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1. Provability Logic (Stanford Encyclopedia Of Philosophy) Provability logic is a modal logic that is used to investigate what arithmetical theories can express in a restricted language about their Provability http://plato.stanford.edu/entries/logic-provability/

Cite this entry Search the SEP Advanced Search Tools ... Please Read How You Can Help Keep the Encyclopedia Free Provability Logic First published Wed 2 Apr, 2003 From a philosophical point of view, provability logic is interesting because the concept of provability in a fixed theory of arithmetic has a unique and non-problematic meaning, other than concepts like necessity and knowledge studied in modal and epistemic logic. Furthermore, provability logic provides tools to study the notion of self-reference. 1. The history of provability logic 2. The axiom system of propositional provability logic 3. Possible worlds semantics 4. Provability logic and Peano arithmetic ... Related Entries 1. The history of provability logic More formally, let A A , the result of assigning a numerical code to A . Let Prov p Proof p x ). Here, Proof is the formalized proof predicate of Peano Arithmetic, and Proof p x p x . (For a more precise formulation, see Smorynski (1985), Davis (1958)). Now, suppose that Peano Arithmetic proves A Prov A A is not provable in Peano Arithmetic, and thus it is true, for in fact the self-referential sentence A Henkin on the other hand wanted to know whether anything could be said about sentences asserting their own provability: supposing that Peano Arithmetic proves B Prov B ), what does this imply about

2. Provability Logic - Wikipedia, The Free Encyclopedia Provability logic is a modal logic, in which the box (or necessity ) operator is interpreted as it is provable that . The point is to capture the notion http://en.wikipedia.org/wiki/Provability_logic

var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikipedia"; Provability logic From Wikipedia, the free encyclopedia Jump to: navigation search Provability logic is a modal logic , in which the box (or "necessity") operator is interpreted as 'it is provable that'. The point is to capture the notion of a proof predicate of a reasonably rich formal theory , such as Peano arithmetic There are a number of provability logics, some of which are covered in the literature mentioned in the References section. The basic system is generally referred to as GL (for G¶del L¶b ) or L or K4W. It can be obtained by adding the modal version of L¶b's theorem to the logic K (or K4). It was pioneered by Robert M. Solovay in 1976. Since then until his passing in 1996 the prime inspirer of the field was George Boolos . Significant contributions to the field have been made by Sergei Artemov, Lev Beklemishev, Giorgi Japaridze, Dick de Jongh, Franco Montagna, Vladimir Shavrukov, Albert Visser and others. Interpretability logics present natural extensions of provability logic.

3. On Provability Logic This is an introductory paper about Provability logic, a modal propositional logic in which necessity is interpreted as formal Provability. http://www.hf.uio.no/ifikk/filosofi/njpl/vol4no2/provlog/index.html

Next: 1 Introduction Up: Contents On Provability Logic Abstract: This is an introductory paper about provability logic, a modal propositional logic in which necessity is interpreted as formal provability. I discuss the ideas that led to establishing this logic, I survey its history and the most important results, and I emphasize its applications in metamathematics. Stress is put on the use of Gentzen calculus for provability logic. I sketch my version of a decision procedure for provability logic and mention some connections to computational complexity. Footnotes The work on this paper was supported partially by grant GA CR 401/98/0383 and partially by grant 162/97 from Charles University. 1 Introduction 2 Arithmetization of logical syntax 3 Arithmetical semantics of modal logic 4 Calculi, Kripke semantics and conclusions ... Bibliography To make your own printed copy of this article, download one of the following files: Postscript: provlog.ps (245862 bytes) Postscript, compressed: provlog-ps.zip (92551 bytes), provlog.ps.gz

4. Open Site - Science: Mathematics: Logic: Proof Theory: Provability Logic Provability logic, or the logic of Provability, is a modal logic where the modal necessity operator is interpreted as Provability in a reasonably rich http://open-site.org/Science/Mathematics/Logic/Proof_Theory/Provability_logic

Enter your search terms Submit search form Web open-site.org home log in apply to edit Top ... Proof Theory : Provability logic Pages Modal Logic Provability logic , or the logic of provability , is a modal logic where the modal "necessity" operator is interpreted as provability in a reasonably rich formal theory such as Peano arithmetic. Historical survey Pioneered by Robert Solovay in 1976. Since then until his passing 1n 1997 the prime inspirer of the field was George Boolos. Significant contributions to the field have been made by Sergei Artemov, Lev Beklemishev, Giorgi Japaridze, Dick de Jongh, Franco Montagna, Vladimir Shavrukov, Albert Visser and others. References George Boolos, The Logic of Provability . Cambridge University Press, 1993. Giorgi Japaridze and Dick de Jongh, The logic of Provability . In: Handbook of Proof Theory , S.Buss, ed., Elsevier, 1998. addthis_url = location.href; addthis_title = document.title; addthis_pub = 'opensite'; All text is available under the terms of the GNU Free Documentation License . (See for details.)

5. On Kripke-style Semantics For The Provability Logic Of Godel's Proof Predicate W Kripkestyle semantics is suggested for the Provability logic with quantifiers on proofs corresponding to the standard Gödel proof predicate. http://logcom.oxfordjournals.org/cgi/content/abstract/15/4/539

@import "/resource/css/hw.css"; @import "/resource/css/logcom.css"; Skip Navigation Oxford Journals Contact Us My Basket ... Volume 15, Number 4 Pp. 539-549 Journal of Logic and Computation 2005 15(4):539-549; doi:10.1093/logcom/exi037 This Article Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Similar articles in ISI Web of Science Alert me to new issues of the journal ... Request Permissions Google Scholar Articles by Yavorskiy, R. Search for Related Content Original Articles Rostislav Yavorskiy V.A. Steklov Mathematical Institute, Gubkina 8, 119991 Moscow, Russia. Email: Kripke-style semantics is suggested for the provability logic proof predicate. It is proved that the set of valid formulas is decidable. The arithmetical completeness is still an open issue. Keywords: Provability logic, logic of proofs, Kripke semantics, decidability Received May 2005. Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department.

