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ﾅ「kasiewicz Logic
In mathematics and philosophy, ﾅ「kasiewicz logic ( , ) is a non-classical, many-valued logic. It was originally defined in the early 20th century by Jan ﾅ「kasiewicz as a three-valued logic;ﾅ「kasiewicz J., 1920, O logice trﾃｳjwartoﾅ嫩iowej (in Polish). Ruch filozoficzny 5:170窶171. English translation: On three-valued logic, in L. Borkowski (ed.), ''Selected works by Jan ﾅ「kasiewicz'', North窶滴olland, Amsterdam, 1970, pp. 87窶88. it was later generalized to ''n''-valued (for all finite ''n'') as well as infinitely-many-valued ( 邃ｵ0-valued) variants, both propositional and first-order.Hay, L.S., 1963Axiomatization of the infinite-valued predicate calculus ''Journal of Symbolic Logic'' 28:77窶86. The 邃ｵ0-valued version was published in 1930 by ﾅ「kasiewicz and Alfred Tarski; consequently it is sometimes called the ﾅ「kasiewiczTarski logic. citing ﾅ「kasiewicz, J., Tarski, A.Untersuchungen ﾃｼber den Aussagenkalkﾃｼl Comp. Rend. Soc. Sci. et Lettres Varsovie Cl. III 23, 30窶5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sentential Calculus
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions. Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic. Explanation Logical connectives are found in natural languages. In English for example, some examples are "and" (conjunction), "or" (disjunction), "not" ( negation) and "if" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ﾅ「kasiewicz窶溺oisil Algebra
ﾅ「kasiewicz窶溺oisil algebras (LM''n'' algebras) were introduced in the 1940s by Grigore Moisil (initially under the name of ﾅ「kasiewicz algebras) in the hope of giving algebraic semantics for the ''n''-valued ﾅ「kasiewicz logic. However, in 1956 Alan Rose discovered that for ''n'' 竕･ 5, the ﾅ「kasiewicz窶溺oisil algebra does not model the ﾅ「kasiewicz logic. A faithful model for the 邃ｵ0-valued (infinitely-many-valued) ﾅ「kasiewicz窶典arski logic was provided by C. C. Chang's MV-algebra, introduced in 1958. For the axiomatically more complicated (finite) ''n''-valued ﾅ「kasiewicz logics, suitable algebras were published in 1977 by Revaz Grigolia and called MV''n''-algebras. MV''n''-algebras are a subclass of LM''n''-algebras, and the inclusion is strict for ''n'' 竕･ 5.Iorgulescu, A.: Connections between MV''n''-algebras and ''n''-valued ﾅ「kasiewicz-Moisil algebras窶祢. Discrete Math. 181, 155窶177 (1998) In 1982 Roberto Cignoli published some additional constraints that a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grigore Moisil
Grigore Constantin Moisil (; 10 January 1906 窶 21 May 1973) was a Romanian mathematician, computer pioneer, and list of members of the Romanian Academy, titular member of the Romanian Academy. His research was mainly in the fields of mathematical logic (ﾅ「kasiewicz窶溺oisil algebra), algebraic logic, MV-algebra, and differential equations. He is viewed as the father of computer science in Romania. Moisil was also a member of the Academy of Sciences of Bologna and of the International Institute of Philosophy. In 1996, the IEEE Computer Society awarded him posthumously the ''Computer Pioneer'' Award. Biography Grigore Moisil was born in 1906 in Tulcea into an intellectual family. His great-grandfather, Grigore Moisil (1814窶1891), a clergyman, was one of the founders of the George Coﾈ冀uc National College (Nﾄピﾄブd), first Romanian high school in Nﾄピﾄブd. His father, Constantin Moisil (1876窶1958), was a history professor, archaeology, archaeologist and numismatics, numismat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a (strongly connected, formerly called total). Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but refers generally to some sort of totally ordered subsets of a given partially ordered set. An extension of a given partial order to a total order is called a linear extension of that partial order. Strict and non-strict total orders A on a set X is a strict partial ord ... [...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 ﾅ「kasiewicz logic; the letters MV refer to the ''many-valued'' logic of ﾅ「kasiewicz. 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] |
Algebraic Semantics (mathematical Logic)
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 ﾅ「kasiewicz logic. See also * Algebraic semantics (computer science) * Lindenbaum窶典arski 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] |
