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Witness (mathematics)
In mathematical logic, a witness is a specific value ''t'' to be substituted for variable ''x'' of an existential statement of the form ∃''x'' ''φ''(''x'') such that ''φ''(''t'') is true. Examples For example, a theory ''T'' of arithmetic is said to be inconsistent if there exists a proof in ''T'' of the formula "0 = 1". The formula I(''T''), which says that ''T'' is inconsistent, is thus an existential formula. A witness for the inconsistency of ''T'' is a particular proof of "0 = 1" in ''T''. Boolos, Burgess, and Jeffrey (2002:81) define the notion of a witness with the example, in which ''S'' is an ''n''-place relation on natural numbers, ''R'' is an ''(n+1)''-place recursive relation, and ↔ indicates logical equivalence (if and only if): :: ''S''(''x''1, ..., ''x''''n'') ↔ ∃''y'' ''R''(''x''1, . . ., ''x''''n'', ''y'') :"A ''y'' such that ''R'' holds of the ''xi'' may be called a 'witness' to the relation ''S'' holding of the ''xi'' (provide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of logic, 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 Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, 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 pr ... [...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 formally as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |