Things that are both true and false

As a matter of logic, there is a very good reason why you can't have statements that are both true and false. If the statement "P", and the statement "Not P" are both true, something bad happens. Let us express this with logic.

• Premise 1: Not P is true.
• Conclusion 1: "If P, then Q" is true, by Premise 1.
• Premise 2: P is true.
• Conclusion 2: Q is true, by Conclusion 1 and Premise 2.

I just proved statement Q. Likewise, I can prove statement R, statement S, statement T, and so forth. All statements are necessarily true within this system of logic. This is called the Principle of Explosion. If we accept a single contradiction, then we must accept absolutely everything as true. Presumably we don't want to do that.

And so, when you hear about a "different kind of truth", which includes statements that are both true and false, the claim is, on the face, very problematic. Perhaps we're talking about a kind of truth which does not operate by logic. If so, it probably doesn't deserve to be called a truth; it should be called something like "truthiness". Or perhaps it only appears that there are statements that are both true and false within the same system. In other words, the contradiction is an illusion, and therefore not something "deep".