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Boolean-valued Semantics
In mathematical logic, algebraic semantics is a formal semantics based on algebras studied as part of algebraic logic. For example, the modal logic S4 is characterized by the class of topological boolean algebras—that is, boolean algebras with an interior operator. Other modal logics are characterized by various other algebras with operators. The class of boolean algebras characterizes classical propositional logic, and the class of Heyting algebras propositional intuitionistic logic. MV-algebras are the algebraic semantics of Łukasiewicz logic. See also * Algebraic semantics (computer science) * Lindenbaum–Tarski algebra Further reading * (2nd published by ASL in 2009open accessat Project Euclid * * * Good introduction for readers with prior exposure to non-classical logics but without much background in order theory and/or universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures thems ... [...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] |
MV-algebra
In abstract algebra, a branch of pure mathematics, an MV-algebra is an algebraic structure with a binary operation \oplus, a unary operation \neg, and the constant 0, satisfying certain axioms. MV-algebras are the algebraic semantics of Łukasiewicz logic; the letters MV refer to the ''many-valued'' logic of Łukasiewicz. MV-algebras coincide with the class of bounded commutative BCK algebras. Definitions An MV-algebra is an algebraic structure \langle A, \oplus, \lnot, 0\rangle, consisting of * a non-empty set A, * a binary operation \oplus on A, * a unary operation \lnot on A, and * a constant 0 denoting a fixed element of A, which satisfies the following identities: * (x \oplus y) \oplus z = x \oplus (y \oplus z), * x \oplus 0 = x, * x \oplus y = y \oplus x, * \lnot \lnot x = x, * x \oplus \lnot 0 = \lnot 0, and * \lnot ( \lnot x \oplus y)\oplus y = \lnot ( \lnot y \oplus x) \oplus x. By virtue of the first three axioms, \langle A, \oplus, 0 \rangle is a commutat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Background and motivation Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the standard order on the natural numbers e.g. "2 is less than 3", "10 is greater than 5", or "Does Tom have fewer cookies than Sally?". This intuitive concept can be extended to orders on other sets of numbers, such as the integers and the reals. The idea of being greater than or less than another number is one of the basic intuitions of number systems (compare with numeral systems) in general (although one usually is also interested in the actual differenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-classical Logic
Non-classical logics (and sometimes alternative logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models of logical consequence and logical truth. Philosophical logic is understood to encompass and focus on non-classical logics, although the term has other meanings as well. In addition, some parts of theoretical computer science can be thought of as using non-classical reasoning, although this varies according to the subject area. For example, the basic boolean functions (e.g. AND, OR, NOT, etc) in computer science are very much classical in nature, as is clearly the case given that they can be fully described by classical truth tables. However, in contrast, some computerized proof methods may not use classical logi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Project Euclid
Project Euclid is a collaborative partnership between Cornell University Library and Duke University Press which seeks to advance scholarly communication in theoretical and applied mathematics and statistics through partnerships with independent and society publishers. It was created to provide a platform for small publishers of scholarly journals to move from print to electronic in a cost-effective way. Through a combination of support by subscribing libraries and participating publishers, Project Euclid has made 70% of its journal articles available as open access. As of 2010, Project Euclid provided access to over one million pages of open-access content. Mission and goals Project Euclid's stated mission is to advance scholarly communication in the field of theoretical and applied mathematics and statistics. Through a "mixture of open access, subscription, and hosted subscription content it provides a way for small publishers (especially societies) to host their math or statisti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Association For Symbolic Logic
The Association for Symbolic Logic (ASL) is an international organization of specialists in mathematical logic and philosophical logic. The ASL was founded in 1936, and its first president was Alonzo Church. The current president of the ASL is Julia F. Knight. Publications The ASL publishes books and academic journals. Its three official journals are: * ''Journal of Symbolic Logic'(website)– publishes research in all areas of mathematical logic. Founded in 1936, . * ''Bulletin of Symbolic Logic'(website)– publishes primarily expository articles and reviews. Founded in 1995, . * ''Review of Symbolic Logic'(website)– publishes research relating to logic, philosophy, science, and their interactions. Founded in 2008, . In addition, the ASL has a sponsored journal: * ''Journal of Logic and Analysis'(website)– publishes research on the interactions between mathematical logic and pure and applied analysis. Founded in 2009 as an open-access journal, open-access success ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lindenbaum–Tarski Algebra
