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tiime, to wit of the problem of the interpretability logic of all reasonable arithmetical theories. 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We present both the arithmetical side and the modal side of the question.</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":113160827,"attachmentType":"pdf","workUrl":"https://www.academia.edu/117255556/The_interpretability_logic_of_all_reasonable_arithmetical_theories"}">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":113160827,"attachmentType":"pdf","workUrl":"https://www.academia.edu/117255556/The_interpretability_logic_of_all_reasonable_arithmetical_theories"}"><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="113160827" data-landing_url="https://www.academia.edu/117255556/The_interpretability_logic_of_all_reasonable_arithmetical_theories" 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="48566965" 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/48566965/The_Interpretability_Logic_of_all_Reasonable_Arithmetical_Theories_The_New_Conjecture">The Interpretability Logic of all Reasonable Arithmetical Theories. The New Conjecture</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="80365520" href="https://uu.academia.edu/albertvisser">albert visser</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Logic Group Preprint Series, 2012</p><p class="ds-related-work--abstract ds2-5-body-sm">This paper is a presentation of a status qu stionis, to wit of the problem of the interpretability logic of all reasonable arithmetical theories. We present both the arithmetical side and the modal side of the question.</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 Interpretability Logic of all Reasonable Arithmetical Theories. The New Conjecture","attachmentId":67111544,"attachmentType":"pdf","work_url":"https://www.academia.edu/48566965/The_Interpretability_Logic_of_all_Reasonable_Arithmetical_Theories_The_New_Conjecture","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/48566965/The_Interpretability_Logic_of_all_Reasonable_Arithmetical_Theories_The_New_Conjecture"><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="117255565" 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/117255565/A_new_principle_in_the_interpretability_logic_of_all_reasonable_arithmetical_theories">A new principle in the interpretability logic of all reasonable arithmetical theories</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="51116229" href="https://independent.academia.edu/JoostJoosten">Joost Joosten</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2020</p><p class="ds-related-work--abstract ds2-5-body-sm">The interpretability logic of a mathematical theory describes the structural behavior of interpretations over that theory. Different theories have different logics. This paper revolves around the question what logic describes the behavior that is present in all theories with a minimum amount of arithmetic; the intersection over all such theories so to say. We denote this target logic by IL(All). In this paper we present a new principle R in IL(All). We show that R does not follow from the logic ILP0W * that contains all previously known principles. This is done by providing a modal incompleteness proof of ILP0W * : showing that R follows semantically but not syntactically from ILP0W *. Apart from giving the incompleteness proof by elementary methods, we also sketch how to work with so-called Generalized Veltman Semantics as to establish incompleteness. To this extent, a new version of this Generalized Veltman Semantics is defined and studied. Moreover, for the important principles the frame correspondences are calculated. After the modal results it is shown that the new principle R is indeed valid in any arithmetically theory. The proof employs some elementary results on definable cuts in arithmetical theories. 1 Technically speaking the property of so-called essential reflexivity is sufficient. A theory is</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 principle in the interpretability logic of all reasonable arithmetical theories","attachmentId":113160838,"attachmentType":"pdf","work_url":"https://www.academia.edu/117255565/A_new_principle_in_the_interpretability_logic_of_all_reasonable_arithmetical_theories","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/117255565/A_new_principle_in_the_interpretability_logic_of_all_reasonable_arithmetical_theories"><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="3292839" 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/3292839/Interpretability_over_Peano_Arithmetic">Interpretability over Peano 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="3804648" href="https://gu-se.academia.edu/ClaesStranneg%C3%A5rd">Claes Strannegård</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Symbolic Logic, 1999</p><p class="ds-related-work--abstract ds2-5-body-sm">We investigate the modal logic of interpretability over Peano arithmetic (PA). Our main result is an extension of the arithmetical completeness theorem for the interpretability logic ILM ! . This extension concerns recursively enumerable sets of formulas of interpretability logic (rather than single formulas). As corollaries we obtain a uniform arithmetical completeness theorem for the interpretability logic ILM and a theorem answering a question of Orey from 1961. All these results also hold for Zermelo-Fraenkel set theory (ZF).