Moonfisher- If by meaningless or invalid you mean something that cannot be true or false, then meaninglessness or invalidity *is* a new truth-value, by definition, since that means that "if a sentence is meaningless, then it's not true or false. If a sentence is not true or false, then it's meaningless."

So, something must be true, false, or meaningless- what about "This statement is *not true*"? If it's true, then it must not be true, so it's true and not true, so it's false or meaningless. If it's false, then what it's saying is false, so it's false that it's not true, so it's true or meaningless. So, it can't be true, it can't be false, so it's just meaningless... yay! And everything works, right?

Well, no. As it turns out, if it's meaningless, then what it's saying is true- it says it's not true, so it says "I'm false or meaningless". Since it's right, it's true. So, if it's meaningless, it's true, and we get a paradox all over again.

Jez- Just like I showed above, you can have 3 truth values, or 4 truth values, or infinitely many truth values, and you'll solve the original paradox, but you can't solve what's known as the strenghtened liar- "This sentence is not true." Because, if it's truth value Q, or whatever else you make up, then it's true, so it's false, etc...

Also, although the 1-2-3 statement is weirder, it's important because many solutions to the original liar or the strengthened liar fail in front of paradox 1-2-3 - aka what I call the "ultimate liar". For example, we could say a statement cannot refer to itself. That solves the original AND the strengthened liar, but the ultimate liar is not self-referential, so that doesn't work. It's just an acid test for any solution that people come up with.

Peterb- Actually, no. A contradiction is something of the form "x and ~x", while a paradox is a proof of a contradiction from acceptable premises. Just writing down "X and ~X" AKA "X = ~X" is not a paradox, since it must be false according to the rules of logic. Think about it- can x ever possibly equal ~x? Of course not, so it's just false.

However, this isn't the case with the liar- if it's false, then it *must be true*. It is not saying it isn't itself. It's a paradox because I can prove a contradiction based off of it, but it itself isn't a contradiction. To see the proof, you'll need to look at the "tarski's proof" document I included.

Bacillus-

I'm glad no logicians heard you. Zeno's paradoxes are helping guide current quantum physics, and have even helped spawn a new area of physics- digital physics. Russel's paradox showed that the original set theory was wrong, and Godel's incompleteness theorem, which is basically a version of the liar paradox that's been proven in math, showed that math must be incomplete or inconsistent. Once (if ever) the liar paradox is solved, it will reveal new fundamental truths about truth itself. So far, it seems that logic itself must simply be incomplete or inconsistent to solve the liar, which itself would be earth-shattering in all fields dealing with logic (aka all fields) if proved.