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).

References in periodicals archive ?
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
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.
which has the properties of the intuitionistic logic, according to a well-known scheme that I only reproduce with some adaptation, below.
In Chapter 6, 'Intuitionistic Logic', Burgess explains Godel's interpretation of the intuitionistic sentential logic I as a modal logic in which the box is interpreted as a provability operator and shows, via Kripke models, that a formula is a theorem of the intuitionistic logic I iff that formula's modal transformation is a theorem of S4 (130-32).
Intuitionistic logic was developed in the beginning of the XX-th century, in search for a basis for constructive mathematics.
After a brief recapitulation of classical logic--to establish notation and set the pattern for the subsequent material--the book covers normal and nonnormal modal logics, the strict conditional, other conditional logics and notions of entailment, intuitionistic logic, many-valued logics, various relevant logics, and fuzzy logic.
Among antirealists, some view it as genuinely paradoxical, while others take it as further evidence that anti-realists should deny (v) and opt for an intuitionistic logic.
Shapiro and Taschek have argued that simply using intuitionistic logic and its Heyting semantics, one can show that there are no gaps in warranted assertability.
It incorporates features that are to be found in both paraconsistent and intuitionistic logic.
The paper responds to Neil Tennant's recent discussion of Fitch's so-called paradox of knowability in the context of intuitionistic logic.