True, False, Neither.....Both?

It's easy to understand that a sentence is either true or false. Somewhat more difficult to understand is when a sentence is neither true nor false. In fact, classical logic simply denies the possibility of such sentences. But there may be some reason for thinking that they exist--consider future contingent sentences like "I will dance the night away tomorrow night." Surely it will be true or false tomorrow night, but does that sentence have any truth value today? Or consider vague sentences like "He's bald" said of some moderately balding person. Perhaps in those cases, there is neither a true nor false answer.

But could there be sentences that are both true and false? In the abstract, that sounds ok, but the problem is, this opens the door to the existence of true contradictions. So, let's say P is true and false. Well, that means ~P is also true. So that means that P & ~P is true. Well, there ya' go. True contradiction.

Graham Priest spoke today and suggested we ought to believe in true contradictions. His evidence? Take the following sentence. "This sentence is false." Well, if it's false, it's true, and if it's true, it's false. Problem. Well, you could say it's neither true nor false. But then what do you say of this sentence: "This sentence is either false or (neither true nor false)". Well, if it's true, it's false. If it's false, it's true, and if it's neither true nor false, it's true. Problem reappears. So why not say it's both?

Interesting argument. But I'm not sure I buy it. I'd rather, I think, go with other solutions to the Liar Paradox, insofar as I know what those are.