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class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":38092038,"attachmentType":"pdf"}"><img alt="First page of “A second philosophy of arithmetic”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/38092038/mini_magick20190226-5267-deam6x.png?1551216908" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">A second philosophy of arithmetic</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="32791720" href="https://uci.academia.edu/PenelopeMaddy"><img alt="Profile image of Penelope Maddy" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/32791720/20731051/102722102/s65_penelope.maddy.jpg" />Penelope Maddy</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2014, Review of Symbolic Logic</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">28 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 13629980; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=13629980"; const getWorkViews = async (workId) => { const response = await fetch(worksViewsPath); if (!response.ok) { throw new Error('Failed to load work views'); } const data = await response.json(); return data.views[workId]; }; // Get the view count for the work - we send this immediately rather than waiting for // the DOM to load, so it can be available as soon as possible (but without holding up // the backend or other resource requests, because it's a bit expensive and not critical). const viewCount = await getWorkViews(workId); const updateViewCount = (viewCount) => { try { const viewCountNumber = parseInt(viewCount, 10); if (viewCountNumber === 0) { // Remove the whole views element if there are zero views. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); return; } const commaizedViewCount = viewCountNumber.toLocaleString(); const viewCountBody = document.getElementById('work-metadata-view-count'); if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">This paper outlines a second-philosophical account of arithmetic that places it on a distinctive ground between those of logic and set theory.</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":38092038,"attachmentType":"pdf","workUrl":"https://www.academia.edu/13629980/A_second_philosophy_of_arithmetic"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":38092038,"attachmentType":"pdf","workUrl":"https://www.academia.edu/13629980/A_second_philosophy_of_arithmetic"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div></div><div data-auto_select="false" data-client_id="331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b" data-doc_id="38092038" data-landing_url="https://www.academia.edu/13629980/A_second_philosophy_of_arithmetic" data-login_uri="https://www.academia.edu/registrations/google_one_tap" data-moment_callback="onGoogleOneTapEvent" id="g_id_onload"></div><div class="ds-top-related-works--grid-container"><div class="ds-related-content--container ds-top-related-works--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="0" data-entity-id="92756599" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/92756599/Philosophy_of_Arithmetic">Philosophy of Arithmetic</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="795663" href="https://utrgv.academia.edu/MelisaVivanco">Melisa Vivanco</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Melisa Vivanco, 2023</p><p class="ds-related-work--abstract ds2-5-body-sm">Some of the most influential programs in the philosophy of mathematics started from the philosophical study of natural numbers. On the one hand, our arithmetic intuitions appear earlier and more direct than other mathematical (and non-mathematical) intuitions. On the other hand, while arithmetic admits one of the first axiomatizations with wide acceptance within the mathematical practice, the study of natural numbers sets a methodological precedent that will later seek to be replicated in other areas, in particular, in the study of the most complex numerical structures. This course will address the main issues in the philosophical discussion on arithmetic. Among these topics are the ideas of the various classical doctrines on the foundations of arithmetic, from the milestone of Gödel’s incompleteness theorems to recent doctrines on the semantics of numerical expressions and arithmetic sentences. The class will cover debates about metaphysics and the epistemology of numbers and arithmetic truths.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Philosophy of Arithmetic","attachmentId":95681402,"attachmentType":"pdf","work_url":"https://www.academia.edu/92756599/Philosophy_of_Arithmetic","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/92756599/Philosophy_of_Arithmetic"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="1" data-entity-id="160314" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/160314/Predicative_foundations_of_arithmetic">Predicative foundations of arithmetic</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="13693" href="https://stanford.academia.edu/SolomonFeferman">Solomon Feferman</a></div><p class="ds-related-work--abstract ds2-5-body-sm">Predicative mathematics in the sense originating with Poincaré and Weyl begins by taking the natural number system for granted, proceeding immediately to real analysis and related fields. On the other hand, from a logicist or set-theoretic standpoint, this appears problematic, for, as the story is usually told, impredicative principles seem to play an essential role in the foundations of arithmetic itself. 