6. Problems The Provability logic of Heyting arithmetic HA decidability, axiomatization. The Provability logic of bounded arithmetics S21 and S2 decidability, http://www.phil.uu.nl/~lev/problems.html

Here is a list of selected open problems in the areas of my interests. They are grouped thematically. Click on a problem to get a comment and possibly some related questions. If you would like to make a comment or contribute a question, please mail to lev@phil.uu.nl . Presumably easier problems suitable, e.g., for a graduate project are marked by (*). 31.07.05. On the basis of part of the list below we have written a paper together with Albert Visser where we give an orderly overview and more extended comments to some questions. See: Beklemishev, L.D. and A. Visser (2005): Problems in the Logic of Provability. Department of Philosophy, Utrecht University, Logic Group Preprint Series 235, May 2005. Problems in Provability Logic Intuitionistic arithmetic The provability logic of Heyting arithmetic HA: decidability, axiomatization. [Markov] The propositional logic of Kleene realizability. [Plisko] Dialectica interpretation. [De Jongh, Visser] Characterization of subalgebras of the Lindenbaum Heyting algebra of HA. The predicate logics of some extensions of HA: HA+MP, HA+ECT The admissible propositional inference rules for HA+MP and HA+ECT Are the Lindenbaum Heyting algebras of HA and HA+RFN(HA) isomorphic? Is the elementary theory of the Lindenbaum Heyting algebra of HA decidable? Which fragments of it are?

7. JSTOR The Logic Of Provability. Many different interlaced logical cultures have participated in the remarkable advances of the past two decades in the area of Provability logic. http://links.jstor.org/sici?sici=0022-4812(199512)60:4<1316:TLOP>2.0.CO;2-7

8. Common Knowledge And Common Rationality Through Provability Logic Author(s) C. Benassi P. Gentilini. 1999 Abstract No abstract is available for this item. http://ideas.repec.org/p/bol/bodewp/350.html

This file is part of IDEAS , which uses RePEc data Papers Articles Software Books ... Help! Common Knowledge and Common Rationality Through Provability Logic Author info Abstract Publisher info Download info ... Statistics Author Info C. Benassi P. Gentilini Additional information is available for the following registered author(s): Corrado Benassi Abstract No abstract is available for this item. Download Info To our knowledge, this item is not available for download . To find whether it is available, there are two options: 1. Check on the provider's web page whether it is in fact available. 2. Perform a search for a similarly titled item that would be available Publisher Info Paper provided by Dipartimento Scienze Economiche, Universit  di Bologna in its series Working Papers with number 350. Download reference. The following formats are available: HTML plain text BibTeX RIS (EndNote), ReDIF Length: Date of creation: Mar 1999 Date of revision: Handle: RePEc:bol:bodewp:350 Keywords: Contact details of provider: Postal: Piazza Scaravilli, 2, Bologna

9. Boolos Bibliography On systems of modal logic with Provability interpretations. Theoria 46 (1980) 718. On notions of Provability in Provability logic. http://web.mit.edu/philos/www/facultybibs/boolos_bib.html

George Boolos: List of Publications 1. (with Hilary Putnam) "Degrees of unsolvability of constructible sets of integers." Journal of Symbolic Logic 2. "Effectiveness and natural languages." In S. Hook, ed., Language and Philosophy . New York University Press, 1969. 3. "On the semantics of the constructible levels." Zeitschrift für mathematische Logik und Grundlagen der Mathematik 4. "A proof of the Löwenheim-Skolem theorem." Notre Dame Journal of Formal Logic 5. "The iterative conception of set." Journal of Philosophy 68 (1971) 215-231. Reprinted in Logic, Logic, and Logic and in Benacerraf, P. and Putnam, H., eds. Philosophy of Mathematics: Selected Readings , second ed. Cambridge: Cambridge University Press, 1984, pp. 486-502. 6. "A note on Beth's theorem." Bulletin de l'Academie Polonaise des Sciences 7. "Arithmetical functions and minimization." Zeitschrift für mathematische Logik und Grundlagen der Mathematik 8. "Reply to Charles Parsons' 'Sets and classes' (1974)" First published in Logic, Logic, and Logic

10. Provability Logic With Operations On Proofs Provability logic with Operations on Proofs. Source, Lecture Notes In Computer Science; Vol. 1234 archive Proceedings of the 4th International Symposium on http://portal.acm.org/citation.cfm?id=664430

11. IngentaConnect Properties Of Intuitionistic Provability And Preservativity Logic We study the modal properties of intuitionistic modal logics that belong to the Provability logic or the preservativity logic of Heyting Arithmetic. http://www.ingentaconnect.com/content/oup/igpl/2005/00000013/00000006/art00615

var tcdacmd="dt"; Home About Ingenta Ingenta Labs Ingenta Blog ... Check our FAQs Or contact us to report problems with: Subscription access Article delivery Registration Library administrator tasks ... Other problems For Publishers Why go online? Why choose IngentaConnect? Beyond Print: enhancing your service Access and authentication ... Contact us For Researchers For Authors About IngentaConnect Search and browse Publications available ... Register For Librarians Resource Zone Why choose IngentaConnect? Why choose IngentaConnect Complete? Activating Subscriptions ... Register CM8ShowAd("HorizontalBanner"); Properties of Intuitionistic Provability and Preservativity Logics Authors: Iemhoff, Rosalie; De Jongh, Dick; Zhou, Chunlai Source: Logic Journal of the IGPL , Volume 13, Number 6, November 2005 , pp. 615-636(22) Publisher: Oxford University Press view table of contents Key: - Free Content - New Content - Subscribed Content - Free Trial Content CM8ShowAd("Skyscraper");

12. Selected Publications Annals of Pure and Applied logic, v. 75 (12)1, 1995. Special issue of papers from the Conference on Proof Theory, Provability logic and Computation, http://www.cs.cornell.edu/Info/People/artemov/publ.html

Selected publications S.Artemov, E.Barzilay , R.Constable, A.Nogin Reflection and Propositions-as-Type s'', submitted to a conference, 2001. S.Artemov and T.Sidon-Yavorskaya On the first order logic of proofs , Technical Report CFIS 99-11, Cornell University, 1999. To appear in the Moscow Mathematical Journal in 2001. S. Artemov, `` Explicit provability and constructive semantics Bulletin of Symbolic Logic, volume 7, No.1, pp.1-36, 2001 S. Artemov, `` Operations on proofs that can be specified by means of modal logic ,'' Technical Report CFIS 99-02, Cornell University, 1999. To appear in Advances in Modal Logic , Volume 2, CSLI Publications, Stanford University, pp.59-72, 2001. S. Artemov, `` Unified semantics for modality and lambda-terms via proof polynomials ,'' Technical Report CFIS 99-03, Cornell University, 1999. To appear in Logic, Language and Computation, CSLI Publications, Stanford University, 2001 J.Alt and S.Artemov, Reflective lambda-calculus , Technical Report CFIS 00-06, Cornell University, 2000 (submitted to a conference in 2001) S.Artemov, E.Kazakov and D.Shapiro

13. Words And Other Things: Proofs And Provability Provability logic is an interpretation of modal logic. Now, the question I have is what is the relation between proof theory and Provability logic? http://indexical.blogspot.com/2007/11/proofs-and-provability.html

skip to main skip to sidebar Words and Other Things Trying to generate more light than heat since 2006 Wednesday, November 07, 2007 Proofs and provability Earlier today I had a thought about proof theory. Proof theory is, roughly, a formal investigation of the properties of proof systems. (I sort of dig proof theory.) Provability logic Posted by Shawn at 8:18 PM Labels: logic 2 comments: Kenny said... I remember learning a little bit of provability logic in Johan van Benthem's modal logic class at Stanford. At any rate, I seem to recall that the characteristic axiom was []([]p->p)->[]p, which they called Loeb's Theorem. There's a proof-theory argument for that. I think once you've got that axiom though, the argument from Godel's first incompleteness theorem to the second is very quick, and can be written in modal terms. At least, I once was familiar enough with this stuff that I remember writing a proof something like that down on my preliminary exam in logic. 1:27 AM Shawn said...

14. Giorgi Japaridze: Research And Publications Semidecidable fragments of first order Provability logic. The arithmetical completeness of Provability logic with quantifier modalities. http://www.csc.villanova.edu/~japaridz/study.html

Giorgi Japaridze Research and Publications NOTE: Before 1994 the English spelling of my name was "Dzhaparidze", the result of a two-step transliteration Georgian -> Russian -> English, while the current "Japaridze" is a direct transliteration from Georgian into English. Don't get confused if you need to look up my name in some earlier literature. Main Contributions to Science Publications Dissertations Research grants ... American Mathematical Society review of my papers Main Contributions to Science Provability and interpretability logics While a student, introduced what is now called " Japaridze's Polymodal Logic " GLP , and proved its arithmetical completeness. This contained a solution of an open problem on the logic of w- provability raised by George Boolos a decade earlier (1985-1988). Introduced Logic D and proved its arithmetical completeness (1987). Extended Solovay's theorems from propositional level to the one-variable predicate level, and introduced the corresponding sound and complete logic GLq Introduced the concepts of cointerpretability tolerance and cotolerance Proved that cointerpretability is equivalent to S -conservativity and tolerance is equivalent to P -consistency. This was an answer to the long-standing open problem regarding the metamathematical meaning of

15. Review Albert Visser, A Course On Bimodal Provability Logic Albert Visser, A Course on Bimodal Provability logic. Fulltext Access via JSTOR (no additional login). Go to this article in JSTOR. Links and Identifiers http://projecteuclid.org/handle/euclid.jsl/1183745258

Log in RSS Title Author(s) Abstract Subject Keyword All Fields FullText more options Home Browse Search ... next Review: Albert Visser, A Course on Bimodal Provability Logic Franco Montagna Source: J. Symbolic Logic Volume 62, Issue 2 (1997), 686-687. Reviewed Items: Albert Visser, A Course on Bimodal Provability Logic. Full-text: Remote access If you are a member of the ASL, log in to Euclid for access. Full-text is available via JSTOR, for JSTOR subscribers. Go to this article in JSTOR. Links and Identifiers Permanent link to this document: http://projecteuclid.org/euclid.jsl/1183745258 JSTOR: links.jstor.org back to Table of Contents previous next Journal of Symbolic Logic Current Issue All Issues Search this Journal Editorial Board ... Log in RSS Cornell University Library

16. Provability Logic - Indopedia, The Indological Knowledgebase Provability logic, or the logic of Provability, is a modal logic where the necessity operator is interpreted as Provability in a reasonably rich formal http://www.indopedia.org/Provability_logic.html

Indopedia Main Page FORUM Help ... Log in The Indology CMS Categories Modal logic Printable version Wikipedia Article Provability logic ज्ञानकोश: - The Indological Knowledgebase Provability logic , or the logic of provability , is a modal logic where the "necessity" operator is interpreted as provability in a reasonably rich formal theory such as Peano arithmetic . It was pioneered by Robert Solovay in 1976. Since then until his passing in 1996 the prime inspirer of the field was George Boolos . Significant contributions to the field have been made by Sergei Artemov, Lev Beklemishev, Giorgi Japaridze, Dick de Jongh, Franco Montagna, Vladimir Shavrukov, Albert Visser and others. Interpretability logics present natural extensions of provability logic. edit References Provability logic http://plato.stanford.edu/entries/logic-provability/ , from the Stanford Encyclopaedia of Philosophy. George Boolos, The Logic of Provability . Cambridge University Press, 1993. Giorgi Japaridze http://www.csc.villanova.edu/~japaridz/ and Dick de Jongh

17. The Logic Of Provability - Cambridge University Press Modal logic within set theory; 14. Modal logic within analysis; 15. The joint Provability logic of consistency and wconsistency; 16. http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521483254

18. Wolter, Frank: All Finitely Axiomatizable Subframe Logics Containing The Provabi All finitely axiomatizable subframe logics containing the Provability logic CSM are decidable, 1997 In this paper we investigate those extensions of the http://lips.informatik.uni-leipzig.de/pub/1997-37/en

Category Value Available via http://lips.informatik.uni-leipzig.de:80/pub/1997-37/en Submitted on 4th of September 1998 Author Wolter, Frank Title All finitely axiomatizable subframe logics containing the provability logic CSM are decidable Date of publication Published in Archive in Mathematical logic (wird erscheinen) Citation Wolter, Frank. All finitely axiomatizable subframe logics containing the provability logic CSM are decidable, 1997 Number of pages Language English Organization The Institute of Computer Science Type Conference or Journal Paper Subject group Computer Science, Data Processing Abstract In this paper we investigate those extensions of the bimodal provability logic CSM0 (alias PRL1 or F?) which are subframe logics, i.e. whose general frames are closed under a certain type of substructures. Most bimodal provability logics are in this class. The main result states that all finitely axiomatizable subframe logics containing CSM0 are decidable. We note that, as a rule, interesting systems in this class do not have the finite model property and are not even complete with respect to Kripke semantics. Source(s) Postscript ( ps ps.gz

