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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. Examples There are a number of provability logics, some of which are covered in the literature mentioned in . The basic system is generally referred to as GL (for Gödel– Löb) or L or K4W (W stands for well-foundedness). It can be obtained by adding the modal version of Löb's theorem to the logic K (or K4). Namely, the axioms of GL are all tautologies of classical propositional logic plus all formulas of one of the following forms: * Distribution axiom: * Löb's axiom: And the rules of inference are: * ''Modus ponens'': From ''p'' → ''q'' and ''p'' conclude ''q''; * Necessitation: From \vdash ''p'' conclude \vdash . History The GL model was pioneered by Robert M. Solovay in 1976. Since then, until his death in 1996, the prime inspi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Modal Logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other systems by adding unary operators \Diamond and \Box, representing possibility and necessity respectively. For instance the modal formula \Diamond P can be read as "possibly P" while \Box P can be read as "necessarily P". Modal logics can be used to represent different phenomena depending on what kind of necessity and possibility is under consideration. When \Box is used to represent epistemic necessity, \Box P states that P is epistemically necessary, or in other words that it is known. When \Box is used to represent deontic necessity, \Box P states that P is a moral or legal obligation. In the standard relational semantics for modal logic, formulas are assigned truth values relative to a '' possible world''. A formula's truth value ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Giorgi Japaridze
Giorgi Japaridze (also spelled Giorgie Dzhaparidze) is a Georgian-American researcher in logic and theoretical computer science. He currently holds the title of Full Professor at the Computing Sciences Department of Villanova University. Japaridze is best known for his invention of computability logic, cirquent calculus, and Japaridze's polymodal logic. Research During 1985–1988 Japaridze elaborated the system GLP, known as Japaridze's polymodal logic. This is a system of modal logic with the "necessity" operators …, understood as a natural series of incrementally weak provability predicates for Peano arithmetic. In "The polymodal logic of provability" Japaridze proved the arithmetical completeness of this system, as well as its inherent incompleteness with respect to Kripke frames. GLP has been extensively studied by various authors during the subsequent three decades, especially after Lev Beklemishev, in 2004, pointed out its usefulness in understanding the proof the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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. Examples There are a number of provability logics, some of which are covered in the literature mentioned in . The basic system is generally referred to as GL (for Gödel– Löb) or L or K4W (W stands for well-foundedness). It can be obtained by adding the modal version of Löb's theorem to the logic K (or K4). Namely, the axioms of GL are all tautologies of classical propositional logic plus all formulas of one of the following forms: * Distribution axiom: * Löb's axiom: And the rules of inference are: * ''Modus ponens'': From ''p'' → ''q'' and ''p'' conclude ''q''; * Necessitation: From \vdash ''p'' conclude \vdash . History The GL model was pioneered by Robert M. Solovay in 1976. Since then, until his death in 1996, the prime inspi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Stanford Encyclopedia Of Philosophy
The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from many academic institutions worldwide. Authors contributing to the encyclopedia give Stanford University the permission to publish the articles, but retain the copyright to those articles. Approach and history As of August 5th, 2022, the ''SEP'' has 1,774 published entries. Apart from its online status, the encyclopedia uses the traditional academic approach of most encyclopedias and academic journals to achieve quality by means of specialist authors selected by an editor or an editorial committee that is competent (although not necessarily considered specialists) in the field covered by the encyclopedia and peer review. The encyclopedia was created in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Rineke Verbrugge
Laurina Christina (Rineke) Verbrugge (born 12 March 1965 in Amsterdam) is a Dutch logician and computer scientist known for her work on interpretability logic and provability logic. She completed her PhD at the University of Amsterdam in 1993 under the supervision of Dick de Jongh, Anne Troelstra, and Albert Visser. She holds the chair of Logic and Cognition at the University of Groningen's Bernoulli Institute of Mathematics, Computer Science and Artificial Intelligence, where she has been the leader of the Multi-Agent Systems working group since 2002. She is particularly known for her work connecting formal logic to cognition and developmental psychology and the role of logic in explaining social behaviour. From 2005 to 2021, she was the President (''voorzitter'') of the ''Nederlandse Vereniging voor Logica & Wijsbegeerte der Exacte Wetenschappen'' (VvL; Dutch Association for Logic and Philosophy of the Exact Sciences). In 2021, she was elected a fellow of the Royal Nethe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Israel Journal Of Mathematics
'' Israel Journal of Mathematics'' is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem (Magnes Press). Founded in 1963, as a continuation of the ''Bulletin of the Research Council of Israel'' (Section F), the journal publishes articles on all areas of mathematics. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.70, and its 2009 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... was 0.754. External links * Mathematics journals Publications established in 1963 English-language journals Bimonthly journals Hebrew University of Jerusalem {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Per Lindström
