exec(T(e2) @ T(e1) @ [add], []) = exec(T(e1) @ [add], exec(T(e2), []))\u003C/tt>\n \u003C/div>\n \u003Cdiv class=\"l\">\n \n \u003C/div>\n \u003C/div>\n\u003Cspan class=\"html-italic\">doesn’t make sense given that\u003C/span> \n \u003Ctt>exec\u003C/tt> \n \u003Cspan class=\"html-italic\">produces a number, not a stack.\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\n\u003Cb>GPT-4\u003C/b>\u003Cspan class=\"html-italic\">: I apologize for the repeated confusion in my previous explanations. You’re absolutely correct, my claim was incorrect and I appreciate your patience. Here’s the corrected line of reasoning:\u003C/span>\n\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>[\u003C/mo>\n \u003Cmo>⋯\u003C/mo>\n \u003Cmrow>\n \u003Cmi>Y\u003C/mi>\n \u003Cmi>e\u003C/mi>\n \u003Cmi>t\u003C/mi>\n \u003Cmo> \u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>t\u003C/mi>\n \u003Cmi>h\u003C/mi>\n \u003Cmi>e\u003C/mi>\n \u003Cmi>r\u003C/mi>\n \u003Cmo> \u003C/mo>\n \u003Cmi>a\u003C/mi>\n \u003Cmi>t\u003C/mi>\n \u003Cmi>t\u003C/mi>\n \u003Cmi>e\u003C/mi>\n \u003Cmi>m\u003C/mi>\n \u003Cmi>p\u003C/mi>\n \u003Cmi>t\u003C/mi>\n \u003Cmo> \u003C/mo>\n \u003Cmi>w\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003Cmi>t\u003C/mi>\n \u003Cmi>h\u003C/mi>\n \u003Cmo> \u003C/mo>\n \u003Cmi>t\u003C/mi>\n \u003Cmi>y\u003C/mi>\n \u003Cmi>p\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003Cmi>g\u003C/mi>\n \u003Cmo> \u003C/mo>\n \u003Cmi>e\u003C/mi>\n \u003Cmi>r\u003C/mi>\n \u003Cmi>r\u003C/mi>\n \u003Cmi>o\u003C/mi>\n \u003Cmi>r\u003C/mi>\n \u003Cmi>s\u003C/mi>\n \u003C/mrow>\n \u003Cmo>⋯\u003C/mo>\n \u003Cmo>]\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>\u003C/div>\n\u003Cdiv class=\"html-p\">As usual, GPT-4 goes through the motions on autopilot, displaying a superficial—if inordinately confident—understanding of the reasoning, but a closer inspection reveals fundamental errors. Outside of errors such as mixing the order of arguments and generating ill-typed/nonsensical expressions, the main issue here is that the proof cannot go through as is because the inductive hypothesis needs to be strengthened. This requires the introduction of an explicitly quantified stack variable \u003Cspan class=\"html-italic\">S\u003C/span> in the correctness result. That modified result can then be derived by a similar structural induction. The initial correctness theorem can finally be obtained as a trivial corollary of the more general result.\u003C/div>\n\u003Cdiv class=\"html-p\">What is more concerning than the inability to strengthen the inductive hypothesis (which is a genuinely tall order, after all, as it requires considerable experience and proof skill) is the inability of GPT-4 to detect its own errors, both flagrant ones (such as type errors) and more subtle ones. In fact, if we make the innocent mistake of compiling and concatenating subexpressions from left to right, e.g., by defining \u003Ctt>T(sum(e1,e2))\u003C/tt> as \u003Ctt>T(e1)@T(e2)@[add]\u003C/tt> (and likewise for the other operators), correctness no longer holds. But GPT-4 happily goes on to claim that the compiler is correct and generates a plausible-sounding but incorrect “proof” for it, oblivious to the fact that \u003Ctt>T(e1)@T(e2)@[op]\u003C/tt> and \u003Ctt>T(e2)@T(e1)@[op]\u003C/tt> have drastically different effects for noncommutative operations (such as division).\u003C/div>\u003C/section>\u003C/section>\u003Csection id=\"sec4-preprints-81518\" type=\"conclusions\">\u003Ch2 data-nested=\"1\" id=\"preprints-h2-4\"> 4. Conclusions\u003C/h2>\n\u003Cdiv class=\"html-p\">\n\u003Ca href=\"#sec3-preprints-81518\" class=\"html-sec\">Section 3\u003C/a> paints a bleak picture of GPT-4’s reasoning ability. It shows that the model is plagued by internal inconsistency, an inability to correctly apply elementary reasoning techniques, and a lack of understanding of concepts that play a fundamental role in reasoning (such as the material conditional). These problems can be loosely viewed as forms of hallucination, but as pointed out in the January article, they present a fundamentally different type of challenge from empirical hallucination, because empirical hallucination concerns \u003Cspan class=\"html-italic\">this particular world\u003C/span> whereas logical properties and relations (such as consistency and entailment) must apply to \u003Cspan class=\"html-italic\">all possible worlds\u003C/span>. It is not unreasonable to believe that search engines and knowledge graphs, using techniques such as retrieval augmentation, can act as guardrails to constrain LLMs from confabulating empirical truths. But ensuring that LLM outputs are \u003Cspan class=\"html-italic\">internally consistent\u003C/span> and \u003Cspan class=\"html-italic\">logically correct\u003C/span> answers to arbitrary problems, especially logico-mathematical problems (and a lot of coding problems fall under this category\u003Ca href=\"#fn027-preprints-81518\" class=\"html-fn\">27\u003C/a>), is a \u003Cspan class=\"html-italic\">much\u003C/span> harder problem. There is nothing to be retrieved from the web or from a knowledge base in response to a brand new problem (and even if there were, there would still be no guarantee of correctness or consistency) that could serve as a sandbox for the LLM.