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 (http://britanica.com/bcom/eb/article/3/0,5716,118173+14+109826,00.html).
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