Per "Pelle" Lindström (9 April 1936 – 21 August 2009, Gothenburg)ASLbr>Newsletter September 2009 was a Swedish logician, after whom Lindström's theorem and the Lindström quantifier are named. (He also independently discovered Ehrenfeucht–Fraïssé games.) He was one of the key followers of Lars Svenonius. Lindström was awarded a PhD from the University of Gothenburg in 1966. His thesis was titled ''Some Results in the Theory of Models of First Order Languages''. A festschrift In academia, a ''Festschrift'' (; plural, ''Festschriften'' ) is a book honoring a respected person, especially an academic, and presented during their lifetime. It generally takes the form of an edited volume, containing contributions from the ... for Lindström was published in 1986. Selected publications * Per Lindström, First Order Predicate Logic with Generalized Quantifiers, ''Theoria'' 32, 1966, 186–195. * Per Lindström, On Extensions of Elementary Logic, ''Theoria'' 35, 1969, 1� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Kripke Semantics
Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Joyal. It was first conceived for modal logics, and later adapted to intuitionistic logic and other non-classical systems. The development of Kripke semantics was a breakthrough in the theory of non-classical logics, because the model theory of such logics was almost non-existent before Kripke (algebraic semantics existed, but were considered 'syntax in disguise'). Semantics of modal logic The language of propositional modal logic consists of a countably infinite set of propositional variables, a set of truth-functional connectives (in this article \to and \neg), and the modal operator \Box ("necessarily"). The modal operator \Diamond ("possibly") is (classically) the dual of \Box and may be defined in terms of necessity like so: \Di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Hilbert–Bernays Provability Conditions
In mathematical logic, the Hilbert–Bernays provability conditions, named after David Hilbert and Paul Bernays, are a set of requirements for formalized provability predicates in formal theories of arithmetic (Smith 2007:224). These conditions are used in many proofs of Kurt Gödel's second incompleteness theorem. They are also closely related to axioms of provability logic. The conditions Let be a formal theory of arithmetic with a formalized provability predicate , which is expressed as a formula of with one free number variable. For each formula in the theory, let be the Gödel number of . The Hilbert–Bernays provability conditions are: # If proves a sentence then proves . # For every sentence , proves # proves that and imply Note that is predicate of numbers, and it is a provability predicate in the sense that the intended interpretation of is that there exists a number that codes for a proof of . Formally what is required of is the above three condition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Japaridze's Polymodal Logic
Japaridze's polymodal logic (GLP) is a system of provability logic with infinitely many provability modalities. This system has played an important role in some applications of provability algebras in proof theory, and has been extensively studied since the late 1980s. It is named after Giorgi Japaridze. Language and axiomatization The language of GLP extends that of the language of classical propositional logic by including the infinite series of necessity operators. Their dual possibility operators are defined by . The axioms of GLP are all classical tautologies and all formulas of one of the following forms: * * * * And the rules of inference are: * From and conclude * From conclude Provability semantics Consider a sufficiently strong first-order theory such as Peano Arithmetic . Define the series of theories as follows: * is * is the extension of through the additional axioms for each formula such that proves all of the formulas For each , let be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Interpretability Logic
Interpretability logics comprise a family of modal logics that extend provability logic to describe interpretability or various related metamathematical properties and relations such as weak interpretability, Π1-conservativity, cointerpretability, tolerance, cotolerance, and arithmetic complexities. Main contributors to the field are Alessandro Berarducci, Petr Hájek, Konstantin Ignatiev, Giorgi Japaridze, Franco Montagna, Vladimir Shavrukov, Rineke Verbrugge, Albert Visser, and Domenico Zambella. Examples Logic ILM The language of ILM extends that of classical propositional logic by adding the unary modal operator \Box and the binary modal operator \triangleright (as always, \Diamond p is defined as \neg \Box\neg p). The arithmetical interpretation of \Box p is “p is provable in Peano arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Dick De Jongh
Dick Herman Jacobus de Jongh (born 19 October 1939, Enschede) is a Dutch logician and mathematician and a retired professor at the University of Amsterdam. He received his PhD degree in 1968 from the University of Wisconsin–Madison under supervision of Stephen Kleene with a dissertation entitled ''Investigations on the Intuitionistic Propositional Calculus''. De Jongh is mostly known for his work on proof theory, provability logic and intuitionistic logic. De Jongh is a member of the group collectively publishing under the pseudonym L. T. F. Gamut. In 2004, on the occasion of his retirement, the Institute for Logic, Language and Computation at the University of Amsterdam published a festschrift In academia, a ''Festschrift'' (; plural, ''Festschriften'' ) is a book honoring a respected person, especially an academic, and presented during their lifetime. It generally takes the form of an edited volume, containing contributions from the ... in his honor.. References Curricu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |