This Statement Is False

Everything I say is a lie. The following statement is false, the previous statement is true. The shortened form: this statement is false. All these rewordings of the liar's paradox show it's one of the most iconic paradoxes in history. Like a good paradox, it's simple, it defies logic, and trying to follow it makes you go round and round in circles forever.

The statement says it's false, so "this statement is false" is false, which means it must be true. Or does it?

Recently, I read a clever solution to the liar's paradox; on Wikipedia of all places!

The classic solution is to say a logical statement is invalid if it references itself. Treat self-reference like dividing by zero. You can write it, but the rules stipulate that you can't make a meaningful statement out of it. That always felt like a bit of a cop-out though, especially because other self-referencial statements work just fine.

According to Arthur Prior the liar's paradox is just false, but to explain why we'll have to do a bit of algebra. I'll avoid using formal logic notation and symbols, but that means my explanation is going to get a bit wordy.

In logic, and in everyday speech, we can reword a statement without changing its meaning or truth value. If I say "I have more than 8 fingers" that means the same thing as "I don't have 8 fingers or fewer". "Not every dog is good" is the same as "at least one dog isn't good." In every situation where the first is true, the second is also true, and if because of some horrible accident one of them is false, the other must be false as well.

Arthur points out that another way to reword a statement is to add "...and this statement is true" to the end of it. "It's raining" is true if and only if "It's raining, and this statement is true" is true. Even if they carry different implications.

What happens if we reword the liar's paradox this way? We end up with "This statement is false, and this statement is true."

This is fairly well known, but if you say "one thing and another", the whole statement is false if either thing is false. And because of the Principle of Bivalence, nothing and its opposite is true. I.e. "It's raining" is true sometimes, "It's not raining" is true sometimes, but "It's raining and it's not raining" can never be true.

If we put this all together, the liar's paradox is false. If it were true, then the first half, "this statement is false" makes the whole thing false. If it were false, then the second half, "this statement is true" would make it false. So either way, it's false.