# "It is possible that this sentence is not true"

This is a "modal liar sentence" that I'm trying to use to illustrate how liar sentences are not well-formed. It seems like a nifty species to consider, and I never came across it before, but so... The illustrative feature seems to be a vicious circle where the whole thing becomes, "It is possible that, 'This sentence is possibly not true,' is not true," and so on, just like feeding the liar sentence into the disquotational scheme results in a warp.

But is there an easier analysis in the literature already, that I'm not aware of?

• All of the otherwise undecidable decision problems become decidedly untrue when we construe the body of analytical knowledge as a recursive language with a membership algorithm. A language L on Σ is said to be recursive if there exists a Turing machine M that accepts L and halts on every w in Σ+. In other words, a language is recursive if and only if there exists a membership algorithm for it. (Linz 1990:288). Linz, Peter 1990. An Introduction to Formal Languages and Automata. Lexington/Toronto: D. C. Heath and Company. May 5 '20 at 21:10
• I am afraid this illustrates not that Liar sentences are ill formed, but rather that having a truth predicate + self-reference device creates an inconsistent language. The modal Liar can be formulated without the truth predicate, with a necessity/possibility operator only, but even they are truth-like enough to do it, see Parent, New Modal Liar. May 5 '20 at 23:22

The sentence is true, but not necessarily so.

It seems clear prima facie that a statement whose truth value you cannot really know might be something other than true. And that the mere possibility of it being something other than true does not imply that it is not actually true. Not everything that is possible, happens. (Note that you have chosen 'not true' and not 'false'. In the realm of modality, those are not necessarily equivalent. A proposition that depends on your moral stance might, when asked in general be neither true nor false.)

If you assume that you know it is false, you have a direct contradiction, because the impossible is not possible. So your option is that either you know it is true, or that you can't really know, so it might not be true, so it is still true.

So the statement is true, but that truth is not necessary -- it remains possible it could be otherwise, or it wouldn't be true in the first place.

What the question forces us to ask, I suppose, is whether what is actual is necessary. I am coming down on the side (opposite to Aristotle) that necessity is a stronger requirement than actuality, so that even what is known to be true for some reason, could still be false in an alternative situation.

There could be a world where only what is necessary is true. In such a world, the modality of necessity and possibility vanishes, (at the expense of a lot of things being meaningless.) So it would devolve down to the original Liar's paradox, which has no truth value. So it could not be true. By at least one common definition of possibility, Kripke's, we do not live in that world, because we possess the ability to define coherent fictional worlds. It is only by this observed incident of our world, which is not necessary, that we can deduce the possibility of the statement's non-truth is not ruled out by the observation of its truth.

Unfortunately, that is hard to make clear. Formalizing it involves hokey nonsense. Here are a couple of approaches:

If we stick to the Kripke notion of 'possible', that something is possible if you can imagine a world where it is true. "It is possible that this sentence is not true." can be true.

Since the whole idea involves reference to text, it involves literal quoting. We can imagine the possible world where the literal text 'true' means 'interesting' instead of meaning true. We can then imagine that the sentence is uninteresting there, because it is here, at least to me. So we have found a world in which the sentence, as a piece of text, interpreted there would be true.

So the sentence "It is possible this sentence is not true." is true, if by 'this sentence' you mean the text of the sentence.

If you don't like the focus on the text, another way to look at it is that "It is possible this sentence is not true" means "It is not necessary that the sentence 'It is possible this sentence is not true.' is true." Well, in at least one logical world, the one where only what is necessary is true, it paradoxical, and therefore not necessarily true. So it is obviously not necessary for it to be true. That means this equivalent sentence is true and therefore the original sentence is true, just not out of necessity.

I will say again here, what I said at the top:

Note that none of this applies to "It is possible this statement is false." If you assume that is true, it is false. If you assume it is false, it is true, so it is false. If you assume it is neither true nor false, it cannot be false, so it cannot be true. In no case is it not in contradiction with itself.

Any argument that assumes these two statements are related, has not paid enough attention.

• "We can imagine the possible world where the literal text 'true' means 'interesting' instead of meaning true." By your same reasoning (swapping word labels for semantic meanings) We could build a five mile long brick wall from "one cute little bunny" because "one cute little bunny" is what we call a very huge pile of bricks. Nice try. May 7 '20 at 2:37
• @polcott Mockery is not an argument unless it involves an actual argument. "Nice try." The cute bunny does not depend entirely upon its name for existence. "This sentence" means what it means only because of the context in which it appears, and the fact the language can refer to itself. So your objection is not an argument. If we did not quote "one cute little bunny", using it this way would be a lie. But 'this sentence' means the quoted sentence, which consists of actual sounds or letters. If it doesn't, then actually propose a credible thing that it would refer to... May 7 '20 at 12:32
• I conceded these arguments' 'hokey' quality in the post (which makes your then attacking it tacky at best). But what is hokey is the idea that all forms of "possible" are created equal and there is a single interpretation of the sentence that can be discussed, to begin with. If there is one, and this is actually a question, why is this not the right interpretation of possible? (Not, "please call me stupid", Why, really, is this not it?) May 7 '20 at 13:02
• In my fundamental frame-of-reference semantic meanings are absolute and immutable, thus changing the word labels is a fallacy of equivocation error. May 7 '20 at 14:48
• May 7 '20 at 16:47

I disagree with the premise that this is a "model liar sentence" and with @Polcott's answer as well. Let me explain:

The sentence says "It is possible that this sentence is not true."

So there is two options:

If the sentence is true, it remains: "It is possible that this sentence is not true." As we cannot logically determine this sentence to be true 100% of the time, this sentence is ok.

If the sentence is false, then the sentence would be: "It is not possible that this sentence is not true." Which is equivolent to "This sentence is true," which again works.

• It's "modal", not "model", and it does generate a contradiction, see Post, Possible Liar. May 6 '20 at 4:18
• Yes @Conifold is correct, the added extraneous complexity does not change a thing. May 6 '20 at 16:08
• To apply the link, one has to assume not true == false, which is not reasonable in a modal context. It surely does not apply in the deontological mode, especially once it intersects the mode of necessity. Things can be not always immoral, yet not always moral. May 7 '20 at 17:57

I am afraid this illustrates not that Liar sentences are ill formed, but rather that having a truth predicate + self-reference device creates an inconsistent language. The modal Liar can be formulated without the truth predicate, with a necessity/possibility operator only, but even they are truth-like enough to do it, see Parent, New Modal Liar. – Conifold

The above is a direct quote of Conifold's reply. I agree with the gist of it except that because it is the Liar Paradox sentence itself that specifies self-reference I disagree:
"not that Liar sentences are ill formed". Liar sentence themselves are ill-formed.