In constructive mathematics even very small sets can be quite a bit more interesting than in classical mathematics. Since you will not believe me that sets with at most one element are very interesting, let us look at the set of truth values, which has “two” elements.

# Category Archives: Tutorial

# The Law of Excluded Middle

Ordinary mathematicians usually posses a small amount of knowledge about logic. They know their logic is *classical* because they believe in the *Law of Excluded Middle (LEM)*:

For every proposition `p`, either `p` or `not p` holds.

To many this is a self-evident truth. Therefore they cannot understand why someone would reject such a law, and a useful one at that, since many neat proofs depend on it. An equivalent law of logic is *reductio ad absurdum* or *proof by contradiction*:

For every proposition `p`, if `not p` does not hold, then `p` holds.

Constructive mathematicians do indeed reject LEM. But this does not mean they accept its negation! Unfortunately, many ordinary mathematicians seem to think precisely that, and so naturally they conclude that constructive mathematics is garbage. In fact, both classical and constructive mathematics prove quite easily that the negation of LEM is false. So what do constructive mathematicians believe?