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Intuitionistic logic, or constructivist logic, is the symbolic logic system originally developed by Arend Heyting to provide a formal basis for Brouwer's programme of intuitionism. The system preserves justification, rather than truth, across transformations yielding derived propositions. From a practical point of view, there is also a strong motivation for using intuitionistic logic, since it has the existence property, making it also suitable for other forms of mathematical constructivism.
Syntax
The syntax of formulæ of intuitionistic logic is similar to propositional logic or first-order logic. The obvious difference is that many tautologies of these classical logics can no longer be proven within intuitionistic logic. Examples include not only the law of excluded middle P ∨ ¬P, but also Peirce's Law ((P → Q) → P) → P, and even double negation elimination. In classical logic, both P → ¬¬P and also ¬¬P → P are theorems. In intuitionistic logic, only the first is a theorem: Double negation can be introduced, but it cannot be eliminated.
The observation that many classically valid tautologies are not theorems of intuitionistic logic leads to the idea of weakening the proof theory of classical logic.
More on [ Intuitionistic logic ]
@almostandy By Curry-de Bruijn-Howard there's no difference between intuitionistic logic and code.
psnively (Paul Snively) Thu, 10 Dec 2009 15:31:52 -0000
@almostandy By Curry-de Bruijn-Howard there's no difference between intuitionistic logic and code.
StackExchange: intuitionistic interpretation of classical logic - http://stackexchangesites.com/c6C
stackexchange (stackexchange) Wed, 09 Dec 2009 04:19:23 -0000
StackExchange: intuitionistic interpretation of classical logic - http://stackexchangesites.com/c6C
A Bibliography of Constructive Mathematics - Compiled by Erik Palmgren.
Confessions of a Formalist, Platonist Intuitionist - Autobiographical article by Fred Richman, describing his encounter with intuitionism.
Constructive Mathematics - Maintained by Fred Richards.
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Intuitionism - A brief hypertext overview of the history of the subject.
Intuitionistic Logic - A short entry in the Stanford Encyclopaedia of Philosophy by Joan R. Moschovakis.
Intuitionistic Logic - A very brief overview of the subject by Alex Sakharov from MathWorld.
Intuitionistic Logic - A very concise introduction to the subject. Includes overview of the syntax, Kripke models, analytic tableau, natural deduction.
Intuitionistic logic - Wikipedia (free encyclopedia) article.
Intuitionistic Topology and Foundations of Constructive Mathematics - Math page of Frank Waaldijk, containing articles and PhD thesis on foundations of constructive mathematics and intuitionistic topology. Also links to other mathemtaicians in this field.
Meta Description: [ frank waaldijk's math page on constructive mathematics, intuitionism and intuitionistic topology ]
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Porgi - Porgi is a Proof-Or-Refutation Generator for Intuitionistic propositional logic, implementated by Allen Stoughton. Given a sequent, Porgi either finds a minimally sized, normal natural deduction of the sequent, or it finds a small, tree-based Kripke countermodel of the sequent. It is written in Standard ML.
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