19. NDJFL Editors Warren Goldfarb, Philosophical logic Proof Theory, Provability logic, Early Analytic Philosophy (Frege, Russell, Wittgenstein, Carnap) http://www.nd.edu/~ndjfl/editors.html

NDJFL Editorial Board The table below shows the areas of interest for each of our editorial board members. Papers submitted for publication should be sent to the Production Editor, Martha Kummerer , not to individual board members. Authors are welcome, however, to suggest potential editors/areas for their submissions. Editors-in-chief Areas of Interest Michael Detlefsen Philosophical Logic: Philosophy of Mathematics, Proof Theory Peter Cholak Mathematical Logic: Computability Theory Editors Areas of Review Peter Aczel Mathematical Logic: Constructive Mathematics, Foundations of Mathematics, Dependent Type Theories G. Aldo Antonelli Philosophical Logic: Knowledge Representation, Foundation of Mathematics, Alternative Set Theories Jeremy Avigad Mathematical Logic: Proof Theory Patrick Blackburn Applied Logic: Modal Logic, Logic and Natural Language, Logic and Computation Patricia Blanchette Philosophical Logic: Philosophy of Logic, Philosophy of Mathematics Sam Buss Mathematical Logic: Complexity Theory, Theories of Arithmetic, Proof Theory Philosophical Logic: Proof Theory, Categorical Logic, Substructural Logics

20. Home Page Of Rostislav Yavorskiy On Kripkestyle Semantics for the Provability logic of Godel s Proof Predicate with Quantifiers on Proofs. Journal of logic and Computation 2005, Vol. http://www.mi.ras.ru/~rey/

R ostislav Y avorskiy Senior Researcher at the Department of Mathematical Logic Steklov Mathematical Institute Russian Academy of Sciences Research interests first order logic; provability logic, logic of proofs; applications of mathematical logic in computer science Selected publications Rostislav Yavorskiy. On Kripke-style Semantics for the Provability Logic of Godel's Proof Predicate with Quantifiers on Proofs. Journal of Logic and Computation 2005, Vol. 15, No. 4, pp. 539-549. Vladimir Filatov and Rostislav Yavorskiy. Scenario based analysis of linear computations. Proceedings of 12th International Workshop on Abstract State Machines ASM'05, March 8-11, 2005, Laboratory of Algorithmics, Complexity and Logic, University Paris 12 - Val de Marne, Creteil, France, pp. 167-174. PDF Andrey Novikov and Rostislav Yavorskiy. Applying formal semantics of an object-oriented language to program invariant checking. Proceedings of 12th International Workshop on Abstract State Machines ASM'05, March 8-11, 2005, Laboratory of Algorithmics, Complexity and Logic, University Paris 12 - Val de Marne, Creteil, France, pp. 305-312. PDF Anton Esin, Andrey Novikov and Rostislav Yavorskiy.

21. Provability Logic (logic) - Philosophy Dictionary And Research Guide Provability logic Provability logic is a modal logic, in which the box (or. http://www.123exp-beliefs.com/t/00804249409/

The Language of Philosophy - Dictionary and Research Guide Provided by Search: Add to Favorites Provability logic Provability logic is a modal logic, in which the box (or "necessity") operator is interpreted as 'it is provable that'. The point is to capture the notion of a proof predicate of a reasonably rich formal theory, such as Peano arithmetic. Wikipedia and Wikis Provability logic - Wikipedia http://en.wikipedia.org/wiki/Provability_logic Other Provability logic http://plato.stanford.edu/entries/logic-provabilit ... Sergei Artemov http://www.cs.gc.cuny.edu/~sartemov/ Lev Beklemishev http://www.phil.uu.nl/~lev/ Giorgi Japaridze http://www.csc.villanova.edu/~japaridz/ ''Provability logic'' http://www.phil.uu.nl/preprints/preprints/PREPRINT ... ''The logic of provability'' http://www.csc.villanova.edu/~japaridz/Text/prov.p ... More topics about: Provability logic Edit this page Add new links, rate and edit existing links, or make suggestions. - Staff Explore related topics: Kripke semantics Interpretability logic Stanford Encyclopedia of Philosophy Robert M. Solovay ... modal logic Some descriptions may have been derived in part from Princeton University WordNet or Wikipedia Last update: December 19, 2007

22. Broadview Press: Logical Options Provability logic. 4.4.1. Arithmetic Provability and Contextual logic. 4.4.2. Frames and Provability logic. 4.4.3. Trees for Provability logic http://www.broadviewpress.com/bvbooksprintable.asp?BookID=237

23. [Iem2001] A Modal Analysis Of Some Principles Of The Provability Logic Of Heytin @inproceedings{Iem01c, volume = {2}, title = {A modal analysis of some principles of the Provability logic of {H}eyting Arithmetic}, year = {2001}, http://www.logic.at/dmgfg2-pub/entry-Iem01c.html

Overview Tree Index Bibliography FRAMES NO FRAME [Iem2001] A modal analysis of some principles of the Provability Logic of Heyting Arithmetic (In proceedings) Author(s) Iemhoff R. Title " A modal analysis of some principles of the Provability Logic of Heyting Arithmetic " Date In Advances in Modal Logic Editor(s) Rijke (de) M., Segerberg K., Wansing H. and Zakharyaschev M. Volume Page(s) Publisher CSLI Publications BibTeX code Overview Tree Index Bibliography FRAMES NO FRAME Submit a bug This document was generated by Stéphane GALLAND (under the GNU General Public License

24. EconPapers: Common Knowledge And Common Rationality Through Provability Logic Common Knowledge and Common Rationality Through Provability logic. Corrado Benassi and P. Gentilini. Working Papers from Dipartimento Scienze Economiche, http://econpapers.repec.org/paper/bolbodewp/350.htm