1 It is the main purpose of this paper to show that this appearance is illusory: as will emerge, a predicatively acceptable axiomatization of the natural number system can be formulated, and both the existence of structures of the relevant type and the categoricity of the relevant axioms can be proved in a predicatively acceptable way.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Predicative foundations of\n arithmetic","attachmentId":68370,"attachmentType":"pdf","work_url":"https://www.academia.edu/160314/Predicative_foundations_of_arithmetic","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/160314/Predicative_foundations_of_arithmetic"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="2" data-entity-id="125701704" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/125701704/From_Arithmetic_to_Metaphysics_A_Path_Through_Philosophical_Logic">From Arithmetic to Metaphysics: A Path Through Philosophical Logic</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="67711" href="https://unicatt.academia.edu/CiroDeFlorio">Ciro De Florio</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2018</p><p class="ds-related-work--abstract ds2-5-body-sm">The distinction between appearance and reality, introduced by Parmenides and Heraclitus, is said to be one of the pillars of Western thought. And this is precisely what this volume is, i.e. a real tribute to its addressee, even though it does not appear as the typical collection of papers in honor of someone. It is indeed an homage to a professor and to his influence in the research areas which the collected essays are a representation of. Sergio Galvan graduated in 1969 under Evandro Agazzi's tuition with a dissertation on Alfred Tarski's semantic conception of truth. At that time Alfred Tarski was still alive and his well-known essay Truth and Proof appeared on Scientific American. Galvan spent a period as visiting researcher in Germany; then, he taught at secondary and high school, bravely explaining to -likely astoundedstudents the theory of syllogism. Finally, he worked at the universities in Milan and Verona, becoming full professor of Logic at Trento University in 1994. Three years later, he returned to the Catholic University of Milan where he previously had studied. At Catholic University he was chair of Logic, Philosophy of Science and Analytic Ontology. In the Sixties, the Italian philosophical scenario, in which Galvan intellectually grew up, was dominated by discussions concerning political philosophy (in particular, the debate between Catholics and Communists), philosophy of existence, aesthetics, and moral philosophy. Logic and philosophy have been just introduced, each of them in his own way and peculiar attitude, by such scholars as Ludovico Geymonat, Ettore Casari, Alberto Pasquinelli, and Galvan's teacher Evandro Agazzi,. Metaphysics lied idle, rather forgotten. However, Catholic University still paid great attention to the science of being, thanks to some important teachers: Amato Masnovo, Gustavo Bontadini and Sofia Vanni Rovighi. Indeed, Gustavo Bontadini was the former teacher of Evandro Agazzi and Galvan himself attended his lectures. Throughout the years of study, Sergio nurtured the passion for the classic metaphysical themes -from the problem of becoming to the ground of reality, from the constitution of entities to the problem of universals -and the interest for them has been a constant in his thought. Sergio Galvan is a logician and all his scientific work embodies Hilbert's motto: clear thought is axiomatic thought. His main areas of research can be summarized as follows: mathematical logic, modal logic, logic of explanation and metaphysics. His first field of research concerns the analysis of Tarskian theory of truth and its connection with the classical correspondentist conception of truth. 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Lesniewski's extensional calculus of names (the system called 'Ontology' by Lesniewski).