In mathematical logic, the Lindenbaum–Tarski algebra (or Lindenbaum algebra) of a logical theory ''T'' consists of the equivalence classes of sentences of the theory (i.e., the quotient, under the equivalence relation ~ defined such that ''p'' ~ ''q'' exactly when ''p'' and ''q'' are provably equivalent in ''T''). That is, two sentences are equivalent if the theory ''T'' proves that each implies the other. The Lindenbaum–Tarski algebra is thus the quotient algebra obtained by factoring the algebra of formulas by this congruence relation. The algebra is named for logicians Adolf Lindenbaum and Alfred Tarski. It was first introduced by Tarski in 1935 as a device to establish correspondence between classical propositional calculus and Boolean algebras. The Lindenbaum–Tarski algebra is considered the origin of the modern algebraic logic.; here: pages 1-2 Operations The operations in a Lindenbaum–Tarski algebra ''A'' are inherited from those in the underlying theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Semantics (computer Science)
In computer science, algebraic semantics is a form of axiomatic semantics based on algebraic laws for describing and reasoning about program specifications in a formal manner. Syntax The syntax of an algebraic specification is formulated in two steps: (1) defining a formal signature of data types and operation symbols, and (2) interpreting the signature through sets and functions. Definition of a signature The signature of an algebraic specification defines its formal syntax. The word "signature" is used like the concept of "key signature" in musical notation. A signature consists of a set S of data types, known as sorts, together with a family \Sigma of sets, each set containing operation symbols (or simply symbols) that relate the sorts. We use \Sigma_ to denote the set of operation symbols relating the sorts s_1,~s_2,~...,~s_n \in S to the sort s \in S. For example, for the signature of integer stacks, we define two sorts, namely, int and stack, and the following famil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Łukasiewicz Logic
In mathematics and philosophy, Łukasiewicz logic ( , ) is a non-classical, many-valued logic. It was originally defined in the early 20th century by Jan Łukasiewicz as a three-valued logic;Łukasiewicz J., 1920, O logice trójwartościowej (in Polish). Ruch filozoficzny 5:170–171. English translation: On three-valued logic, in L. Borkowski (ed.), ''Selected works by Jan Łukasiewicz'', North–Holland, Amsterdam, 1970, pp. 87–88. it was later generalized to ''n''-valued (for all finite ''n'') as well as infinitely-many-valued ( ℵ0-valued) variants, both propositional and first-order.Hay, L.S., 1963Axiomatization of the infinite-valued predicate calculus ''Journal of Symbolic Logic'' 28:77–86. The ℵ0-valued version was published in 1930 by Łukasiewicz and Alfred Tarski; consequently it is sometimes called the ŁukasiewiczTarski logic. citing Łukasiewicz, J., Tarski, A.Untersuchungen über den Aussagenkalkül Comp. Rend. Soc. Sci. et Lettres Varsovie Cl. III 23, 30–5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intuitionistic Logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for L. E. J. Brouwer's programme of intuitionism. From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed. Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, but do not hold universally as they do with classical logic. The standard explanation of intuitionistic logic is the BHK interpreta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Semantics (logic)
In logic, the semantics of logic or formal semantics is the study of the semantics, or interpretations, of formal and (idealizations of) natural languages usually trying to capture the pre-theoretic notion of entailment. Overview The truth conditions of various sentences we may encounter in arguments will depend upon their meaning, and so logicians cannot completely avoid the need to provide some treatment of the meaning of these sentences. The semantics of logic refers to the approaches that logicians have introduced to understand and determine that part of meaning in which they are interested; the logician traditionally is not interested in the sentence as uttered but in the proposition, an idealised sentence suitable for logical manipulation. Until the advent of modern logic, Aristotle's '' Organon'', especially ''De Interpretatione'', provided the basis for understanding the significance of logic. The introduction of quantification, needed to solve the problem of multip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heyting Algebra
In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' of ''implication'' such that (''c'' ∧ ''a'') ≤ ''b'' is equivalent to ''c'' ≤ (''a'' → ''b''). From a logical standpoint, ''A'' → ''B'' is by this definition the weakest proposition for which modus ponens, the inference rule ''A'' → ''B'', ''A'' ⊢ ''B'', is sound. Like Boolean algebras, Heyting algebras form a variety axiomatizable with finitely many equations. Heyting algebras were introduced by to formalize intuitionistic logic. As lattices, Heyting algebras are distributive. Every Boolean algebra is a Heyting algebra when ''a'' → ''b'' is defined as ¬''a'' ∨ ''b'', as is every complete distributive lattice satisfying a one-sided infinite distributive law when ''a'' → ''b'' is taken to be the supremum of the set of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