</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":"Interpretability over Peano Arithmetic","attachmentId":50362267,"attachmentType":"pdf","work_url":"https://www.academia.edu/3292839/Interpretability_over_Peano_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/3292839/Interpretability_over_Peano_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="3" data-entity-id="80112869" 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/80112869/The_interpretability_logic_of_Peano_arithmetic">The interpretability logic of Peano 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="18337933" href="https://independent.academia.edu/AlessandroBerarducci">Alessandro Berarducci</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Symbolic Logic, 1990</p><p class="ds-related-work--abstract ds2-5-body-sm">PA is Peano arithmetic. The formula InterpPA(α, β) is a formalization of the assertion that the theory PA + α interprets the theory PA + β (the variables α and β are intended to range over codes of sentences of PA). We extend Solovay&#39;s modal analysis of the formalized provability predicate of PA, PrPA(x), to the case of the formalized interpretability relation InterpPA(x, y). The relevant modal logic, in addition to the usual provability operator ‘□’, has a binary operator ‘⊳’ to be interpreted as the formalized interpretability relation. We give an axiomatization and a decision procedure for the class of those modal formulas that express valid interpretability principles (for every assignment of the atomic modal formulas to sentences of PA). Our results continue to hold if we replace the base theory PA with Zermelo-Fraenkel set theory, but not with Gödel-Bernays set theory. This sensitivity to the base theory shows that the language is quite expressive. Our proof uses in an ess...</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 interpretability logic of Peano arithmetic","attachmentId":86603412,"attachmentType":"pdf","work_url":"https://www.academia.edu/80112869/The_interpretability_logic_of_Peano_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/80112869/The_interpretability_logic_of_Peano_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="4" data-entity-id="21016613" 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/21016613/Interpretability_in">Interpretability in</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="42250913" href="https://independent.academia.edu/DickJongh">Dick Jongh</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Annals of Pure and Applied Logic, 2009</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper we study IL(PRA), the interpretability logic of PRA. As PRA is neither an essentially reflexive theory nor finitely axiomatizable, the two known arithmetical completeness results do not apply to PRA: IL(PRA) is not ILM or ILP. IL(PRA) does of course contain all the principles known to be part of IL(All), the interpretability logic of the principles common to all reasonable arithmetical theories. In this paper, we take two arithmetical properties of PRA and see what their consequences in the modal logic IL(PRA) are. These properties are reflected in the so-called Beklemishev Principle B, and Zambella's Principle Z, neither of which is a part of IL(All). Both principles and their interrelation are submitted to a modal study. In particular, we prove a frame condition for B. Moreover, we prove that Z follows from a restricted form of B. Finally, we give an overview of the known relationships of IL(PRA) to important other interpetability principles. We thank Lev Beklemishev for his help and suggestions. Evan Goris did a thorough proofread of an early draft and suggested a simplification of the notion of Bsimulation. We thank Albert Visser for fruitful discussions and challenges. We also thank Franco Montagna for his many contributions to the subject. Two unknown referees improved our paper considerably with their remarks and suggestions.</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":"Interpretability in","attachmentId":41672780,"attachmentType":"pdf","work_url":"https://www.academia.edu/21016613/Interpretability_in","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/21016613/Interpretability_in"><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="117255566" 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/117255566/Provability_and_interpretability_logics_with_restricted_realizations">Provability and interpretability logics with restricted realizations</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="51116229" href="https://independent.academia.edu/JoostJoosten">Joost Joosten</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2020</p><p class="ds-related-work--abstract ds2-5-body-sm">The provability logic of a theory T is the set of modal formulas, which under any arithmetical realization are provable in T. We slightly modify this notion by requiring the arithmetical realizations to come from a specified set Γ. We make an analogous modification for interpretability logics. This is a paper from 2012. We first studied provability logics with restricted realizations, and show that for various natural candidates of theory T and restriction set Γ, where each sentence in Γ has a well understood (meta)-mathematical content in T , the result is the logic of linear frames. However, for the theory Primitive Recursive Arithmetic (PRA), we define a fragment that gives rise to a more interesting provability logic, by capitalizing on the well-studied relationship between PRA and IΣ 1. We then study interpretability logics, obtaining some upper bounds for IL(PRA), whose characterization remains a major open question in interpretability logic. Again this upper bound is closely relatively to linear frames. The technique is also applied to yield the non-trivial result that IL(PRA) ⊂ ILM. 2 Σ 1-sound is sufficient here. 3 There is a paper by de Jongh, Jumelet and Montagna [22] where an alternative proof of Solovay's theorem is given. In that proof, using the diagonal lemma, one finds some sentences with the required properties rather than defining the sentences and then proving the necessary properties.