EconPapers Home About EconPapers Working Papers Journal Articles ... Format for printing EconPapers has moved to http://econpapers.repec.org! Please update your bookmarks. Common Knowledge and Common Rationality Through Provability Logic Corrado Benassi and P. Gentilini Working Papers from Dipartimento Scienze Economiche, Universit  di Bologna Date: There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: This item may be available elsewhere in EconPapers: Search for items with the same title. Access Statistics for this paper More papers in Working Papers from Dipartimento Scienze Economiche, Universit  di Bologna Contact information at EDIRC Series data maintained by Piera Appoggi, Dolve Bolzoni ( This site is part of RePEc and all the data displayed here is part of the RePEc data set. Is your work missing from RePEc? Here is how to contribute Questions or problems? Check the EconPapers FAQ or send mail to Econpapers is hosted by the Department of Business, Economics, Statistics and Informatics at –rebro University. Page updated 2007-12-20 Handle: RePEc:bol:bodewp:350

25. Martin Löb (1921-2006) The inspiration of Löb s Theorem made Amsterdam one of the places where the first results in Provability logic were obtained, a program started by de http://www.illc.uva.nl/Obituaries/Loeb.html

Martin Löb (1921-2006) Home About the ILLC News and Events People ... Search Martin Löb, Amsterdam, 14 November 1978 On Monday August 21th 2006, our esteemed former colleague Professor Martin Löb, holder of the chair of Mathematical Logic from 1971 to 1985 at the University of Amsterdam, passed away in Annen (Drente). Obituary by Stan Wainer (Mathematics, University of Leeds) to appear in the Guardian Martin Löb: A Pioneer of Mathematical Logic Martin was an intensely private, cultured and quietly strong-willed person, devoted to his Dutch wife Caroline and their daughters Maryke and Stefani. After his retirement from Amsterdam they moved to a quiet spot in the north of Holland, and there he stayed until his death, Caroline having sadly pre-deceased him. He valued his students highly and was concerned as much for their welfare as their academic progress. Only a few had the good fortune to complete their doctorates under him but two of them now head the departments where he worked, and he influenced many others in the early stages of their careers. He is remembered with affection, as a profound and dedicated logician and teacher, and a man of great inner strength and integrity. Supplementary statement by Dick de Jongh (ILLC, University of Amsterdam):

26. CiteULike: On Provability Logic msakai s tags for this article. logic modallogic Provability game haskell ilp intuitionistic lcm linear-logic logic machine-learning maude modal-logic http://www.citeulike.org/user/msakai/article/1662361

Register Log in FAQ CiteULike News Discussion Journals Browse current issues Groups Search groups Profile Library Watchlist ... Import On Provability Logic Authors à  vejdar V Online Article URL: http://www.hf.uio.no/ifikk/filosofi/njpl/... Note: You or your institution must have access rights to this article. CiteULike will not help you view an online article which you aren't authorized to view. Copy-and-Pasteable Citation Nordic Journal of Philosophical Logic , Vol. 4, No. 2., pp. 95-116. Citation format: Plain APA Chicago Elsevier Harvard MLA Nature Oxford Science Turabian Vancouver msakai's tags for this article logic modal-logic provability Everyone's tags for this article logic modal-logic provability BibTeX Note: You may cite this page as: http://www.citeulike.org/user/msakai/article/1662361 EndNote BibTeX msakai's tags All tags in msakai's library Filter: category-theory coalgebra concurrency continuation ... virtual-machine CiteULike organises scholarly (or academic) papers or literature and provides bibliographic (which means it makes bibliographies) for universities and higher education establishments. It helps undergraduates and postgraduates. People studying for PhDs or in postdoctoral (postdoc) positions. The service is similar in scope to EndNote or RefWorks or any other reference manager like BibTeX, but it is a social bookmarking service for scientists and humanities researchers.

27. Matematicheskie Zametki Provability logic without Craig s interpolation property L. D. Beklemishev UDC 510.65 Received 26.10.1988 Citation L. D. Beklemishev, Provability logic http://www.mathnet.ru/php/journal.phtml?wshow=paper&jrnid=mzm&paperid=3498&year=

28. CSLI Calendar, 19 November 1997, Vol. 13:10 logic of Proofs provides an intended Provability semantics for the Goedel Provability logic S4, as well as for some other constructions in logic and http://www-csli.stanford.edu/Archive/calendar/1997-98/msg00010.html

[Prev] [Next] [Index] [Thread] CSLI Calendar, 19 November 1997, vol. 13:10 To friends@Csli.Stanford.EDU Subject : CSLI Calendar, 19 November 1997, vol. 13:10 From emma@csli.Stanford.EDU Date : Wed, 19 Nov 1997 12:47:43 -0800 (PST) C S L I C A L E N D A R O F P U B L I C E V E N T S 19 November 1997 Stanford Vol. 13, No. 10 A weekly publication of the Center for the Study of Language and Information (CSLI) Stanford University, Ventura Hall, Stanford, CA 94305-4115 ACTIVITIES DURING 19 NOVEMBER TO 28 NOVEMBER 1997 WEDNESDAY, 19 NOVEMBER 3:45pm Psychology Department Colloquium Jordan Hall (420:050) Pathways to early conscience Grazyna Kochanska University of Iowa 4:15pm EE380: Computer Systems Laboratory Colloquium Gates B03 (NEC Auditorium) The AVR Family of Embedded Processors Jim Panfil [ http://www.stanford.edu/class/ee380/contents.html

29. Project: Constructive And Intensional Logic (www.onderzoekinformatie.nl) Another theme is Provability and interpretability logic of arithmetic, Some new results on the Provability logic of Heyting arithmetic were obtained, http://www.onderzoekinformatie.nl/en/oi/nod/onderzoek/OND1280272/

Login English KNAW Research Information NOD - Dutch Research Database ... Research entire www.onderzoekinformatie.nl site fuzzy match Project: Constructive and intensional logic Print View Titel Constructieve en intensionele logica Abstract - Characterization: This project continues the long-standing Amsterdam tradition in mathematical logic and the foundations of mathematics. Over the years, the original core theme of constructivism has widened to become general proof theory and provability logic, and on the other hand modal and dynamic logic. Thus, the two main 'trademarks' of mathematical logic at Amsterdam fit together in their efforts to create a general framework for reasoning and information flow. - Main themes: The first theme is concerned with foundations of constructivism, and more general proof theories emanating from that tradition. Current interests here include intuitionistic logic, type theories, linear logic and other substructural resource logics. Semantic foundations of these theories, in terms of Kripke models or categorial models, are also actively investigated. The eventual aim is a general formulation of the constructivist program as a practical general-purpose tool for the working mathematician, computer scientist, and computational linguist. Another theme is provability and interpretability logic of arithmetic, with its current ramifications into recursion theory and complexity theory, as well as 'weak arithmetics'. Topics here include interpolation properties, axiomatic completeness, and modal-style formalizations of further proof-theoretic notions. This research has close ties with the projects Algorithmics and Information Processing and Computational and Applied Logic. It also serves as a test-bed for more discriminating notions of complexity that may increase our theoretical understanding of the actual workings of automated deduction on large-scale input sets.