\u003C/div>\n\u003Cdiv class=\"html-p\">Could LLMs make progress by outsourcing reasoning problems to external systems? That might work for toy problems where the type of reasoning needed is obvious and can be handled by a single call to an external system, although even in those cases the LLM would have to (a) decide \u003Cspan class=\"html-italic\">which\u003C/span> reasoning system is most appropriate; \u003Ca href=\"#fn028-preprints-81518\" class=\"html-fn\">28\u003C/a> (b) decide whether the problem is indeed simple enough that it can be handled by the chosen system in one fell swoop; (c) correctly translate the problem into whatever formal notation is used by the chosen reasoner; and eventually also (d) translate the reasoner’s output into appropriate text. Even these tasks are far from straightforward.\u003Ca href=\"#fn029-preprints-81518\" class=\"html-fn\">29\u003C/a> But the real challenge lies in harder problems that call for the right type of formulation (which is a craft by itself), decomposition, iteration, heuristics, and repeated calls to external systems. After all, automated reasoning systems, particularly those for expressive logics, are themselves of limited power, precisely due to the computational complexity issues mentioned in the introduction. That is why many computer-based proof efforts to this day are guided by humans, with automated reasoners only filling in tedious details at the leaves of the proof tree. The challenges here are similar to those for the general “plug-in” approach discussed in \u003Ca href=\"#sec3dot1-preprints-81518\" class=\"html-sec\">Section 3.1\u003C/a>. Tackling complex problems requires planning, and planning itself requires reasoning.\u003C/div>\n\u003Cdiv class=\"html-p\">Given that GPT-4 is currently the most capable LLM, I draw three main conclusions from these findings:\u003C/div>\n\u003Cdiv class=\"html-p\">\u003Cul class=\"html-order\">\n\u003Cli>\u003Cdiv class=\"html-p\">Use of generative AI in software development (or in science and engineering in general) for anything other than tedious tasks (as a sort of turbo-charged autocomplete for knowledge-heavy coding questions) is fraught with serious risks. Normative standards of correctness are of paramount importance in these fields, and current LLMs cannot meet such standards. Just like generative AI is already starting to \u003Ca href=\"https://www.technologyreview.com/2023/06/26/1075504/junk-websites-filled-with-ai-generated-text-are-pulling-in-money-from-programmatic-ads/\" target=\"_blank\">pollute the web with badly written ads\u003C/a>,\u003Ca href=\"#fn030-preprints-81518\" class=\"html-fn\">30\u003C/a> it has the potential to proliferate buggy code at scale.\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">If LLM reasoning continues to improve, rigorous proof checking is likely to become increasingly important. Confidence in the correctness of a system’s reasoning is imperative for applications, particularly in science, medicine, and engineering, and proof checking is a technology that can deliver such confidence. This approach could be implemented by requiring LLMs to formalize their reasoning (express it in a symbolic notation that is amenable to proof checking), or potentially by training other LLMs to check a stretch of reasoning expressed in natural language.\u003C/div>\u003C/li>\n\u003Cli>\u003Cdiv class=\"html-p\">As things stand, dystopian scenarios involving a rogue AI that subjugates humankind, or even other humans using AI for sinister purposes, are exceedingly far-fetched, often to the point of absurdity.\u003Ca href=\"#fn031-preprints-81518\" class=\"html-fn\">31\u003C/a> When the most advanced AI system cannot tell left from right (literally, see \u003Ca href=\"#sec3dot12-preprints-81518\" class=\"html-sec\">Section 3.12\u003C/a>), it is at best comically premature to call for policies and institutions to protect humanity from it or its descendants (often by appeal to the latest “scaling law”). At worst, it is a misuse of human time and capital that could be better channeled into addressing much more pressing challenges.\u003C/div>\u003C/li>\n\u003C/ul>\u003C/div>\n\u003Cdiv class=\"html-p\">Inevitably, some will say that these results are “cherry-picking” data. But that would indicate a misconception of what cherry-picking is about and when it is a relevant consideration. We are not evaluating a statistical claim over a population of individuals. Cherry-picking, insofar as it underscores certain pieces of evidence while ignoring other divergent findings, can be perfectly innocuous—and indeed \u003Cspan class=\"html-italic\">necessary\u003C/span>—depending on the logical structure of the proposition in question and on the overall context. Debugging a computer program with a view to discovering and understanding its weaknesses, trying to falsify a scientific theory, kicking the tires of a new car, trying to find countermodels to a putative theorem, all of these activities are fundamentally cherry-picking (though “lemon-picking” might be more apt), and there is nothing wrong with any of them. If I find that the car I’m thinking of buying has a flat tire, it won’t carry much weight for the dealer to protest that I’m cherry-picking the data, and that I should take into account how beautifully inflated the other three tires are (that’s a 75% success rate after all). Likewise, applications in science, medicine, and engineering, particularly software engineering, have stringent standards. Just as we don’t want a bridge that is 90% likely to stand up, we need sorting algorithms that work on all inputs, not just most of them, we need Amazon’s cart to charge customers the right amount every time, not just most of the time, and so on. Computation-heavy and reasoning-heavy applications are not like recommendation engines. They need to be \u003Cspan class=\"html-italic\">sound\u003C/span>.