</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":"Provability and interpretability logics with restricted realizations","attachmentId":113160839,"attachmentType":"pdf","work_url":"https://www.academia.edu/117255566/Provability_and_interpretability_logics_with_restricted_realizations","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/117255566/Provability_and_interpretability_logics_with_restricted_realizations"><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="76406862" 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/76406862/Modal_Matters_in_Interpretability_Logics">Modal Matters in Interpretability Logics</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="51116229" href="https://independent.academia.edu/JoostJoosten">Joost Joosten</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2020</p><p class="ds-related-work--abstract ds2-5-body-sm">This paper from 2008 is the first in a series of three related papers on modal methods in interpretability logics and applications. In this first paper the foundations are laid for later results. These foundations consist of a thorough treatment of a construction method to obtain modal models. This construction method is used to reprove some known results in the area of interpretability like the modal completeness of the logic IL. Next, the method is applied to obtain new results: the modal completeness of the logic IL M_0, and modal completeness of IL( W^*).</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":"Modal Matters in Interpretability Logics","attachmentId":84125096,"attachmentType":"pdf","work_url":"https://www.academia.edu/76406862/Modal_Matters_in_Interpretability_Logics","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/76406862/Modal_Matters_in_Interpretability_Logics"><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="48566718" 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/48566718/Interpretability_degrees_of_finitely_axiomatized_sequential_theories">Interpretability degrees of finitely axiomatized sequential theories</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="80365520" href="https://uu.academia.edu/albertvisser">albert visser</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Archive for Mathematical Logic, 2014</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper we show that the degrees of interpretability of finitely axiomatized extensions-in-the-same-language of a finitely axiomatized sequential theory-like Elementary Arithmetic EA, IΣ 1 , or the Gödel-Bernays theory of sets and classes GBhave suprema. This partially answers a question posed by VítěslavŠvejdar in his paper [Šve78]. The partial solution ofŠvejdar's problem follows from a stronger fact: the convexity of the degree structure of finitely axiomatized extensions-in-the-same-language of a finitely axiomatized sequential theory in the degree structure of the degrees of all finitely axiomatized sequential theories. In the paper we also study a related question: the comparison of structures for interpretability and derivability. In how far can derivability mimic interpretability? We provide two positive results and one negative result. Dedicated to Dirk van Dalen on the occasion of his 80th birthday.</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":"Interpretability degrees of finitely axiomatized sequential theories","attachmentId":67111731,"attachmentType":"pdf","work_url":"https://www.academia.edu/48566718/Interpretability_degrees_of_finitely_axiomatized_sequential_theories","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/48566718/Interpretability_degrees_of_finitely_axiomatized_sequential_theories"><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="69310050" 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/69310050/Formalized_Interpretability_in_Primitive_Recursive_Arithmetic">Formalized Interpretability in Primitive Recursive 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="51116229" href="https://independent.academia.edu/JoostJoosten">Joost Joosten</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2003</p><p class="ds-related-work--abstract ds2-5-body-sm">Interpretations are a natural tool in comparing the strength of two theories. In this paper we give a brief introduction to the topic of interpretability and interpretability logics. We will focus on the, so far, unknown interpretability logic of PRA. One research technique will be treated. This technique can be best described as restricting the realizations in the arithmetical semantics. 1 What are interpretations and why study them? How to interpret “Eli, Eli, lama sabachtani”? Let us consider the concept of interpretation in the previous phrase1. What does it actually mean to interpret something. Or more specifically, what do we mean when we say that T interprets some utterance φ of S? Well, in this case T can first translate φ to its own language, then place it in an adequate context and then somehow make sense of it. The mathematical notion of interpretation is somewhat similar. We say that a theory T interprets another theory S whenever there is some translation such that all ...