</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A New Path to the Logicist Construction of Numbers","attachmentId":54901850,"attachmentType":"pdf","work_url":"https://www.academia.edu/35039964/A_New_Path_to_the_Logicist_Construction_of_Numbers","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/35039964/A_New_Path_to_the_Logicist_Construction_of_Numbers"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="4" data-entity-id="117464484" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/117464484/The_Relationship_of_Arithmetic_As_Two_Twin_Peano_Arithmetic_s_and_Set_Theory_A_New_Glance_From_the_Theory_of_Information">The Relationship of Arithmetic As Two Twin Peano Arithmetic(s) and Set Theory: A New Glance From the Theory of Information</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="4716755" href="https://bas.academia.edu/VasilPenchev">Vasil Penchev</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Social Science Research Network, 2020</p><p class="ds-related-work--abstract ds2-5-body-sm">The paper introduces and utilizes a few new concepts: "nonstandard Peano arithmetic", "complementary Peano arithmetic", "Hilbert arithmetic". They identify the foundations of both mathematics and physics demonstrating the equivalence of the newly introduced Hilbert arithmetic and the separable complex Hilbert space of quantum mechanics in turn underlying physics and all the world. That new both mathematical and physical ground can be recognized as information complemented and generalized by quantum information. A few fundamental mathematical problems of the present such as Fermat's last theorem, four-color theorem as well as its new-formulated generalization as "four-letter theorem", Poincaré's conjecture, "P vs NP" are considered over again, from and within the newfounding conceptual reference frame of information, as illustrations. Simple or crucially simplifying solutions and proofs are demonstrated. The link between the consistent completeness of the system mathematics-physics on the ground of information and all the great mathematical problems of the present (rather than the enumerated ones) is suggested.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"The Relationship of Arithmetic As Two Twin Peano Arithmetic(s) and Set Theory: A New Glance From the Theory of Information","attachmentId":113314768,"attachmentType":"pdf","work_url":"https://www.academia.edu/117464484/The_Relationship_of_Arithmetic_As_Two_Twin_Peano_Arithmetic_s_and_Set_Theory_A_New_Glance_From_the_Theory_of_Information","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/117464484/The_Relationship_of_Arithmetic_As_Two_Twin_Peano_Arithmetic_s_and_Set_Theory_A_New_Glance_From_the_Theory_of_Information"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" data-entity-id="88346837" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/88346837/A_general_framework_for_a_Second_Philosophy_analysis_of_set_theoretic_methodology">A general framework for a Second Philosophy analysis of set-theoretic methodology</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="27959640" href="https://konstanz.academia.edu/DeborahKant">Deborah Kant</a></div><p class="ds-related-work--abstract ds2-5-body-sm">Penelope Maddy’s Second Philosophy is one of the most well-known ap- proaches in recent philosophy of mathematics. She applies her second-philosophical method to analyze mathematical methodology by reconstructing historical cases in a setting of means-ends relations. However, outside of Maddy’s own work, this kind of methodological analysis has not yet been extensively used and analyzed. In the present work, we will make a first step in this direction. We develop a general framework that allows us to clarify the procedure and aims of the Second Philosopher’s investigation into set-theoretic methodology; pro- vides a platform to analyze the Second Philosopher’s methods themselves; and can be applied to further questions in the philosophy of set theory</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A general framework for a Second Philosophy analysis of set-theoretic methodology","attachmentId":92336678,"attachmentType":"pdf","work_url":"https://www.academia.edu/88346837/A_general_framework_for_a_Second_Philosophy_analysis_of_set_theoretic_methodology","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/88346837/A_general_framework_for_a_Second_Philosophy_analysis_of_set_theoretic_methodology"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="9727886" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/9727886/On_Arithmetic_Formulated_Connexively">On Arithmetic Formulated Connexively</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="9896443" href="https://rpi.academia.edu/ThomasFerguson">Thomas M Ferguson</a></div><p class="ds-related-work--abstract ds2-5-body-sm">One of the richest and most salient applications of a non-classical logic is the matter of how mathematics operates within its province. Historically, this is most evident in the case of intuitionism, insofar as the intuitionistic standpoints with respect to deduction and mathematical practice are tightly bound together. Yet even in the case of Robert Meyer's relevant arithmetic R#, that a robust and compelling theory of arithmetic can be erected on relevant foundations speaks to the maturity of relevant logics. Accordingly, as connexive logic matures as a field, the topography of mathematics against a connexive backdrop becomes more and more compelling. The contraclassicality of connexive logics entails that the development