\u003C/div>\n\u003Cdiv class=\"html-p\">The bone of contention here is the thesis that GPT-4 is capable of reasoning. This claim can be understood in two ways. The weak interpretation is that GPT-4 has the same functional reasoning competence as an average human reasoner. The strong interpretation is that GPT-4 can reason well enough to be used as an off-the-shelf component in practical applications in science, medicine, and engineering. The evidence presented in this article refutes both interpretations. \u003Ca href=\"#sec3-preprints-81518\" class=\"html-sec\">Section 3\u003C/a> lists a significant number of diverse but elementary reasoning problems (some to the point of triviality) on which GPT-4 doesn’t simply fail, but repeatedly reveals itself to be deeply confused about key reasoning concepts.\u003C/div>\n\u003Cdiv class=\"html-p\">Performance statistics on appropriate reasoning datasets could also be informative, but, as stressed in the introduction, such datasets must be constructed with extraordinary care. To the best of my knowledge, the only recent work that focuses specifically on evaluating the reasoning ability of GPT-4 is an April paper by Liu et al. [\u003Ca href=\"#B7-preprints-81518\" class=\"html-bibr\">7\u003C/a>]. However, their tests are largely based on pre-existing benchmarks (LogiQA, ReClor, ConTRoL, MED, ConjNLI, and TaxiNLI). The only two “out of distribution” datasets are AR-LSAT, a set of analytical reasoning LSAT questions released in 2022; and LogiQA, which contains questions from the 2022 Chinese Civil Servant Exam. However, these appear to be quite similar to other datasets that predate 2021.\u003C/div>\n\u003Cdiv class=\"html-p\">Moreover, all of these tests are multiple-choice questions or binary classification problems. This is problematic because, as stressed in the introduction, deductive reasoning is an inherently generative activity, whereby the reasoner emits a \u003Cspan class=\"html-italic\">derivation\u003C/span> of a conclusion that can be understood as a rationale or an explanation; it is not a simple discriminative task. The reasoner must be able to produce a sequence of steps that are appropriately connected to one another via the right logical relations. But derivations expressed in natural language are not easy to evaluate automatically, as all available metrics that can be computed by machine (such as BLEU, ROUGE, and even semantic-similarity measures based on embeddings) are entirely unsuitable for that purpose. This means that LLM outputs have to be scrutinized manually, which is infeasible at scale. Accordingly, smaller-scale but deeper manual investigations, such as the one undertaken in this article, will be necessary in gaining better insight into the reasoning abilities of LLMs.\u003C/div>\u003C/section>\n \n \u003Csection id=\"html-references_list\">\u003Ch2 id=\"preprints-h2-5\">References\u003C/h2>\n\u003Col class=\"html-xx\">\n\u003Cli id=\"B1-preprints-81518\" class=\"html-x\" data-content=\"1.\">Arkoudas, K. and Musser, D., \u003Cspan class=\"html-italic\">Fundamental Proof Methods in Computer Science\u003C/span>, MIT Press, 2017.\u003C/li>\n\u003Cli id=\"B2-preprints-81518\" class=\"html-x\" data-content=\"2.\">Barwise, J. and Perry, J., \u003Cspan class=\"html-italic\">Situations and Attitudes\u003C/span>, MIT Press, 1983.\u003C/li>\n\u003Cli id=\"B3-preprints-81518\" class=\"html-x\" data-content=\"3.\">Karpas, E., Abend, O., Belinkov, Y., Lenz, B., Lieber, O., Ratner, N., \u003Cspan class=\"html-italic\">…\u003C/span>, Tenenholtz, M., \u003Cspan class=\"html-italic\">MRKL Systems: A modular, neuro-symbolic architecture that combines large language models, external knowledge sources and discrete reasoning\u003C/span>, 2022.\u003C/li>\n\u003Cli id=\"B4-preprints-81518\" class=\"html-x\" data-content=\"4.\">Yao, S., Zhao, J., Yu, D., Du, N., Shafran, I., Narasimhan, K., Cao, Y., \u003Cspan class=\"html-italic\">ReAct: Synergizing Reasoning and Acting in Language Models\u003C/span>, \u003Ctt>\u003Ca href=\"https://arxiv.org/abs/2210.03629\" target=\"_blank\">https://arxiv.org/abs/2210.03629\u003C/a>\u003C/tt>, 2023. [\u003Ca href=\"https://doi.org/10.48550/arXiv.2210.03629\" class=\"cross-ref\" target=\"_blank\" rel=\"noopener noreferrer\">CrossRef\u003C/a>]\u003C/li>\n\u003Cli id=\"B5-preprints-81518\" class=\"html-x\" data-content=\"5.\">Planken, L., \u003Cspan class=\"html-italic\">Temporal Reasoning Problems and Algorithms for Solving Them: Literature Survey\u003C/span>, 2008.\u003C/li>\n\u003Cli id=\"B6-preprints-81518\" class=\"html-x\" data-content=\"6.\">McCoy, T., Pavlick, E., Linzen, T., \u003Cspan class=\"html-italic\">Right for the Wrong Reasons: Diagnosing Syntactic Heuristics in Natural Language Inference\u003C/span>, Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, 2019.\u003C/li>\n\u003Cli id=\"B7-preprints-81518\" class=\"html-x\" data-content=\"7.\">Liu H., Ning R., Teng, Z., Liu, J., Zhou, Q., Zhang, Y., \u003Cspan class=\"html-italic\">Evaluating the Logical Reasoning Ability of ChatGPT and GPT-4\u003C/span>, 2023. [\u003Ca href=\"https://doi.org/10.48550/arXiv.2304.03439\" class=\"cross-ref\" target=\"_blank\" rel=\"noopener noreferrer\">CrossRef\u003C/a>]\u003C/li>\n\u003Cli id=\"B8-preprints-81518\" class=\"html-x\" data-content=\"8.\">OpenAI, \u003Cspan class=\"html-italic\">GPT-4 Technical Report\u003C/span>, 2023.\u003C/li>\n\u003Cli id=\"B9-preprints-81518\" class=\"html-x\" data-content=\"9.