TITLE: Systems without the law of excluded middle
QUESTION [5 upvotes]: If my understanding is correct (indeed, I think Wikipedia says this same thing) there are systems of logic in which the law of excluded middle does not hold. I can see how in some kind of generalization of logic we could have a system without the law of excluded middle, but then are these just abstract notions?
By that I mean, it seems that the 'standard' way of 'using' logic seems to be with the law of excluded middle, indeed if someone makes a sensible statement (i.e. not intentionally trying to be tricky) it seems to be either true or false. So do these systems which reject the law of excluded middle have a use, or deeper implication, or are they just generalizations  in some sense?

REPLY [6 votes]: There is both a use and a deeper implication. Let me start with the deeper implication. At the beginning of the 20th century, there was a so-called foundational crisis in mathematics. Out of this, several schools of philosophy of mathematics emerged, one of which was intuitionism, fronted by L. E. J. Brouwer. Brouwer's idea was that mathematics is a creation of the humand mind and exists only in people's minds. Hence, he only accepted those mathematical truths which could be constructed in some sense. Now, the problem with the law of excluded middle (LEM) from this point of view, is that it allows you to make use of the reductio ad absurdum (RAA) proof technique which in turn allows you to prove things without actually constructing the thing you are proving. Essentially, RAA argues that since it can not be the case that $\neg P$, using LEM it must be the case that $P$, but it does not actually give a way to construct $P$.
This is a fairly esoteric and philosophical discussion, but it turns out to actually have some use. When we have a constructive proof, i.e. a proof that actually constructs what is to be proven, the proof also gives a procedure or an algorithm for doing this construction. Think of the Gram-Schmidt process, for example. The theorem says that every finite-dimensional inner product space has an orthonormal basis, but since the proof is constructive, it also gives an algorithm for actually generating such an orthonormal basis. This turns out to be extremely useful in many applications.