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Verbrugge, On the provability logic of bounded arithmetic, Annals of Pure and Applied Logic 61 (1993) 75-93.","publication_date":"1993,,","publication_name":"Annals of Pure and Applied Logic","grobid_abstract_attachment_id":"50643683"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"On the Provability Logic of Bounded Arithmetic","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true,"seo_quality":null}}["work"]; window.loswp.workCoauthors = [41333,18337933]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "full_page_mobile_sutd_modal"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":50643683,"attachmentType":"pdf"}"><img alt="First page of “On the Provability Logic of Bounded Arithmetic”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/50643683/mini_magick20190128-10991-1gs91cc.png?1548695472" /><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">On the Provability Logic of Bounded 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="41333" href="https://rug.academia.edu/RinekeVerbrugge"><img alt="Profile image of Rineke Verbrugge" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Rineke Verbrugge</a><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="18337933" href="https://independent.academia.edu/AlessandroBerarducci"><img alt="Profile image of Alessandro Berarducci" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/18337933/22540702/21734456/s65_alessandro.berarducci.jpg" />Alessandro Berarducci</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">1993, Annals of Pure and Applied 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">19 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 = 2405515; 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Verbrugge, On the provability logic of bounded arithmetic, Annals of Pure and Applied Logic 61 (1993) 75-93.</p></div></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="112245398" 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/112245398/The_Modal_Logic_of_Consistency_Assertions_of_Peano_Arithmetic">The Modal Logic of Consistency Assertions 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="34208831" href="https://unipd.academia.edu/SilvioValentini">Silvio Valentini</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 1983</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 Modal Logic of Consistency Assertions of Peano Arithmetic","attachmentId":109534497,"attachmentType":"pdf","work_url":"https://www.academia.edu/112245398/The_Modal_Logic_of_Consistency_Assertions_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/112245398/The_Modal_Logic_of_Consistency_Assertions_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="1" data-entity-id="3084012" 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/3084012/A_note_on_some_explicit_modal_logics">A note on some explicit modal 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="809285" href="https://umcp.academia.edu/EricPacuit">Eric Pacuit</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2006</p><p class="ds-related-work--abstract ds2-5-body-sm">Artemov introduced the Logic of Proofs (LP) as a logic of explicit proofs. 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One captured formal provability and resulted in the logic GL and Solovay's Completeness Theorem. The other was based on the modal logic S4 and led to Artemov's Logic of Proofs LP. In this paper, we study introduced by the author logic GLA, which is a fusion of GL and LP in the union of their languages. GLA is supplied with a Kripke-style semantics and the corresponding completeness theorem. 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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 class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="41133178" 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/41133178/A_Proof_Theory_for_the_Logic_of_Provability_in_True_Arithmetic">A Proof Theory for the Logic of Provability in True 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="512336" href="https://mlit.academia.edu/HirohikoKushida%E4%B8%B2%E7%94%B0%E8%A3%95%E5%BD%A6">Hirohiko Kushida　串田裕彦</a></div><p class="ds-related-work--metadata ds2-5-body-xs">STUDIA LOGICA, vol.108(4), pp 857-875, 2020</p><p class="ds-related-work--abstract ds2-5-body-sm">[Attached is an accepted version.] In a classical 1976 paper, Solovay proved the arithmetical completeness of the modal logic GL; provability of a formula in GL coincides with provability of its arithmetical interpretations in Peano Arithmetic. In that paper, he also provided an axiomatic system GLS and proved arithmetical completeness for GLS; provability of a formula in GLS coincides with truth of its arithmetical interpretations in the standard model of arithmetic. Proof theory for GL has been studied intensively up to the present day. However, it might sound somewhat strange that no proof theory for GLS was ever developed nor even suggested thus far, except for the axiomatic system offered by Solovay. In this paper, we develop a proof theory for GLS based on the sequent calculus method. We provide a sequent calculus for GLS and prove the cut- elimination and some standard consequences of it: the interpolation and de- finability theorems. As another consequence of cut-elimination, we also prove the equivalence of GL and GLS with respect to a special form of formulas which we call G ̈odel sentences, using a purely proof-theoretical method.</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 Proof Theory for the Logic of Provability in True Arithmetic","attachmentId":65120834,"attachmentType":"pdf","work_url":"https://www.academia.edu/41133178/A_Proof_Theory_for_the_Logic_of_Provability_in_True_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/41133178/A_Proof_Theory_for_the_Logic_of_Provability_in_True_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="7" data-entity-id="17714192" 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/17714192/On_Some_Completeness_Theorems_in_Modal_Logic">On Some Completeness Theorems in Modal 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="31418790" href="https://204.academia.edu/DavidMakinson">David Makinson</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1966</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 Some Completeness Theorems in Modal Logic","attachmentId":39670180,"attachmentType":"pdf","work_url":"https://www.academia.edu/17714192/On_Some_Completeness_Theorems_in_Modal_Logic","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/17714192/On_Some_Completeness_Theorems_in_Modal_Logic"><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="19269804" 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/19269804/Proving_unprovability_in_some_normal_modal_logics">Proving unprovability in some normal modal 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="39410570" href="https://su-se.academia.edu/ValentinGoranko">Valentin Goranko</a></div><p class="ds-related-work--metadata ds2-5-body-xs">1991</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":"Proving unprovability in some normal modal logics","attachmentId":40527294,"attachmentType":"pdf","work_url":"https://www.academia.edu/19269804/Proving_unprovability_in_some_normal_modal_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/19269804/Proving_unprovability_in_some_normal_modal_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="9" data-entity-id="2815285" 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/2815285/On_propositional_quantifiers_in_provability_logic">On propositional quantifiers in provability 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="33366" href="https://gc-cuny.academia.edu/SergeiArtemov">Sergei Artemov</a></div><p class="ds-related-work--metadata ds2-5-body-xs">1993</p><p class="ds-related-work--abstract ds2-5-body-sm">Abstract The first order theory of the Diagonalizable Algebra of Peano Arithmetic (DA (PA)) represents a natural fragment of provability logic with propositional quantifiers. We prove that the first order theory of the O-generated subalgebra of DA (PA) is decidable but not elementary recursive; the same theory, enriched by a single free variable ranging over DA (PA), is already undecidable.</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 propositional quantifiers in provability logic","attachmentId":30767868,"attachmentType":"pdf","work_url":"https://www.academia.edu/2815285/On_propositional_quantifiers_in_provability_logic","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/2815285/On_propositional_quantifiers_in_provability_logic"><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":50643683,"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":50643683,"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_50643683" 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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