\">Wang, J., Hu, X., Hou, W., Chen, H., Zheng, R., Wang, Y., \u003Cspan class=\"html-italic\">…\u003C/span>, Xie, X., \u003Cspan class=\"html-italic\">On the Robustness of ChatGPT: An Adversarial and Out-of-distribution Perspective\u003C/span>, 2023. [\u003Ca href=\"https://doi.org/10.48550/arXiv.2302.12095\" class=\"cross-ref\" target=\"_blank\" rel=\"noopener noreferrer\">CrossRef\u003C/a>]\u003C/li>\n\u003Cli id=\"B10-preprints-81518\" class=\"html-xx\" data-content=\"10.\">Niven, T., Kao, H.-Y. \u003Cspan class=\"html-italic\">Probing Neural Network Comprehension of Natural Language Arguments\u003C/span>, Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, 2019. [\u003Ca href=\"https://doi.org/10.48550/arXiv.1907.07355\" class=\"cross-ref\" target=\"_blank\" rel=\"noopener noreferrer\">CrossRef\u003C/a>]\u003C/li>\n\u003Cli id=\"B11-preprints-81518\" class=\"html-xx\" data-content=\"11.\">Johnson-Laird, P.N., \u003Cspan class=\"html-italic\">How We Reason\u003C/span>, Oxford University Press, 2006.\u003C/li>\n\u003C/ol>\u003C/section>\u003Csection class=\"html-fn_group\">\u003Ctable>\n\u003Ctr id=\"fn001-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">1\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">A modified version of that is being published in the journal \u003Cspan class=\"html-italic\">Philosophy & Technology\u003C/span>.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn002-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">2\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">The notion of an emergent property is clear enough, at least at a high enough level. What is not clear is the relationship between such properties and LLM architectures, their basic configurations (number of parameters, compute budget, dataset size, and so on), and more importantly, important tasks such as reasoning.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn003-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">3\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">Or with perfect precision and recall, to put it—more loosely—in ML-like terms.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn004-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">4\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">Of which there are many: propositional logic, the two-variable fragment of first-order logic, the Ackerman fragment, the guarded fragment, various quantifier-prefix fragments, and so on.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn005-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">5\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">Understanding that structure and rigorously characterizing its relationship with algorithm performance (e.g., via different problem parameterizations, such as clause/variable ratios in the case of SAT) is a key open problem in theoretical computer science, but that is another matter.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn006-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">6\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">Humans do not seem to solve problems by predicting the most likely sequence of tokens to generate. They think, explore, experiment, engage in protracted conversation with the people who posed the problem (sometimes over weeks, months, or even years), refine, generalize, come up with new concepts and terminology, prove results, make and refute conjectures, apply heuristics, execute algorithms, analyze and synthesize, and iterate. But \u003Cspan class=\"html-italic\">how\u003C/span> solutions are generated is one thing and \u003Cspan class=\"html-italic\">what\u003C/span> solutions are generated is another, and that’s why it’s not incoherent to speak of a model whose reasoning performance is roughly at the same level as that of an average human engineer. Such a claim can be understood operationally, to mean that a given LLM is able to produce roughly the same solutions that we might reasonably expect an average human engineer to produce (though obviously on a very different time scale).\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn007-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">7\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">According to the analysis carried out by the \u003Ca href=\"https://hitz-zentroa.github.io/lm-contamination/\" target=\"_blank\">\u003Ctt>lm-contamination index\u003C/tt>\u003C/a>, well-known NLP datasets such as Squad, CoNLL03, MNLI, and others, are indeed contaminated, while several others are at best suspicious.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn008-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">8\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">In fact, the substring checks carried out by OpenAI were not even applied on the entire problem instance, only on 3 randomly selected substrings of 50 characters each. This is not enough to ensure disjointness for long (or even moderately long) problems, which are quite common in tests like the UBE (Uniform Bar Exam).\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn009-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">9\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">Models have been shown to leverage the presence of certain cue words (especially negation words) and to formulate quick-and-dirty (i.e., unsound) heuristics such as lexical overlap, subsequence, and constituency [\u003Ca href=\"#B6-preprints-81518\" class=\"html-bibr\">6\u003C/a>]. Most of these results are from 2019 and revolve around BERT, but more recent work [\u003Ca href=\"#B9-preprints-81518\" class=\"html-bibr\">9\u003C/a>] has shown that while larger foundational models such as ChatGPT are more robust to input perturbations and OOD (out-of-distribution) samples, these continue to be challenges, suggesting that even ChatGPT-scale models learn unsound shortcuts.