# 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 ?
For the author an "intuitionistic logic" means a logic that invalidates the double negation law {equation (3>}.
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.
Compactness, meanwhile, has to be confronted with the learnability arguments Dummett developed to recommend intuitionistic logic. The claim that FOL-theorems are implied by all theories once we have formalized them in FOL is almost vacuous in the debate, as that very formalization is the issue of contention.
Intuitionistic Logic and Elementary Rules, LLOYD HUMBERSTONE and DAVID MAKINSON
Our setting is Bishop's constructive mathematics ([3], [4], [5], [6], [8] and [15]), mathematics developed with Constructive logic (or Intuitionistic logic) - logic without the Law of Excluded Middle P [disjunction][logical not]P.
(B) But to refer to intuitionistic logic, which means incomplete information on a variable proposition or event one has T +1 + F < 1.
13.) [logical not]K[phi] [right 11, by implication in arrow] [logical not][phi] intuitionistic logic And either result is devastating.
From the intuitionistic logic, paraconsistent logic, dialetheism, faillibilism, paradoxes, pseudoparadoxes, and tautologies we transfer the "adjectives" to the sets, i.e.
After all, an instance of the negation of LEM is inconsistent even in intuitionistic logic. Intuitionistic logic merely holds that LEM is not an axiom yet cannot be counterinstanced.
Following a biographical sketch and summary of Grzegorczyk's contributions to the foundations of mathematics and view that rational standards can be applied to moral issues, 21 essays treat topics relating to his contributions in logic, e.g., the Grzegorczyk hierarchy, the geometry of solids, results about indecidability, recursive computabiliy, and the S4Grz system or semantics for intuitionistic logic. This book version of a special edition of Fundamenta Informaticae, v.
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.

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