of connexive mathematics will be more complex---and, arguably, more interesting---than intuitionistic or relevant accounts. For example, although formally undecidable sentences in classical Peano arithmetic remain independent of its intuitionistic and relevant counterparts, there exist undecidable sentences of classical arithmetic that will become decidable modulo any reasonable connexive arithmetic. In, e.g., Peano arithmetic, the Gödel sentence G is undecidable. Classically, this entails that the sentence ~(G->~G) is likewise undecidable. Of course, in a connexive logic L and connexive arithmetic L#, L# will prove ~(G->~G), witnessing that some classically undecidable statements in number theory become decidable connexively. Although this example is extremely simple, it demonstrates that there are many subtle questions that uniquely arise in a connexive mathematics. In this paper, I wish to make a few comments on how mathematics---in particular, arithmetic---must behave if formulated connexively. We will first consider some relevant historical and philosophical topics, such as Łukasiewicz' number-theoretic argument against Aristotle's Thesis, before taking a foray into the formalization of modest subsystems of arithmetic in Richard Angell's PA1 and PA2, observing some of the pathologies that will greet arithmetic in these settings.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On Arithmetic Formulated Connexively","attachmentId":35913338,"attachmentType":"pdf","work_url":"https://www.academia.edu/9727886/On_Arithmetic_Formulated_Connexively","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/9727886/On_Arithmetic_Formulated_Connexively"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="113731321" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/113731321/Predicative_Logic_and_Formal_Arithmetic">Predicative Logic and Formal Arithmetic</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="117828974" href="https://independent.academia.edu/JohnBurgess31">John Burgess</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Notre Dame Journal of Formal Logic, 1998</p><p class="ds-related-work--abstract ds2-5-body-sm">After a summary of earlier work it is shown that elementary or Kalmar arithmetic can be interpreted within the system of Russell's Principia Mathematica with the axiom of infinity but without the axiom of reducibility. 1 Historical introduction After discovering the inconsistency in Frege's Grundgesetze der Arithmetik, Russell proposed two changes: first, dropping the assumption that to every higher-order entity there corresponds a first-order entity; and second, restricting the assumptions on the existence of higher-order entities, so that instead of a simple hierarchy of first-order, second-order, third-order, and so on, one has a ramified hierarchy in which each order is subdivided into various types in such a way that a condition involving quantification over all entities of one type is never assumed to determine another entity of the same type, but only of a higher type. But Russell found that with these two changes he could not derive classical mathematics, so in Principia Mathematica he partially compensated for the first change by assuming the axiom of infinity and, for all mathematical purposes, wholly undid the second change by assuming his axiom of reducibility. The predicativist tradition from Weyl [21] to Feferman [2] and beyond accepts infinity but rejects reducibility and is willing to give up parts of classical mathematics. However, predicativists have been unable to derive classical arithmetic and unwilling to give it up and so have simply assumed it as axiomatic. This assumption has its defenders, as with Feferman and Hellman [3], and also its detractors, as with C. Parsons [15]. It is, therefore, of some philosophical as well as historical interest to ask how large a fragment of classical arithmetic can be developed within the Russellian system of Principia Mathematica with infinity but without reducibility. Now many subsystems of classical or Peano arithmetic have been recognized since the work of Skolem [18], Kalmar [9], Grzegorczyk [4], and other pioneers. Among these the most studied have been the subprimitive or Grzegorczyk arithmetics n. These agree in allowing definitions by primitive recursion, but only when the function F being defined recursively is bounded by some function already given; or</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Predicative Logic and Formal Arithmetic","attachmentId":110619161,"attachmentType":"pdf","work_url":"https://www.academia.edu/113731321/Predicative_Logic_and_Formal_Arithmetic","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/113731321/Predicative_Logic_and_Formal_Arithmetic"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="85680722" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/85680722/Arithmetic_Formulated_in_a_Logic_of_Meaning_Containment">Arithmetic Formulated in a Logic of Meaning