30. NORM Since Löb s announcement of his solution to Henkin s problem (LÖB54, LÖB55) there has been successful and fruitful research on Provability logic tied up http://nb.vse.cz/kfil/elogos/logpoint/93-2/KIM.htm

Paradoxes as Analogues to G¶del's Second Incompleteness Theorem SANG MUN KIM As L¶b's paradox was uncovered by L.Henkin as an analogue to L¶b's theorem, paradoxes as analogues to G¶del's Second Incompleteness Theorem are derived in this paper from an inconsistent set of sentences arising from G¶del's Second Incompleteness Theorem. As the proof of G¶del's Second Theorem ( ) is largely a formalization of the proof of G¶del's First Theorem ( might be said to have provided a cross check on proposed consistency proofs, despite G¶del's remark on the irrelevance of to any sensible consistency problem - " if Con(F) is in doubt, why should it proved in F and not in an incomparable system? " [KREISEL]. Since L¶b's announcement of his solution to Henkin's problem ([L–B54], [L–B55]) there has been successful and fruitful research on provability logic tied up with modal logic. L¶b's theorem ( LT ) is of far-reaching significance in the metamathematical and philosophical sense. In particular, LT plus some additional properties of the modal box r standing for provability axiomatize completely provability logic theory. Thus, the proofs of the provable equivalence between

31. DBLP: Albert Visser 3, Albert Visser An Inside View of EXP; or, The Closed Fragment of the Provability logic of I Delta0+Omega1 with a Propositional Constant for EXP. http://www.informatik.uni-trier.de/~ley/db/indices/a-tree/v/Visser:Albert.html

Albert Visser List of publications from the DBLP Bibliography Server FAQ Coauthor Index - Ask others: ACM DL Guide CiteSeer CSB ... EE Albert Visser, Maartje de Jonge : No Escape from Vardanyan's theorem. Arch. Math. Log. 45 EE Processes, Terms and Cycles 2005 EE ... Lev D. Beklemishev , Albert Visser: On the limit existence principles in elementary arithmetic and Sigma n -consequences of theories. Ann. Pure Appl. Logic 136 Albert Visser: Substitutions of Sigma - sentences: explorations between intuitionistic propositional logic and intuitionistic arithmetic. Ann. Pure Appl. Logic 114 Albert Visser: The Donkey and the Monoid. Dynamic Semantics with Control Elements. Journal of Logic, Language and Information 11 Marco Hollenberg , Albert Visser: Dynamic Negation, the One and Only. Journal of Logic, Language and Information 8 Albert Visser: Rules and Arithmetics. Notre Dame Journal of Formal Logic 40 Albert Visser: Contexts in Dynamic Predicate Logic. Journal of Logic, Language and Information 7 Albert Visser: Dynamic Relation Logic Is the Logic of DPL-Relations. Journal of Logic, Language and Information 6

32. Švejdar Švejdar, Vít zslav The decision problem of Provability logic with only one atom; In Archive for Mathematical logic. 2003, ro . 42, . 8, s. 763768. http://svi.ff.cuni.cz/sm_sy/svejdar.htm

Švejdar, Vítìzslav zpìt na seznam Švejdar, Vítìzslav: Note on Inter-Expressibility of Logical Connectives in Finitely-Valued Godel-Dummett Logics ; In: Soft Computing. 2006, roè. 10, è. 7, s. 629-630. Poznámka o vzájemné vyjádøitelnosti logických spojek v koneènìhodnotových Godelových-Dummettových logikách) ISSN 1432-7643. Anotace: Švejdar, Vítìzslav: On Modal Systems with Rosser Modalities editoøi: Tomala, O., Bílková, M.; In: The Logica Yearbook 2005: Proc. of the Logica '05 International Conference . 1. vyd. 2006, Praha: Filosofia; s. 203-214. O modálních systémech s rosserovskými modalitami) ISBN 80-7007-229-6. Anotace: Modal logics with Rosser modalities are discussed. Alternative axioms involving these symbols that are sound w.r.t. generalized proof predicates are presented. Švejdar, Vítìzslav: On Sequent Calculi for Intuitionistic Propositional Logic ; In: Commentationes Mathematicae Universitatis Carolinae. 2006, roè. 47, è. 1, s. 159-173. O sekventových kalkulech pro intuicionistickou výrokovou logiku) ISSN 0010-2628. Anotace: Single- and multi-conclusion calculi for intuitionistic propositional logic are discussed or presented, complexity of decision procedures based on these calculi is explored.

33. Anders Moen On Numerals, Variables and Quantifiers in Provability logic abstract logic is the scientific language of Computer Science. Instead of investigating logic http://www.nr.no/~andersmo/personal.html

Anders Moen Born 24.04.1969 BA: Mathemathics, Philosophy and Computer Science (SLI) Master Degrees: Philosophy: "Models for Delgrandes NP" (1996), Theoretical Computer Science (SLI): "Proof Theoretical Investigations into Gödel's second Incompleteness Theorem" (1999) Founder of NJPL (Nordic Journal of Philosophical Logic) together with Johan W. Klüwer and Arild B. Torjussen 1995-1996. Coeditor of NJPL 1995-1997. Referee for Nordic Journal of Computing 2001. University Lecturer (20%) at Department of Informatics PMA group from 01/08-20o1 Scientific publications: "The concept 'session' modelled from below" NWPT'2000 Working on the papers: "On Numerals, Variables and Quantifiers in Provability Logic" abstract: Logic is the scientific language of Computer Science. Instead of investigating logic in computer science, we apply concepts of computer science in provability logic. The way variables are treated in provability logic is similar to the concept of local and global variable in block-structured programming languages. This insight can be used to give an understanding of the notion of provability 'perspective', one missing link in the foundations of provability logic. "Non-termination of existing algorithms for Cut-elemination in Provability Logic" abstract: To find an appropriate axiomatization of provability logic in a Gentzen type of sequent calculus, still remains an open problem. We prove that the existing algorithms presented by D. Leivant and S. Valentini, for eliminating cuts one by one, does not terminate. But we prove a normal form theorem, that every proof-tree can rewritten to a prooftree on Sambin Normal form. The theorem can be generalized to modal logical rules with only boxed formulas in the consequent sequent.

