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Psychohistorian comments on Unsolved Problems in Philosophy Part 1: The Liar's Paradox - Less Wrong
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Comments (130)
I remember being bothered by this problem, and feeling like I had resolved it as an undergrad. Calling it a "true contradiction" seems absurd; you've just drawn a circle around it and said, "Nothing to see here! Move along!"
I think the solution is related to modal logic. "This sentence is false" creates a self-referential universe devoid of meaning, and thus has no truth value. It refers only to the world of itself, and there are no rules that it can be evaluated against, nor are there any observations that can confirm or disconfirm it. It is, in a sense, epiphenomenal, as there is no actual thing which it corresponds, predicts, or relates to. It is, in a sense, a one-sentence universe that cannot be tied to anything in any other universe.
This concept seems more robust in my mind; I suspect I am either making a mistake or failing to explain myself. Criticism or questions would be appreciated.
I'm highly sympathetic to the intuition that the liar sentence is devoid of meaning in some important respect, but I don't think we can just declare the liar sentence meaningless and then call it a day. Because in another respect, it definitely seems meaningful. I understand what a sentence is, and I feel like I understand what it is for a sentence to be true or false. If someone wrote on a blackboard "The thing written on the blackboard of room 428 is false," I feel like I would understand what this is saying before I went to check out room 428. Hence I must understand the sentence if it turns out that we're in room 428 already.
Also consider the Strengthened Liar: "This sentence is not true." According to your solution, that sentence should also be dismissed as meaningless, right? But surely meaningless sentences a fortiori aren't true. But that's precisely what the sentence asserts, hence it is true.
If it's meaningless, it doesn't assert anything.
A sharper formulation of the paradox just came to my mind. Consider the statements X = "X is not true" and Y = "X isn't true". (The difference in spelling is intentional.) If X is meaningless, then X isn't true, therefore Y is true. But it's a very weird state of affairs if replacing "isn't" by "is not" can make a true sentence meaningless!
The apostrophe in this sentence isn't needed for comprehension.
When you state ~~A(S) iff ~S, you are formally substituting S for ~A(S), but the meaning of "~~A(S) iff ~S" is "the set of truth-conditions for ~~A(S) is the same as the set of truth-conditions for ~S". But this assumes that there exists a set of truth-conditions for ~S, which assumes that there exists a set of truth-conditions for S, i.e. that S is meaningful, by your definition.
O.K., I don't know how to italicize here.
Next time you comment, try the Help link (lower right).
Ah, thanks.
Interesting idea. But what is it that shifts us into a new universe? A clause of the form "___ is true"? The use of an indicative "this"? I like the idea of a universe disconnected from the rest of reality. But what puts us there, and what can we talk about while in residence?
You might enjoy Vicious Circles which sketches a resolution of the Liar which seems similar to what you are suggesting. Your idea may also be very similar to the "relevance logic" and "paraconsistency" approach sketched in the article linked by the OP.