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn010-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">10\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">Here we understood premises and conclusions as syntactic objects (sentences or diagrams), but there are alternative approaches. For instance, a semanticist might think of premises and conclusions as \u003Cspan class=\"html-italic\">propositions\u003C/span>, abstract objects capable of being true or false. A sentence then \u003Cspan class=\"html-italic\">expresses\u003C/span> or \u003Cspan class=\"html-italic\">represents\u003C/span> a proposition. Propositions are handy theoretical entities for many reasons. For example, they can serve as the objects of psychological attitudes such as beliefs and desires. What do I mean when I claim to believe that Obama won the 2012 presidential election? Surely I don’t believe a particular \u003Cspan class=\"html-italic\">sentence\u003C/span>, i.e., a specific syntactic object like “Obama won the 2012 US presidential election” (I). Rather, I believe something about the way the world actually is. That something can be understood as a proposition, a unique entity that can expressed by many different equivalent sentences. Propositions can be cashed out in modal terms, as \u003Cspan class=\"html-italic\">sets of possible worlds\u003C/span> (or as “situations” in situation-theoretic semantics [\u003Ca href=\"#B2-preprints-81518\" class=\"html-bibr\">2\u003C/a>]). A possible world is a way in which things might have been, but described completely, down to the most minute detail (unlike situations, which can be thought of as partial specifications of worlds). So the proposition that Obama won the 2012 US presidential election is identified with the set of all possible worlds in which Obama won that election. This set becomes the \u003Cspan class=\"html-italic\">information content\u003C/span> of sentences such as (I). Propositions can also serve to analyze fundamental semantic notions such as entailment. A set of premises \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>{\u003C/mo>\n \u003Cmsub>\n \u003Cmi>p\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>…\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmsub>\n \u003Cmi>p\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003Cmo>}\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> entails a conclusion \u003Cspan class=\"html-italic\">p\u003C/span> iff the intersection of the sets of possible words represented by all the \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmsub>\n \u003Cmi>p\u003C/mi>\n \u003Cmi>i\u003C/mi>\n \u003C/msub>\n\u003C/semantics>\u003C/math> is a superset of the set of worlds represented by \u003Cspan class=\"html-italic\">p\u003C/span>. This is another way of understanding the claim that the conclusion of a valid deductive argument does not introduce any information that is not already contained in the premises. Note, however, that while the possible-worlds approach to propositions is very powerful, it also suffers from severe defects, as it is notoriously coarse-grained, meaning that it cannot distinguish between propositions that we intuitively regard as quite distinct. This is perhaps easier to see in the case of mathematical truths, which, being necessary (true in all possible worlds), are collapsed into one and the same object, the set of all possible worlds (and dually, of course, all contradictions are identified with the empty set of worlds). As a result, the proposition that 1 + 1 = 2 and Fermat’s theorem become identical, as they have the exact same information content. There have been attempts to address these issues (structured propositions and impossible worlds being two of the most prominent), but the interested reader will have to consult the literature for more details.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn011-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">11\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">This can be made more precise using information-theoretic notions, at least in the case of propositional logic, where we have an infinite supply of formulas that are either atomic (propositional variables) or else Boolean combinations of formulas. Instead of imposing the usual Kolmogorov axioms on a probability measure defined over a set of events (a \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmi>σ\u003C/mi>\n\u003C/semantics>\u003C/math>-field) from a sample space \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmi mathvariant=\"sans-serif\">Ω\u003C/mi>\n\u003C/semantics>\u003C/math>, we impose the same axioms (non-negativity, finite additivity, and the axiom that assigns a measure of 1 to every tautology—the analogue of \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi mathvariant=\"script\">P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi mathvariant=\"sans-serif\">Ω\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>=\u003C/mo>\n \u003Cmn>1\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>) on a probability measure defined over the set of all formulas. Then truth and falsity become the extreme probabilities of 1 and 0, respectively. This allows us to associate a probability \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi mathvariant=\"script\">P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>ϕ\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> with any sentence (event) \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmi>ϕ\u003C/mi>\n\u003C/semantics>\u003C/math>, and hence every sentence \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmi>ϕ\u003C/mi>\n\u003C/semantics>\u003C/math> automatically gets an information content in the usual way: \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmrow>\n \u003Cmi>I\u003C/mi>\n \u003Cmi>C\u003C/mi>\n \u003C/mrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>ϕ\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>=\u003C/mo>\n \u003Cmo>−\u003C/mo>\n \u003Cmo form=\"prefix\">log\u003C/mo>\n \u003Cmi mathvariant=\"script\">P\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>ϕ\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>. To say that the information content of a valid deductive argument with premises \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmo>{\u003C/mo>\n \u003Cmsub>\n \u003Cmi>p\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmo>,\u003C/mo>\n \u003Cmo>…\u003C/mo>\n \u003Cmo>,\u003C/mo>\n \u003Cmsub>\n \u003Cmi>p\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003Cmo>}\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> and conclusion \u003Cspan class=\"html-italic\">p\u003C/span> is zero is simply to say that the conditional \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmsub>\n \u003Cmi>p\u003C/mi>\n \u003Cmn>1\u003C/mn>\n \u003C/msub>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmo>∧\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmo>⋯\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmo>∧\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmsub>\n \u003Cmi>p\u003C/mi>\n \u003Cmi>n\u003C/mi>\n \u003C/msub>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmo>⇒\u003C/mo>\n \u003Cmspace width=\"0.222222em\" />\n \u003Cmi>p\u003C/mi>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> is a tautology. By definition, a tautology \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmi>ϕ\u003C/mi>\n\u003C/semantics>\u003C/math> has probability 1, and therefore \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmrow>\n \u003Cmi>I\u003C/mi>\n \u003Cmi>C\u003C/mi>\n \u003C/mrow>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>ϕ\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003Cmo>=\u003C/mo>\n \u003Cmn>0\u003C/mn>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn012-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">12\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">At this point the reader might ask: If deductive arguments convey zero information, why bother with them? Indeed, if all mathematical proofs are proofs of tautologies, with zero information content, what is their point? The thinking is that arguments with no information content are not useful, so if all deductive arguments (including all mathematical results) have zero information content, then they are not useful. This is, in brief, the so-called “scandal of deduction” (named by parity to the “scandal of induction,” i.e., Hume’s problem of induction). There have not been any widely accepted resolutions of this ostensible paradox. But few of course doubt that mathematical results are actually informative and extend our knowledge. (Surely if we woke up tomorrow and read that someone proved \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>P\u003C/mi>\n \u003Cmo>≠\u003C/mo>\n \u003Cmrow>\n \u003Cmi>N\u003C/mi>\n \u003Cmi>P\u003C/mi>\n \u003C/mrow>\n \u003C/mrow>\n\u003C/semantics>\u003C/math>, that would be tremendously informative.) It’s also clear that the word “information” has a number of informal meanings that are not captured by the canonical definition of information content (as the negative logarithm of probability), and most efforts to resolve the “scandal of deduction” have attempted to formalize distinct notions of informational gain that would render deductive arguments informative.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn013-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">13\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">Several other types of reasoning are often discussed in the literature, such as analogical reasoning (which includes, for instance, case-based reasoning), Bayesian reasoning, causal reasoning, and so on, but these are usually subsumed under one of the three main categories I have described, most often under induction. (But there is no consensus, for instance, some thinkers, from Aristotle to recent authors, have tried to assimilate analogical reasoning under deduction.)\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn014-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">14\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">We are assuming of course that the car model whose mpg we are predicting was \u003Cspan class=\"html-italic\">not\u003C/span> included in the given data, otherwise there would be no prediction or generalization involved.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn015-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">15\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">The training of deep neural networks, too, works by trying to discover values for various weights that are “optimal” for a given training dataset (in that they minimize loss), except that in their case the relationship between the inputs, outputs, and weights can be much more complicated (non-linear) and the training algorithm might not converge to the optimal weight values.