Containment</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="693815" href="https://illinois.academia.edu/RossBrady">Ross Brady</a></div><p class="ds-related-work--metadata ds2-5-body-xs">The Australasian Journal of Logic, 2021</p><p class="ds-related-work--abstract ds2-5-body-sm">We assess Meyer’s formalization of arithmetic in his [21], based on the strong relevant logic R and compare this with arithmetic based on a suitable logic of meaning containment, which was developed in Brady [7]. We argue in favour of the latter as it better captures the key logical concepts of meaning and truth in arithmetic. We also contrast the two approaches to classical recapture, again favouring our approach in [7]. We then consider our previous development of Peano arithmetic including primitive recursive functions, finally extending this work to that of general recursion.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Arithmetic Formulated in a Logic of Meaning Containment","attachmentId":90304942,"attachmentType":"pdf","work_url":"https://www.academia.edu/85680722/Arithmetic_Formulated_in_a_Logic_of_Meaning_Containment","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/85680722/Arithmetic_Formulated_in_a_Logic_of_Meaning_Containment"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="54099043" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/54099043/In_defense_of_epistemic_arithmetic">In defense of epistemic arithmetic</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="142973211" href="https://uni-konstanz.academia.edu/leonhorsten">leon horsten</a></div><p class="ds-related-work--metadata ds2-5-body-xs">1998</p><p class="ds-related-work--abstract ds2-5-body-sm">This paper presents a defense of Epistemic Arithmetic as used for a formalization of intuitionistic arithmetic and of certain informal mathematical principles. First, objections by Allen Hazen and Craig Smorynski against Epistemic Arithmetic are discussed and found wanting. Second, positive support is given for the research program by showing that Epistemic Arithmetic can give interesting formulations of Church's Thesis.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"In defense of epistemic arithmetic","attachmentId":70627578,"attachmentType":"pdf","work_url":"https://www.academia.edu/54099043/In_defense_of_epistemic_arithmetic","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/54099043/In_defense_of_epistemic_arithmetic"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":38092038,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":38092038,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_38092038" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. 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href="https://www.academia.edu/24422946/The_principle_of_constructive_mathematizability_of_any_theory_A_sketch_of_formal_proof_by_the_model_of_reality_formalized_arithmetically">The principle of constructive mathematizability of any theory: A sketch of formal proof by the model of reality formalized arithmetically</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="4716755" href="https://bas.academia.edu/VasilPenchev">Vasil Penchev</a></div><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"The principle of constructive mathematizability of any theory: A sketch of formal proof by the model of reality formalized 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data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/97982632/A_Substitutional_Framework_for_Arithmetical_Validity">A Substitutional Framework for Arithmetical Validity</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="11326288" href="https://lisboa.academia.edu/FernandoFerreira">Fernando Ferreira</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Grazer Philosophische Studien, 1998</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A Substitutional Framework for Arithmetical 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data-author-id="5937540" href="https://ucsd.academia.edu/SamBuss">Sam Buss</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Studies in Logic and the Foundations of Mathematics, 1998</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"First-Order Proof Theory of Arithmetic","attachmentId":41904055,"attachmentType":"pdf","work_url":"https://www.academia.edu/21480115/First_Order_Proof_Theory_of_Arithmetic","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/21480115/First_Order_Proof_Theory_of_Arithmetic"><span class="ds2-5-text-link__content">View PDF</span><span 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ds2-5-body-xs">Annals of Pure and Applied Logic, 1985</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Fragments of arithmetic","attachmentId":42334579,"attachmentType":"pdf","work_url":"https://www.academia.edu/17065648/Fragments_of_arithmetic","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/17065648/Fragments_of_arithmetic"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" 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