intuitionistic logic oor Sjinees
intuitionistic logic
en
(mathematics, logic) A type of logic which rejects the axiom law of excluded middle or, equivalently, the law of double negation and/or Peirce's law. It is the foundation of intuitionism.
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直觉主义逻辑
en
logic
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voorbeelde
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It is sound and complete with respect to intuitionistic logic and admits a similar cut-elimination proof.
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Modified realizability is one way to show that Markov's principle is not derivable in intuitionistic logic.
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Correspondingly, classical Boolean logic arises by adding the law of double negation to intuitionistic logic.
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For example, S4 itself is the smallest modal companion of the intuitionistic logic (IPC).
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For example, the preserved property could be justification, the foundational concept of intuitionistic logic.
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Most of the classical identities are only theorems of intuitionistic logic in one direction, although some are theorems in both directions.
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Building upon his work on semantics of modal logic, Saul Kripke created another semantics for intuitionistic logic, known as Kripke semantics or relational semantics.
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The primary motivation for Lorenzen and Kuno Lorenz was to find a game-theoretic (their term was "dialogical" Dialogische Logik) semantics for intuitionistic logic.
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In contrast, propositional formulae in intuitionistic logic are not assigned a definite truth value and are only considered "true" when we have direct evidence, hence proof.
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It can be thought of as a variant of intuitionistic logic with star and with a noncommutative conjunction whose identity need not be the top element.
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One reason that this particular aspect of intuitionistic logic is so valuable is that it enables practitioners to utilize a wide range of computerized tools, known as proof assistants.
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The Curry–Howard isomorphism implies a connection between logic and programming: every proof of a theorem of intuitionistic logic corresponds to a reduction of a typed lambda term, and conversely.
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Gentzen discovered that a simple restriction of his system LK (his sequent calculus for classical logic) results in a system which is sound and complete with respect to intuitionistic logic.
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The relation between Heyting algebras and interior algebras reflects the relationship between intuitionistic logic and S4, in which one can interpret theories of intuitionistic logic as S4 theories closed under necessity.
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Heyting was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic logic on a footing where it could become part of mathematical logic.
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It was later discovered that the logic in categories of sheaves is intuitionistic logic (this observation is now often referred to as Kripke–Joyal semantics, but probably should be attributed to a number of authors).
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Kurt Gödel in 1932 showed that intuitionistic logic is not a finitely-many valued logic, and defined a system of Gödel logics intermediate between classical and intuitionistic logic; such logics are known as intermediate logics.
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The converse can be proven as well: if a formula always has the value 1, then it is deducible from the laws of intuitionistic logic, so the intuitionistically valid formulas are exactly those that always have a value of 1.
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The principle of explosion (Latin: ex falso (sequitur) quodlibet (EFQ), "from falsehood, anything (follows)", or ex contradictione (sequitur) quodlibet (ECQ), "from contradiction, anything (follows)"), or the principle of Pseudo-Scotus, is the law of classical logic, intuitionistic logic and similar logical systems, according to which any statement can be proven from a contradiction.
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In intuitionistic first-order logic both quantifiers ∃, ∀ are needed.
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Hence meaningful concepts of "intuitionistic truth", "linear-logic truth" and "IF-logic truth" can be derived from the semantics of CoL.
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It was originally proved by Gerhard Gentzen in his landmark 1934 paper "Investigations in Logical Deduction" for the systems LJ and LK formalising intuitionistic and classical logic respectively.
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He doesn't give inference rules for epistemic logic, and so cannot give a completeness proof; Richard Montague had many of the key ideas contained in Kripke's work, but he did not regard them as significant, because he had no completeness proof, and so did not publish until after Kripke's papers had created a sensation in the logic community; Evert Willem Beth presented a semantics of intuitionistic logic based on trees, which closely resembles Kripke semantics, except for using a more cumbersome definition of satisfaction.
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Simply typed lambda calculus which is a higher-order logic; intuitionistic type theory; system F; LF is often used to define other type theories; calculus of constructions and its derivatives.
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For simplicity, the logics presented so far have been intuitionistic.
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