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	Provability logic - Wikipedia, the free encyclopedia
		Provability logic is a modal logic where the box (or "necessity") operator is interpreted as 'it is provable that', capturing 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.
		Giorgi Japaridze and Dick de Jongh, The logic of provability.
	en.wikipedia.org /wiki/Provability_logic (219 words)

	Logic
		Aristotelian logic is sometimes referred to as formal logic because it specifically deals with forms of reasoning, but is not formal in the sense we use it here or as is common in current usage.
		Mathematical logic refers to two distinct areas of research: The first, primarily of historical interest, is the use of formal logic to study mathematical reasoning, and the second, in the other direction, the application of mathematics to the study of formal logic.
		Logics such as fuzzy logic have since been devised with an infinite number of "degrees of truth", e.g., represented by a real number between 0 and 1.
	www.brainyencyclopedia.com /encyclopedia/l/lo/logic_1.html (1472 words)

	Boolos Bibliography
		Reprinted in Logic, Logic, and Logic and in Benacerraf, P. and Putnam, H., eds.
		"Provability in arithmetic and a schema of Grzegorczyk." Fundamenta Mathematicae 106 (1980): 41-45.
		Reprinted in Logic, Logic, and Logic and in William Demopoulos, ed., Frege's Philosophy of Mathematics Cambridge, Mass: Harvard University Press, 1995, pp.
	web.mit.edu /philos/www/facultybibs/boolos_bib.html (1077 words)

	Modal Logic
		Modal logic is, strictly speaking, the study of the deductive behavior of the expressions ‘it is necessary that’ and ‘it is possible that’.
		Deontic logics introduce the primitive symbol O for ‘it is obligatory that’, from which symbols P for ‘it is permitted that’ and F for ‘it is forbidden that’ are defined: PA = ~O~A and FA = O~A. The deontic analog of the modal axiom (M): OA→A is clearly not appropriate for deontic logic.
		is a constant of provability logic denoting a contradiction.
	plato.stanford.edu /entries/logic-modal (7308 words)

	Provability logic - Encyclopedia, History, Geography and Biography
		It is obtained by adding the modal version of Löb's theorem to the logic K4.
		Provability logic (http://plato.stanford.edu/entries/logic-provability/), from the Stanford Encyclopedia of Philosophy.
		Sergei Artemov and Lev Beklemishev (http://www.phil.uu.nl/~lev/), Provability logic (http://www.phil.uu.nl/preprints/preprints/PREPRINTS/preprint234.pdf), In: Handbook of Philosophical Logic, D. Gabbay and F. Guenthner, eds., vol.
	www.arikah.com /encyclopedia/Provability_logic (210 words)

	Read about Provability logic at WorldVillage Encyclopedia. Research Provability logic and learn about Provability logic ... (Site not responding. Last check: )
		modal logic where the box (or "necessity") operator is interpreted as 'it is provable that', capturing the notion of a proof predicate of a reasonably rich
		Giorgi Japaridze (http://www.csc.villanova.edu/~japaridz/) and Dick de Jongh, The logic of Provability.
		Provability logic (http://www.phil.uu.nl/preprints/preprints/PREPRINTS/preprint234.pdf), In: Handbook of Philosophical Logic, D. Gabbay and F. Guenthner, eds., vol.
	encyclopedia.worldvillage.com /s/b/Provability_logic (171 words)

	Provability logic (Site not responding. Last check: )
		Interpretability logic s present natural extensions of provability logic.
		logic avance logic logic controller logic control programmable logic logic audio ladder logic fuzzy logic crayon logic logic test basic logic symbios logic boolean logic
		Logic is the basis of many principles including the scientific method.
	www.serebella.com /encyclopedia/article-Provability_logic.html (390 words)

	Provability Logic
		Provability logic is a modal logic that is used to investigate what arithmetical theories can express in a restricted language about their provability predicates.
		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.
		Even though propositional provability logic is a modal logic with a kind of “necessity” operator, it withstands Quine's (1976) critique of modal notions as unintelligible, because of its clear and unambiguous arithmetic interpretation.
	plato.stanford.edu /entries/logic-provability (4840 words)

	Problems
		Classification of bimodal provability logics for pairs of r.e.
		[Shavrukov] Are the provability algebras of PA and PA+Con(PA) isomorphic?
		The interpretability and the Pi_1-conservativity logic for PRA.
	www.phil.uu.nl /~lev/problems.html (1032 words)

	[No title]
		The discovery of a fundamental connection between modal logic and typed lambda-calculi according to which the major modal logics (K, K4, S4, S5) are nothing but implicit specifications of a families of lambda-terms standing for operations on proofs.
		Provability has the format of a logical connective: if F is a sentence then F is provable is again a sentence.
		Note that the logic of formal provability introduced by Solovay and Boolos in the 1970s addresses a different issue and it is not compatible with Goedel's provability calculus.
	www.cs.cornell.edu /Info/People/artemov/accom.html (981 words)

	Cointerpretability - Wikipedia, the free encyclopedia
		In mathematical logic, cointerpretability is a binary relation on formal theories: a formal theory T is cointerpretable in another such theory S, when the language of S can be translated into the language of T in such a way that S proves every formula whose translation is a theorem of T.
		G.Japaridze, A generalized notion of weak interpretability and the corresponding logic.
		G.Japaridze and D. de Jongh, The logic of provability.
	en.wikipedia.org /wiki/Cointerpretability (152 words)

	Provability logic - Definition, explanation
		Provability logic, from the Stanford Encyclopedia of Philosophy.
		Giorgi Japaridze and Dick de Jongh, The logic of Provability.
		Sergei Artemov and Lev Beklemishev, Provability logic, In: Handbook of Philosophical Logic, D. Gabbay and F. Guenthner, eds., vol.
	www.calsky.com /lexikon/en/txt/p/pr/provability_logic.php (324 words)

	Provability logic -- Facts, Info, and Encyclopedia article (Site not responding. Last check: )
		It is obtained by adding the modal version of (Click link for more info and facts about Löb's theorem) Löb's theorem to the logic K4.
		and Dick de Jongh, The logic of Provability.
		Sergei Artemov and,, In: Handbook of Philosophical Logic, D. Gabbay and F. Guenthner, eds., vol.
	www.absoluteastronomy.com /encyclopedia/p/pr/provability_logic.htm (183 words)

	The Logic of Provability - Cambridge University Press (Site not responding. Last check: )
		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.
		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.
	www.cambridge.org /catalogue/print.asp?isbn=0521433428&print=y (252 words)

	Group in Logic and the Methodology of Science -
		Students in this program acquire a good understanding of the mathematical theory known as mathematical logic, which deals in a rigorous way with such central concepts as truth, definability, provability, and computability.
		The program in Logic and the Methodology of Science is intended for students whose interests lie in more than one of these fields.
		The program is administered by the Group in Logic and the Methodology of Science, an interdepartmental agency which cooperates closely with the Department of Mathematics, the Department of Philosophy, and the Department of Electrical Engineering and Computer Sciences.
	logic.berkeley.edu (432 words)

	Is There a Logic of Provability for Nonmonotonic Reasoning? - Amati, Pirri (ResearchIndex) (Site not responding. Last check: )
		One of the central issue in the area is to establish a notion of nonmonotonic provability, which is usually characterized via some modal logic.
		29 Provability interpretations of modal logic (context) - Solovay - 1976
		22 Self-Reference and Modal Logic (context) - Smorynski - 1985
	citeseer.ist.psu.edu /amati96is.html (640 words)

	Courses: Artemov
		In particular, Goedel's provability logic S4 turned out to be a forgetful projection of a more general logic of proofs.
		Modal logic S4 together with the simple procedures recovering explicit proof terms from S4-derivations subsumes intuitionistic and combinatory logic, typed lambda-calculus and modal lambda-calculus.
		Explicit provability: the intended semantics for intuitionistic and modal logic.
	www.let.uu.nl /esslli/Courses/artemov.html (289 words)

	Readings in Logic
		These questions, though logically the first in arithmetic, were, of course, historically the last to be dealt with.
		I was startled to see the connections among the players in modern logic, and also be reminded of the terrible events of and between the World Wars and how this led to the great dispersal in which Princeton's Institute for Advanced Study arose as a safe haven.
		It is plain here and in (a) that explorations of mathematical logic moved into more-abstracted and intricate realms, seemingly far removed from the naive origins of axiomatic set theory.
	nfocentrale.net /orcmid/readings/logic.htm (4287 words)

	Giorgi Japaridze: Research and Publications
		In: Notre Dame Journal of Formal Logic 41 (2000), N 2, pp.171-183.
		In: Intensional Logics and Logical Structure of Theories.
		"On the classification of propositional logics of provability".
	www.csc.villanova.edu /~japaridz/study.html (896 words)

	Logic on the Web -- Web architecture
		In higher order logic, intuition suggests we should be able to to construct such a proof of validity in the logic itself.
		There are sets of logics which use sets, but limit them in some way - for example, by requiring sets to be "well formed", meaning they are either empty or contain at least one element which is disjoint with the set itself.
		These tricks limit the logic so that you can't in fact have the Russel paradox set (of all sets which are not members of themselves) as you exclude such things as not well-formed.
	www.w3.org /DesignIssues/Logic.html (3114 words)

	Sergei Artemov (Site not responding. Last check: )
		Logic, provability, proofs and types, modal and epistemic logics, knowledge representation and artificial intelligence
		Artemov, "Explicit provability and constructive semantics," Bulletin of Symbolic Logic, v.
		Artemov, "Operations on proofs that can be specified by means of modal logic," Advances in Modal Logic, Volume 2, CSLI Publications, Stanford University, pp.
	web.gc.cuny.edu /philosophy/artemov.htm (150 words)

	On Provability Logic (Site not responding. Last check: )
		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.
	www.hf.uio.no /filosofi/njpl/vol4no2/provlog (121 words)

