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Finite Model Property
In mathematical logic, a logic L has the finite model property (fmp for short) if any non-theorem of L is falsified by some ''finite'' model of L. Another way of putting this is to say that L has the fmp if for every formula ''A'' of L, ''A'' is an L-theorem if and only if ''A'' is a theorem of the theory of finite models of L. If L is finitely axiomatizable (and has a recursive set of inference rules) and has the fmp, then it is decidable. However, the result does not hold if L is merely recursively axiomatizable. Even if there are only finitely many finite models to choose from (up to isomorphism) there is still the problem of checking whether the underlying frames of such models validate the logic, and this may not be decidable when the logic is not finitely axiomatizable, even when it is recursively axiomatizable. (Note that a logic is recursively enumerable if and only if it is recursively axiomatizable, a result known as Craig's theorem.) Example A first-order formula with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory sho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Quantification
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable. It is usually denoted by the turned A (∀) logical operator symbol, which, when used together with a predicate variable, is called a universal quantifier ("", "", or sometimes by "" alone). Universal quantification is distinct from ''existential'' quantification ("there exists"), which only asserts that the property or relation holds for at least one member of the domain. Quantification in general is covered in the article on quantification (logic). The universal quantifier is encoded as in Unicode, and as \forall in LaTeX a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Philosophical Logic
The ''Journal of Philosophical Logic'' is a bimonthly peer-reviewed academic journal covering all aspects of logic. It was established in 1972 and is published by Springer Science+Business Media. The editors-in-chief are Rosalie Iemhoff (Utrecht University), Reinhard Muskens (University of Amsterdam), and Kai Wehmeier (University of California, Irvine). Abstracting and indexing The journal is abstracted and indexed in: * Arts and Humanities Citation Index *Current Contents/Arts & Humanities * EBSCO databases *International Bibliography of Periodical Literature * Linguistic Bibliography/Bibliographie Linguistique *Modern Language Association Database *Philosopher's Index * ProQuest databases *Scopus *Zentralblatt MATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastruct ... References ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alasdair Urquhart
Alasdair Ian Fenton Urquhart (; born 20 December 1945) is a Scottish–Canadian philosopher and emeritus professor of philosophy at the University of Toronto. He has made contributions to the field of logic, especially non-classical logic. One of his ideas is proving the undecidability of the relevance logic R. He also published papers in theoretical computer science venues, mostly on mathematical logic topics of relevance to computer science. Early life Urquhart is a native of Scotland. He received his MA in philosophy from the University of Edinburgh in 1967. He then attended the University of Pittsburgh, receiving an MA and Ph.D. in 1973 under the supervision of Alan Ross Anderson and Nuel Belnap. Career From 1973 to 1975, Urquhart was an assistant professor at Erindale College, University of Toronto Mississauga. He became an associate professor there in 1975. Starting in 1986, Urquhart was a professor of philosophy at the University of Toronto. From 1983 to 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maarten De Rijke
Maarten de Rijke (born 1 August 1961) is a Dutch computer scientist. His work initially focused on modal logic and knowledge representation, but since the early years of the 21st century he has worked mainly in information retrieval. His work is supported by grants from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO), public-private partnerships, and the European Commission (under the Sixth and Seventh Framework programmes). Biography Maarten de Rijke was born in Vlissingen. He studied philosophy (MSc 1989) and mathematics (MSc 1990) and wrote a PhD thesis, defended in 1993, on extended modal logics, under the supervision of Johan van Benthem. De Rijke worked as a postdoc at the Centrum Wiskunde & Informatica, before becoming a Warwick Research Fellow at the University of Warwick. He joined the University of Amsterdam in 1998, and was appointed professor of Information Processing and Internet at the Informatics Institute of the University of Amsterdam in 200 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Leonid Libkin
Leonid Libkin is a computer scientist who works in data management, in particular in database theory, and in logic in computer science. Libkin is a professor at the University of Edinburgh, where he is chair of Foundations of Data Management in the School of Informatics, and at the École Normale Supérieure in Paris. He previously worked at the University of Toronto, and at Bell Labs. Libkin is the author of standard textbooks on finite model theory and on data exchange. He is an ACM Fellow, a Fellow of the Royal Society of Edinburgh, and a member of Academia Europaea. He won best paper awards at the Symposium on Principles of Database Systems (ACM PODS) in 1999, 2003, and 2005, at International Conference on Database Theory (ICDT) in 2011, at the Principles of Knowledge Representation and Reasoning Conference in 2014 and 2018., and at the ACM SIGMOD SIGMOD is the Association for Computing Machinery's Special Interest Group on Management of Data, which specializes in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Existential Quantification
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, when used together with a predicate variable, is called an existential quantifier ("" or "" or "). Existential quantification is distinct from universal quantification ("for all"), which asserts that the property or relation holds for ''all'' members of the domain. Some sources use the term existentialization to refer to existential quantification. Basics Consider a formula that states that some natural number multiplied by itself is 25. : 0·0 = 25, or 1·1 = 25, or 2·2 = 25, or 3·3 = 25, ... This would seem to be a logical disjunction because of the repeated use of "or". However, the ellipses make this impossible to integrate and to interpret it as a disjunction in formal logic. Instead, the statement could be rephrased more forma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signature (logic)
In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes. They are rarely made explicit in more philosophical treatments of logic. Definition Formally, a (single-sorted) signature can be defined as a 4-tuple , where ''S''func and ''S''rel are disjoint sets not containing any other basic logical symbols, called respectively * ''function symbols'' (examples: +, ×, 0, 1), * ''relation symbols'' or ''predicates'' (examples: ≤, ∈), * ''constant symbols'' (examples: 0, 1), and a function ar: ''S''func \cup ''S''rel → \mathbb N which assigns a natural number called ''arity'' to every function or relation symbol. A function or relation symbol is called ''n''-ary if its arity is ''n''. Some authors define a nullary (0-ary) function symbol as ''constant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists''"'' is a quantifier, while ''x'' is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A theory about a topic is usually a first-order logic together with a specified domain of discourse (over which the quantified variables range), finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Craig's Theorem
In mathematical logic, Craig's theorem states that any recursively enumerable set of well-formed formulas of a first-order language is (primitively) recursively axiomatizable. This result is not related to the well-known Craig interpolation theorem, although both results are named after the same logician, William Craig. Recursive axiomatization Let A_1,A_2,\dots be an enumeration of the axioms of a recursively enumerable set T of first-order formulas. Construct another set T* consisting of :\underbrace_i for each positive integer ''i''. The deductive closures of T* and T are thus equivalent; the proof will show that T* is a recursive set. A decision procedure for T* lends itself according to the following informal reasoning. Each member of T* is either A_1 or of the form :\underbrace_j. Since each formula has finite length, it is checkable whether or not it is A_1 or of the said form. If it is of the said form and consists of ''j'' conjuncts, it is in T* if the (reoccurring ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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