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 ?
Because the logic background of our approach is the intuitionistic propositional calculus, we refer to our approach as propositional logic programming.
Restricting logical specifications of program modules to the formulas of form (2), a precise representation of the semantics of specifications can be given in a simple language which is still universal language of the intuitionistic propositional calculus [6].