34. LICS Newsletter 10 PPC provides a platform for the presentation of recent results in the areas of Proof Theory, Provability logic, and Computation where these are interrelated http://www2.informatik.hu-berlin.de/lics/newsletters/10.html

Newsletter 10, September 16, 1993

35. Essays In Philosophy -- Book Review Boolos is best known for his work on modal Provability logic. Boolos’ The logic of Provability (1993) is regarded as the single best volume on the subject, http://www.humboldt.edu/~essays/marrev.html

Essays in Philosophy A Biannual Journal Vol. 1 No. 2, June 2000 Book Review Logic, Logic and Logic , George Boolos. Harvard University Press, 1998. ix + 443 pages. Hardcover $45, paperback $22.95. ISBN 0-674-53767-X. Cancer robbed our community of an outstanding philosopher. One is tempted to say “philosopher and logician,” but as Richard Cartwright remarked in his eulogy for George Boolos, “he would have not been altogether happy with the description: accurate, no doubt, but faintly redundant—a little like describing someone as ‘mathematician and algebraist.’” While the title of this book is redundant, its content is not. Completed by colleagues, students and friends, this posthumous collection is not only a testament to Boolos’ legacy but also to his logical virtuosity. George Boolos made significant contributions in every area of logic in which he worked. The volume is a treasure trove of insightful observations and elegant formal contributions to central questions in the philosophy of logic. The book is divided into three sections: Studies on Set Theory and the Nature of Logic, Frege Studies, and Various Logical Studies and Lighter Papers. The first section contains Boolos’ seminal essay “The Iterative Conception of Set,” (article 1) published in the early 1970s, which introduced philosophers to the underlying intuitive conception of set articulated by Zermelo’s axioms. Most philosophers had regarded these axioms as

36. Introduction The idea of Provability logic arose in the seventies in work of G. The Provability logic then consists of modal formulas, which are “valid” in every http://www.math.cas.cz/~jerabek/papers/prlast.html

Provability logic of the Alternative Set Theory Emil Jeà ™¡bek Master’s thesis, Faculty of Philosophy and Arts, Charles University, Prague, 2001, 44 pp. PS PDF BIB The idea of provability logic arose in the seventies in work of G. Boolos, R. Solovay, and others, as an attempt to explore certain “modal effects” in the metamathematics of the first order arithmetic. Namely, the formal provability predicate Pr x , originally constructed by G¶del, has several properties resembling the necessity operator of common modal logics: the L¶b’s derivability conditions, T T ⊢ Pr T ⊢ Pr ⌝)→(Pr ⌝)→Pr T ⊢ Pr ⌝)→Pr (⌜Pr look just like an axiomatization of a subsystem of S4: We may form “arithmetical semantics” for formulas in the propositional modal language as follows: we substitute arithmetical sentences for propositional atoms, Pr for boxes, and we ask whether the resulting sentence (the “arithmetical realization” or “provability interpretation” of the modal formula) is provable in our arithmetic T . The provability logic then consists of modal formulas, which are “valid” in every such “model”.

37. Abbeys Bookshop - Logic Of Provability Its subject is the relation between Provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually http://www.abbeys.com.au/items/08/77/58/

Could not build the menu javascript is turned off Quick Search Title/Keywords Author ( Advanced ) Categories Art, Architecture and Photography (3256 Titles) Music and Performing Arts (2155 Titles) Language Learning, Foreign Language, ESL and ELT (10252 Titles) Biography (2997 Titles) Literature (7686 Titles) Fiction (31298 Titles) Reference (746 Titles) History and Archeology (8763 Titles) Philosophy and Religion (5205 Titles) Social sciences and Law (9757 Titles) Economics and Business (1894 Titles) Science and Medicine (6534 Titles) Computing and information technology (654 Titles) Health, Cooking, Gardening and Practical Interests (2606 Titles) Travel and Leisure Interests (3347 Titles) Children's and educational (11904 Titles) Gift Vouchers (11 Titles) The Logic of Provability Author : GEORGE BOOLOS Edition : New ed Format : Paperback ISBN : Publisher : Cambridge University Press Publication Date (UK) : July 1995 Pages : Imprint : UCAMBR Usually ships within 24 hours Web Price AUD$89.95 Description This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency (1979). Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians. Here it receives its first scientific application since its invention. Modal logic is concerned with the notions of necessity and possibility. What George Boolos does is to show how the concepts, techniques, and methods of modal logic shed brilliant light on the most important logical discovery of the twentieth century: the incompleteness theorems of Kurt Godel and the 'self-referential' sentences constructed in their proof. The book explores the effects of reinterpreting the notions of necessity and possibility to mean provability and consistency.

38. Springer Online Reference Works Modal logic. This interpretation is related to the normal extensions of Provability logic, because there exists a natural isomorphism between the lattice of http://eom.springer.de/M/m110020.htm

Encyclopaedia of Mathematics M Article refers to Magari algebra, diagonalizable algebra A Boolean algebra enriched with a unary operation . In the so-expanded signature, the Magari algebra is defined by the axioms of Boolean algebra and the following three specific axioms: Here, denotes complementation and the unit is the greatest element of the Magari algebra with respect to the relation . The notation is often employed instead of One sometimes regards the Magari algebra with the dual operation ), defined by the axioms: Here, the zero is the least element of the Magari algebra. In order to distinguish the Boolean part of a Magari algebra, one writes the Magari algebra as the pair or , where is a Boolean algebra. Magari algebras arose as an attempt to treat diagonal phenomena (cf. the diagonalization lemma in ) in the formal Peano arithmetic Arithmetic, formal ) in an algebraic manner. Indeed, the Lindenbaum sentence algebra of Peano arithmetic, , equipped with defined by is an example of a Magari algebra. Here is the of the sentence and means the equivalence class of the arithmetical sentences formally equivalent in Peano arithmetic to the sentence ). The diagonalization lemma is simulated with the following