	Amazon.com: Books: Computability and Logic (Site not responding. Last check: )
		Like other good logic texts-Jeffrey's Formal Logic or Pollock's Technical Method's (out of print, but available in PDF on his website)-there is very little commentary in the brief chapters, so it is useful if you are already very familiar with the material or if you have a very worthy guide.
		Unfortunately there is little on provability logic for which he was one of the pioneering logicians.
		People who are veterans with logic and logicians may easily spot typos, but for a first time student of the subject, I was confused as hell at some parts simply because there was a typo.
	www.amazon.com /exec/obidos/tg/detail/-/0521389232?v=glance (2001 words)

	Provability Logic (Site not responding. Last check: )
		See the entry modal logic for more information about possible worlds semantics in general.
		In this section, some recent trends in research on provability logic are discussed.
		Left out in the above discussion are many other important strands of research in provability logic and its extensions.
	www.science.uva.nl /~seop/archives/sum2003/entries/logic-provability (4840 words)

	NDJFL Editors
		Philosophical Logic: Philosophy of Logic, Philosophy of Mathematics
		Philosophical Logic: Philosophy and History of Mathematics, Foundation of Mathematics, Philosophy of Logic, Categorical Logic
		Philosophical Logic: Philosophy of Mathematics, Philosophy of Logic, Philosophy of Language, Epistemic Logic, Vagueness
	www.nd.edu /~ndjfl/editors.html (271 words)

	[No title] (Site not responding. Last check: )
		Sieg Status: Work in Progress For: Logic and Computation Thesis Problem: Implementing Modal Logics into AProS AProS: AProS ("A Proof System") generates natural deduction (ND) proofs in sentential and first order predicate logic.some meta-reasoning, in the form of automatic proofs for the Goedel Incompleteness Theorems is also included.
		In terms of possible worlds, S4 is the logic of a set of possible worlds such that the accessibility relationship among the worlds is reflexive and transitive.
		In terms of possible worlds, S5 is the logic of a set of possible worlds such that the accessibility relationship among the worlds is reflexive,symmetric and transitive.
	www.andrew.cmu.edu /user/rramyaa/research.doc (1149 words)

	Contents Page (Site not responding. Last check: )
		· Modal logic - study of deductive behavior of the expressions ‘necessarrilyt’ and ‘possibly’.
		o In terms of possible worlds, S4 is the logic of a set of possible worlds such that the accessibility relationship among the worlds is reflexive and transitive.
		o In terms of possible worlds, S5 is the logic of a set of possible worlds such that the accessibility relationship among the worlds is reflexive,symmetric and transitive.
	www.andrew.cmu.edu /user/rramyaa/research.htm (1159 words)

	[No title]
		Artemov, ``Operations on proofs that can be specified by means of modal logic,'' Technical Report CFIS 99-02, Cornell University, 1999.
		Artemov, ``Explicit provability: the intended semantics for intuitionistic and modal logic,'' Technical Report CFIS 98-10, Cornell University, 1998
		``Logic of proofs: a unified semantics for modality and lambda-terms,'' Technical Report CFIS 98-06, Cornell University, 1998.
	www.cs.cornell.edu /Info/People/artemov/publ.html (737 words)

	Bublos.com: Compare Book Prices ›› Logic of Provability - George S. Boolos - Paperback
		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 (CUP, 1979).
		It contains the first application of quantified modal logic to formal provability, and shows the results of applying modal logic to formal provability, and shows the results of applying modal logic of simple and omega consistency.
		Introduction to Logic and to the Methodology of the Deductive Sciences Alfred Tarski, Olaf Helmer (Translator)
	www.bublos.com /isbn/0521483255.html (683 words)

	Uniform provability in classical logic -- Nadathur 8 (2): 209 -- Journal of Logic and Computation
		Uniform provability in classical logic -- Nadathur 8 (2): 209 -- Journal of Logic and Computation
		of the top-level logical symbol of that formula.
		provability is equivalent to uniform provability admits of a
	logcom.oupjournals.org /cgi/content/abstract/8/2/209 (221 words)

	The Logic of Provability - Japaridze, de Jongh (ResearchIndex) (Site not responding. Last check: )
		30 Journal of Symbolic Logic (context) - of, of et al.
		2 a provability logical study of systems with built-in consist..
		1 Giorgi Japaridze and Dick de Jongh J (context) - logic, provability et al.
	citeseer.ist.psu.edu /75754.html (1221 words)

	Amazon.com: Books: The Logic of Provability (Site not responding. Last check: )
		Boolos, a pre-eminent philosopher of mathematics, investigates the relationship between provability and modal logic.
		We are going to investigate a system of propositional modal logic, which we call 'GL', for Godel and Lob.
		There is only one other book on the subjectâ"C.Smory'nski's "Self-Reference and Modal Logic", now out-of-print, which provides a nice complement to Boolos with surprisingly small overlap.
	www.amazon.com /exec/obidos/tg/detail/-/0521483255?v=glance (883 words)

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