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn016-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">16\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">Some desired properties of explanations are obvious. Truth is one of them—a good explanation cannot be based on a false hypothesis. But other desired properties, such as parsimony and generality (explaining as much as possible while assuming as little as possible) are much harder to explicate.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn017-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">17\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">Even from a purely linguistic viewpoint, it doesn’t seem appropriate to say that I have “concluded” or “derived” or “inferred” anything at all in the swan or in the plumber examples. I have simply made a tentative \u003Cspan class=\"html-italic\">hypothesis\u003C/span> (or \u003Cspan class=\"html-italic\">conjecture\u003C/span>), which might be refuted.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn018-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">18\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">In the same way that even the process of discovering deductions is not itself deductive, at least not entirely so. Both are fundamentally search processes, though they are almost certainly informed and generally penetrated by deduction.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn019-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">19\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">This viewpoint assumes a functional-programming stance, but computation can be readily reduced to deduction in any other style of programming (e.g., imperative) by an appropriate axiomatic formulation of the relevant semantics (e.g., operational semantics using stores).\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn020-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">20\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">In addition, of course, different versions of GPT-4 might get deployed at any time.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn021-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">21\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">An unrealistic assumption given that the Internet is filled with an unbounded number of agents (millions of them, from completely arbitrary computer programs to smart-phone apps to travel-booking APIs to games and beyond) that provide an open-ended and constantly changing array of functionality.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn022-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">22\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">By concrete counting I mean counting a number of specific object tokens instantiated in space and time, as in the coins in one’s pocket or the number of lines in a text file. By contrast, abstract counting based on combinatorial principles, search procedures, and logical constraints (like the scheduling problem in \u003Ca href=\"#sec3dot9-preprints-81518\" class=\"html-sec\">Section 3.9\u003C/a>) is indeed a reasoning activity.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn023-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">23\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">In the same way that the numbers 100000 and 1000000 only differ in one zero, but if we are talking about your bank balance that one zero makes a huge difference.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn024-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">24\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">Usually the quantifier variables range explicitly over a sort such as \u003Ctt>Man\u003C/tt>, but this is not essential for the derivation.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn025-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">25\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">Formally, this problem belongs to a class of temporal-reasoning problems literally known as STP (“Simple Temporal Problems”) [\u003Ca href=\"#B5-preprints-81518\" class=\"html-bibr\">5\u003C/a>]. This class is of limited expressivity and there exist very efficient algorithms for solving STPs (e.g., consistency can be decided in \u003Cmath display=\"inline\">\u003Csemantics>\n \u003Cmrow>\n \u003Cmi>O\u003C/mi>\n \u003Cmo>(\u003C/mo>\n \u003Cmi>n\u003C/mi>\n \u003Cmo>·\u003C/mo>\n \u003Cmi>m\u003C/mi>\n \u003Cmo>)\u003C/mo>\n \u003C/mrow>\n\u003C/semantics>\u003C/math> where \u003Cspan class=\"html-italic\">n\u003C/span> is the number of events described in a given STP and \u003Cspan class=\"html-italic\">m\u003C/span> is the number of constraints between the events).\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn026-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">26\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">This particular version is taken from Chapter 18 of the textbook \u003Cspan class=\"html-italic\">Fundamental Proof Methods in Computer Science\u003C/span> by [\u003Ca href=\"#B1-preprints-81518\" class=\"html-bibr\">1\u003C/a>].