intuitionistic logic


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intuitionistic logic

(logic, mathematics)
Brouwer's foundational theory of mathematics which says that you should not count a proof of (There exists x such that P(x)) valid unless the proof actually gives a method of constructing such an x. Similarly, a proof of (A or B) is valid only if it actually exhibits either a proof of A or a proof of B.

In intuitionism, you cannot in general assert the statement (A or not-A) (the principle of the excluded middle); (A or not-A) is not proven unless you have a proof of A or a proof of not-A. If A happens to be undecidable in your system (some things certainly will be), then there will be no proof of (A or not-A).

This is pretty annoying; some kinds of perfectly healthy-looking examples of proof by contradiction just stop working. Of course, excluded middle is a theorem of classical logic (i.e. non-intuitionistic logic).

History.
References in periodicals archive ?
is just the assumption that has brought intuitionism to life, the neutrosophic logic could not be a generalization of any Intuitionistic logic.
Neutrosophic Logic is a general framework for unification of many existing logics, such as fuzzy logic (especially intuitionistic fuzzy logic), paraconsistent logic, intuitionistic logic, etc.
In Chapter 13 of my (2006a), I considered a somewhat primitive paraconsistent system, the direct dual of intuitionistic logic (in which the law B [?
Compactness, meanwhile, has to be confronted with the learnability arguments Dummett developed to recommend intuitionistic logic.
Intuitionistic Logic and Elementary Rules, LLOYD HUMBERSTONE and DAVID MAKINSON
B) But to refer to intuitionistic logic, which means incomplete information on a variable proposition or event one has T +1 + F < 1.
logical not]K[phi] [right 11, by implication in arrow] [logical not][phi] intuitionistic logic
4<1; then B is a NS but is not an IFS; we can call it intuitionistic set (from intuitionistic logic, which deals with incomplete information).
Intuitionistic logic merely holds that LEM is not an axiom yet cannot be counterinstanced.
the Grzegorczyk hierarchy, the geometry of solids, results about indecidability, recursive computabiliy, and the S4Grz system or semantics for intuitionistic logic.
Intuitionistic logic is an important subsystem of the classical one.
In his 1963 paper, entitled 'Semantical Analysis of Intuitionistic Logic I', Kripke provides a model-theoretic interpretation of intuitionist logic.