39. Magnus Boman - Older Publications Towards the end of my thesis work, one of my supervisors PerErik Malmnäs told me I had to choose between Provability logic and all my other interests. http://www.sics.se/~mab/Publ.html

Annotated list of select publications Magnus Boman Feel free to enquire about hardcopies and downloading problems regarding papers in this highly personal selection, annotated by Magnus Boman Skip to fairly recent work Skip to current research I start with stuff moved from my recent papers page because they stopped being recent: Laaksolahti, J. and Boman, M. (2003) Anticipatory Guidance of Plot In Butz, M.; Sigaud, O. and Gerard, P., eds., Anticipatory Behavior in Adaptive Learning Systems . Springer-Verlag, LNAI2684 pp. 243261 Rasmusson, L. and Boman, M. (2002) Analytical expressions for Parrondo games Fluctuation and Noise Letters Now to a fairly chronological account which I wrote in 2001. My masters thesis Boman, M.: Formal representation of the reasoning of intelligent agents by means of epistemic and doxastic modal propositional logic (DSV Report WP153, Stockholm Univ, 1989) looked at Robert Moore's work, and combined my undergraduate studies in logic with my attempt to delve into the artificial intelligence literature. There were not many papers with "intelligent agent" in the title around this time, and I was thrilled to find other people interested in 1990, when I attended a highly inspiring conference in Keele, UK. There, I met several people that helped found the area of multi-agent systems; among these were Eric Werner, Cristiano Castelfranchi, and Julia Galliers. Julia bravely invited me to Cambridge, and Eric got me involved with MAAMAW, the first European agent event, which I still like a lot.

40. DMG-FG2: Projects It has resulted in the description of the Provability fragment of a logic which is conjectured to be the preservativity (a constructive analogue to http://www.dmg.tuwien.ac.at/fg2/index.php?id=40

41. PlanetMath: Gödel's Incompleteness Theorems The second incompleteness theorem is best presented by means of a Provability logic. Consider an arithmetic theory $ T$ which is p.r. axiomatised by http://planetmath.org/encyclopedia/GodelsIncompletenessTheorems.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... RSS Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About G¶del's incompleteness theorems (Theorem) logic , one can formulate properties of theories and sentences as arithmetical properties of the corresponding , thus allowing 1st order arithmetic to speak of its own consistency, provability of some sentence and so forth. On Formally Undecidable Propositions in Principia Mathematica and Related Systems can be stated as Theorem No theory axiomatisable in the type system of PM (i.e., in Russell's theory of types ) which contains Peano-arithmetic and is -consistent proves all true theorems of arithmetic (and no false ones). Stated this way, the theorem is an obvious corollary of Tarski's result on the undefinability of truth formula , and by Tarski's result it isn't definable by any arithmetic formula. But assume there's a theory

42. Oxford Scholarship Online: Interpolation And Definability This chapter shows that actions of interpolation in extensions of the Provability logic G differs from that over S4. A logic G 0x0003b3 http://www.oxfordscholarship.com/oso/public/content/maths/9780198511748/acprof-9

About OSO What's New Subscriber Services Help ... Interpolation and Definability - Modal and Intuitionistic Logics Gabbay, Dov M. Department of Computer Science, King's College London Maksimova, Larisa Institute of Mathematics, Siberian Branch of Russian Academy of Science, Novosibirsk, Russia Print publication date: 2005 (this edition) Published to Oxford Scholarship Online: September 2007 Print ISBN-13: 978-0-19-851174-8 doi:10.1093/acprof:oso/9780198511748.003.0012 12 EXTENSIONS OF THE PROVABILITY LOGIC D.M. Gabbay L. Maksimova This chapter shows that actions of interpolation in extensions of the provability logic G differs from that over S4. A logic G is the greatest among the infinite-slice logics over G with interpolation property. A continuum of logics with CIP is constructed, and a short proof of the Beth property in logics over G is found. Keywords: provability logic G infinite slice logic diagonalizable algebra ... Beth property doi:10.1093/acprof:oso/9780198511748.003.0012 Quick Search Form Quick Search search entire site search this title only Advanced Search Bibliography Search Contents Full Book Contents Preface 1 INTRODUCTION AND DISCUSSION 2 MODAL AND SUPERINTUITIONISTIC LOGICS: BASIC CONCEPTS ... 7 INTERPOLATION, DEFINABILITY, AMALGAMATION

43. The Discreteness Of Time Interestingly, GL logic is sound and complete for the class of conversewellfounded Kripke frames. The subject of Provability logic has been thoroughly http://www.dcs.ed.ac.uk/home/pgh/dummet.html

The discreteness of time This note concerns the following axiom (scheme) of temporal logic: In words, if eventually p holds forever, and p is precessive in a certain sense, then p must hold already. If one makes the abbreviation p ~> q =def [](p -> q) , one can also write it []p -> []([](p ~> []p) ~> []p) ~> []p On the web, one finds numerous slight variants. These are all() equivalent over KT4. (Rajeev Gore) []p -> []([](p -> []p) -> []p) -> []p []p -> []([](p -> []p) -> p) -> []p []p -> []([](p -> []p) -> []p) -> p []p -> []([](p -> []p) -> p) -> p A classical equivalent that is quite enlightening is: [](p -> p)) -> p -> [] p In essence, Dummett's scheme is a weakening of Grzegorcyk's by " []p ->". It is the only scheme of linear-time discrete temporal logic in which modalities occur nested three deep. The axiom comes from some subset of Grzegorczyk, Prior, Geach, Dummett, Bull, Kripke and Lemmon. (See Hughes and Cresswell, An Introduction to Modal Logic, round pages 261-262, for some references. Also, p. lvii of Truth and Other Enigmas, by Dummett.) It is sometimes called axiom D , perhaps because it says that time is discrete, in the sense that a certain principle of `reverse' temporal induction holds, which argues from something's being inevitable in the future, back towards its holding in the present (providing that something is precessive). (Or perhaps

44. Rineke Verbrugge On the Provability logic of bounded arithmetic,. Annals of Pure and Applied logic. ,. 61. (1993) 7593. Contributions to edited booksback to top http://www.ai.rug.nl/~rineke/topics/topic.php?topic=Publications