\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn027-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">27\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">Many shallow coding problems these days are essentially knowledge problems. What library or API can I use to do such and such? What configuration parameters are available and how can they be set? How do I zip or unzip files in Python? How do I read and write JSON or XML? How do I compute quantiles for a frequency table? Knowledge-heavy problems of this sort tend to be widely discussed on the web, and LLMs can be very effective productivity boosters for such problems (at least as long as this data remains freely available to companies such as OpenAI for pretraining purposes, something that might well change in the near future). Even conventional search engines like Google were already effective for these types of problems, prior to LLMs (and remain more effective than LLMs in many cases). But most interesting coding problems are reasoning-heavy. How can I make sure that this program produces \u003Cspan class=\"html-italic\">correct\u003C/span> outputs? How can I improve the asymptotic complexity of this program (where the program might contain many thousands of line of code)? And so on. If we are talking about self-contained and cookie-cutter components, like sorting algorithms, then these questions can often be reduced to knowledge-based questions. But the minute we start straying into unique situations with arbitrary specifications and code bases, we start facing the curse of general reasoning.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn028-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">28\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">Can this be posed as a simple SAT problem? Is it an SMT problem? Does it need quantifier reasoning? If so, is it of the sort that SMT solvers can handle or does it need a full first-order prover? Does the problem quantify over infinite functions or sets? If so, higher-order logic might be needed. Does it have any temporal or epistemic operators that might call for a modal-logic reasoner? And so on.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn029-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">29\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">For instance, a state-of-the-art automated theorem prover might generate a proof, but the proof would be incomprehensible to the LLM user, as it would be expressed in the resolution calculus and would operate on CNF versions of the input formulas. It is an open problem to convert resolution proofs into fluid natural-deduction proofs (e.g., proofs that avoid references to Skolem constants introduced during the CNF conversion).\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn030-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">30\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">A \u003Ca href=\"https://www.wsj.com/articles/chatgpt-already-floods-some-corners-of-the-internet-with-spam-its-just-the-beginning-9c86ea25?st=pp6it5z67lhxnvx&reflink=desktopwebshare_permalink\" target=\"_blank\">recent Wall Street Journal article\u003C/a> interviewed editors who are “seeing a growing amount of AI-generated content that is so far beneath their standards that they consider it a new kind of spam”, a trend that is “growing exponentially.” The publishers interviewed for the article said that their publications “reject all AI-written submissions” and that these “are easy to identify.” They have “perfect spelling and grammar, but a completely incoherent story.” Another said “They’re all written in a rather bland and generic way. They are all grammatically correct. They just feel very formulaic, and they are really useless to us.”\u003C/div>\u003C/td>\n\u003C/tr>\n\u003Ctr id=\"fn031-preprints-81518\">\n\u003Ctd>\u003Cspan class=\"html-label\">31\u003C/span>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">The former scenarios would be absurd even if AI technology had already attained superhuman intelligence, as LLMs do not have \u003Cspan class=\"html-italic\">desires\u003C/span>, in the same way that they don’t have beliefs or any other mental states. They do not actually \u003Cspan class=\"html-italic\">want\u003C/span> anything. To think otherwise is akin to thinking that a laptop that is simulating a hurricane will get wet (or, as Stephen Pinker has put it, thinking that because airplanes have now exceeded the flight ability of birds, they will suddenly start acting like eagles, swooping down from the sky to grab rabbits and squirrels). Genuine mental states can only be produced by brains, or by systems that have the same \u003Cspan class=\"html-italic\">causal powers\u003C/span> that brains have. Digital computers executing DNNs are not such systems.\u003C/div>\u003C/td>\n\u003C/tr>\n\u003C/table>\u003C/section>\u003Csection class=\"html-fn_group\">\u003Ctable>\u003Ctr id>\n\u003Ctd>\u003C/td>\n\u003Ctd>\u003Cdiv class=\"html-p\">\n\u003Cb>Disclaimer/Publisher’s Note:\u003C/b> The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.\u003C/div>\u003C/td>\n\u003C/tr>\u003C/table>\u003C/section>\n \u003Csection id=\"html-copyright\">\u003Cbr>© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (\u003Ca href=\"http://creativecommons.org/licenses/by/4.0/\" target=\"_blank\">http://creativecommons.org/licenses/by/4.0/\u003C/a>).\u003C/section>\n ",[177,179],{"version":126,"hash_key":128,"id":127,"change":178},"Fixed a few typos and spelling errors that had slipped through the cracks in the first version.",{"version":7,"hash_key":180,"id":127,"change":50},"8489d95ecc40b82aaa1f5906866ea384","Arkoudas, K. 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