</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":"Formalized Interpretability in Primitive Recursive Arithmetic","attachmentId":79455307,"attachmentType":"pdf","work_url":"https://www.academia.edu/69310050/Formalized_Interpretability_in_Primitive_Recursive_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/69310050/Formalized_Interpretability_in_Primitive_Recursive_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="9" data-entity-id="59275271" 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/59275271/On_the_complexity_of_arithmetical_interpretations_of_modal_formulae">On the complexity of arithmetical interpretations of modal formulae</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="98896487" href="https://scopus.academia.edu/LevBeklemishev">Lev Beklemishev</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Archive for Mathematical Logic, 1993</p><p class="ds-related-work--abstract ds2-5-body-sm">It is a well-known fact that for any arithmetic sentence A A c Z1PA ~ PA k A-+ PrrA ~. Here Pr stands for G6del's formula expressing provability in Peano Arithmetic PA and Z~ A denotes the class of sentences PA-equivalent to those in ~l-f~ Kent [1] showed that the converse implication does not hold. Moreover, he found that for each natural number r~ there exists an arithmetic sentence A such that PAFA-~Pff-A = and A f{ A PA. Guaspari [2] rediscovered (a sharpened version of) this result applying his own techniques of partially conservative sentences. He also showed that arithmetically complex sentences implying their own provability cannot be constructed by some class of restricted means. Guaspari posed a few problems generalizing the one solved by Kent and himself, which are formulated in terms of provability interpretations of propositional modal logic. Definition. Let c~ be the language consisting of propositional variables p, q, ...; boolean connectives A, V,-% +-+, ~ and • modal operator []. An arithmetical interpretation f is a mapping of ~C<formulae to arithmetic sentences which commutes with boolean connectives and translates [] as provability, i.e. for every modal formula q~ f([]~) = Prrf(~)" .</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 the complexity of arithmetical interpretations of modal formulae","attachmentId":73282230,"attachmentType":"pdf","work_url":"https://www.academia.edu/59275271/On_the_complexity_of_arithmetical_interpretations_of_modal_formulae","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/59275271/On_the_complexity_of_arithmetical_interpretations_of_modal_formulae"><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":113160827,"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":113160827,"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_113160827" 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. You can download the paper by clicking the button above.</p></div></div></div></div><div class="ds-sidebar--container js-work-sidebar"><div class="ds-related-content--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="0" data-entity-id="11399538" 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/11399538/On_Interpretations_of_Arithmetic_and_Set_Theory">On Interpretations of Arithmetic and Set Theory</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="27506117" href="https://independent.academia.edu/TinLokWong">Tin-Lok Wong</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Notre Dame Journal of Formal Logic, 2007</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link 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href="https://www.academia.edu/99372891/Interpretability_in_PRA">Interpretability in PRA</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="51116229" href="https://independent.academia.edu/JoostJoosten">Joost Joosten</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2020</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":"Interpretability in PRA","attachmentId":100481054,"attachmentType":"pdf","work_url":"https://www.academia.edu/99372891/Interpretability_in_PRA","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/99372891/Interpretability_in_PRA"><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" data-collection-position="3" data-entity-id="2405515" 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/2405515/On_the_Provability_Logic_of_Bounded_Arithmetic">On the Provability Logic of Bounded Arithmetic</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="41333" href="https://rug.academia.edu/RinekeVerbrugge">Rineke Verbrugge</a><span>, </span><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="18337933" href="https://independent.academia.edu/AlessandroBerarducci">Alessandro Berarducci</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Annals of Pure and Applied Logic, 1993</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 the Provability Logic of Bounded Arithmetic","attachmentId":50643683,"attachmentType":"pdf","work_url":"https://www.academia.edu/2405515/On_the_Provability_Logic_of_Bounded_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/2405515/On_the_Provability_Logic_of_Bounded_Arithmetic"><span class="ds2-5-text-link__content">View PDF</span><span 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Shore</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Proceedings of the …, 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":"Interpretability and definability in the recursively enumerable degrees","attachmentId":76900193,"attachmentType":"pdf","work_url":"https://www.academia.edu/65195709/Interpretability_and_definability_in_the_recursively_enumerable_degrees","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/65195709/Interpretability_and_definability_in_the_recursively_enumerable_degrees"><span class="ds2-5-text-link__content">View PDF</span><span 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