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.

(2001-03-18)