Countable Set
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements. In more technical terms, assuming the axiom of countable choice, a set is ''countable'' if its cardinality (its number of elements) is not greater than that of the natural numbers. A countable set that is not finite is said countably infinite. The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers. A note on terminology Although the terms "countable" and "countably infinite" as defined here are quite co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite-valued Logic
In logic, a finite-valued logic (also finitely many-valued logic) is a propositional calculus in which truth values are discrete. Traditionally, in Aristotle's logic, the bivalent logic, also known as binary logic was the norm, as the law of the excluded middle precluded more than two possible values (i.e., "true" and "false") for any proposition. Modern three-valued logic (ternary logic) allows for an additional possible truth value (i.e. "undecided"). The term finitely many-valued logic is typically used to describe many-valued logic having three or more, but not infinite, truth values. The term finite-valued logic encompasses both finitely many-valued logic and bivalent logic. Fuzzy logics, which allow for degrees of values between "true" and "false"), are typically not considered forms of finite-valued logic. However, finite-valued logic can be applied in Boolean-valued modeling, description logics, and defuzzification of fuzzy logic. A finite-valued logic is decidable (sure ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Variable
In mathematical logic, a propositional variable (also called a sentential variable or sentential letter) is an input variable (that can either be true or false) of a truth function. Propositional variables are the basic building-blocks of propositional formulas, used in propositional logic and higher-order logics. Uses Formulas in logic are typically built up recursively from some propositional variables, some number of logical connectives, and some logical quantifiers. Propositional variables are the atomic formulas of propositional logic, and are often denoted using capital roman letters such as P, Q and R. ;Example In a given propositional logic, a formula can be defined as follows: * Every propositional variable is a formula. * Given a formula ''X'', the negation ''ﾂｬX'' is a formula. * Given two formulas ''X'' and ''Y'', and a binary connective ''b'' (such as the logical conjunction 竏ｧ),the expression ''(X b Y)'' is a formula. (Note the parentheses.) Through this const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tautology (logic)
In mathematical logic, a tautology (from el, ﾏﾎｱﾏﾏﾎｿﾎｻﾎｿﾎｳﾎｯﾎｱ) is a formula or assertion that is true in every possible interpretation. An example is "x=y or x竕y". Similarly, "either the ball is green, or the ball is not green" is always true, regardless of the colour of the ball. The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921, borrowing from rhetoric, where a tautology is a repetitive statement. In logic, a formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. In other words, it cannot be false. It cannot be untrue. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formula that is neither a tautology nor a contradiction is said to be Contingency (philosophy), logically contingent. Such a formula can be made either true or false based on the values assigned to its propositi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
T-norm
In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. A t-norm generalizes intersection in a lattice and conjunction in logic. The name ''triangular norm'' refers to the fact that in the framework of probabilistic metric spaces t-norms are used to generalize the triangle inequality of ordinary metric spaces. Definition A t-norm is a function T: , 1× , 1竊 , 1that satisfies the following properties: * Commutativity: T(''a'', ''b'') = T(''b'', ''a'') * Monotonicity: T(''a'', ''b'') 竕､ T(''c'', ''d'') if ''a'' 竕､ ''c'' and ''b'' 竕､ ''d'' * Associativity: T(''a'', T(''b'', ''c'')) = T(T(''a'', ''b''), ''c'') * The number 1 acts as identity element: T(''a'', 1) = ''a'' Since a t-norm is a binary algebraic operation on the interval , 1 infix algebraic notation is also common, with the t-nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
