intuitionistic logic
直觉主义逻辑
intuitionist oor Sjinees
intuitionist
en
A person who studies intuitionistic mathematics
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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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It is sound and complete with respect to intuitionistic logic and admits a similar cut-elimination proof.
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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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In intuitionistic first-order logic both quantifiers ∃, ∀ are needed.
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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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Lambda calculi with dependent types are the base of intuitionistic type theory, the calculus of constructions and the logical framework (LF), a pure lambda calculus with dependent types.
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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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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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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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The Calculus of Constructions extends this isomorphism to proofs in the full intuitionistic predicate calculus, which includes proofs of quantified statements (which we will also call "propositions").
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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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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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Correspondingly, classical Boolean logic arises by adding the law of double negation to intuitionistic 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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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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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 inclusion of Brouwer's name in the Brouwer–Heyting–Kolmogorov interpretation is largely honorific, as Brouwer was opposed in principle to the formalisation of certain intuitionistic principles (and went as far as calling Heyting's work a "sterile exercise").
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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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However, intuitionistic connectives are not definable in terms of each other in the same way as in classical logic, hence their choice matters.
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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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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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Even when the logic under study is intuitionistic, entailment is ordinarily understood classically as two-valued: either the left side entails, or is less-or-equal to, the right side, or it is not.
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For example, S4 itself is the smallest modal companion of the intuitionistic logic (IPC).
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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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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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