See Liar Paradox; according to the logic tradition, from Aristotle on, a sentence that expresses a proposition (that corresponds to a fact ) must have a definite truth value. The Liar sentence does not satisfy this criteria: if we assume that it is true, than what the sentence express is a fact, and thus the sentence must be false, and vice versa.

The most common solution is to assume "regimented" languages, like those of formal logic. In natural language we have rules of grammar that define what is a "correct" expression: This elephant is false" is not. In the same way, the standard approach to Liar for formal languages is that a formal language cannot "speak about itself". In this way, the formal counterpart of Liar sentence is not "correct" (not well-formed).

This type of solution was that used by W&R to avoid Russell's Paradox in Type Theory.

This type of solution was taken to extrems by Wittgenstein in the Tractatus where all metaphysical sentences was deemed non-sense because un-grammatical (according to W's vies of a perfect language).

Regarding your perplexities, we may formulate the paradox in many ways: "Consider a sentence named ‘FLiar’, which says of itself (i.e., says of FLiar) that it is false. FLiar: FLiar is false." The simple form of the paradox is due to the "simple" nature of natural language: natural language is "circular", i.e. we can define everything without stepping outside language, and with it we can "speak of" everything, also of language itself.

Do you think "this elephant is an elephant" is a false statement because the subphrase "this elephant" is not an animal, but a subphrase? In language, "this elephant" is a placeholder for an entity about which to talk. So the subphrase "this sentence" is not itself something that can be true or false, it is a reference to something else that can be true or false (the whole sentence).

@Mauro - Are you saying that the Liar contradicts the PBV? This may be so, but where is the paradox? I see no need for a solution.

@tkruse - Exactly!

The "self-referring" statement is not an issue: we can consider original Epimenides version: here we have an assertion made by a speaker involving the speaker itself. It is obviously syntactically correct because it has the same syntactical form of "All Cretean are males" or (for the self-referential version) "I'm a Cretean".

The article is making the same mistake you are (it's a common mistake, I made myself for a long time)... saying that there is no entity "This sentence". We can reformulate the liar as simply referring to properties of "strings of characters" itself. Say a string of characters translates to a true proposition in english. I'll define that as "brue" and in all other cases the string of characters is "balse". So all strings of characters are either "brue" or "balse". String A: String A is balse. We get the same liar paradox now. If String A is brue then it is balse. If it is balse it is brue.

"In an effort to clear up this antinomy it has been protested that the phrase `This sentence', so used, refers to nothing. This is claimed on the ground that you cannot get rid of the phrase by supplying a sentence that is referred to... we can do so thus: ``Yields a falsehood when appended to its own quotation' yields a falsehood when appended to its own quotation'... This is a genuine antinomy", Quine, Ways of Paradox. Any "solution" to the Liar is defeated by a variation of the Liar, as with Gettier

@Conifold and Ameet - I wonder why I cannot understand this issue. For the life of me I cannot see a paradox. Could you give an example of a genuine antinomy? I'd be quite happy if you can, I have no horse in the race. Not strings of logical symbols but natural language if possible. In the case of "(This sentence is false) is false" my objection still seems to stand.

@PeterJ How do you feel about "This sentence has five words?" This has the same reference structure but there's nothing complicated about its meaning. (And I truly don't understand the reference issue: in "This sentence is false," the "this" clearly refers to the whole sentence "This sentence is false." I can't understand your or the cited essay's objects to this, other than that it is unhelpful in unpacking the sentence's meaning in some way, but I don't get why that's an argument against it.)

@NoahSchweber - I see what you're saying and in a way I agree. But this reduces the paradox to a trivial artefact of language. I;d be okay with "By the time it is finished this sentence will have twelve words". (We can't chat much here but I'll ask a follow-up question when this one is exhausted). .

I suspect that the issue is with the objection. It seems vaguely plausible at first glance but falls apart when thought through, and Quine's reformulation removes its target altogether. Perhaps the root of it is the intuitive idea that "meaning" should be extracted by breaking up sentences into parts and understanding the "meaning" of parts. This idea is deeply problematic and, in a way, things are the other way around. It is use in sentences that gives "meaning" to words, not words combine their "meanings", and the usual rules of use applied to this sentence lead to a contradiction.

Read 'Liars and Outliers' by Bruce Schneier. He goes into the Liars paradox in detail.

@Conifold - If the Liar is a legitimate dialectical proposition having a truth-value. then it has a contrary stating "This sentence is true". So how do we decide between them? Is this sentence true or false? It seems obvious the Liar does not meet the conditions for a dialectical proposition.

How and what we decide is the question for one's semantic theory of propositions to answer. If it can not handle the Liar so much the worse for the theory. It is a theory that needs to meet conditions of a language, not the language that needs to meet conditions of the theory. The paradox exposes that the folk semantics of sentences combined with classical logic are inadequate to the task, hence the value of intuitive seemings is low.

@Conifold Right. So we formulate a proposition that breaks Aristotle's rules for dialectical propositions and use this to prove that Aristotle's dialectical logic is faulty. And this is supposed to be good philosophy? .

We know from many independent linguistic phenomena that Aristotle's logic and semantics are limited. And yes, if a theory encounters a phenomenon it can not handle it is a good philosophy to amend/extend it. Indeed, this is what Aristotle himself advised.

@Conffold Yes, but first we have to prove his logic is faulty. I can see no fault with it. The case for its faultiness has not been made. Making up illegitimate propositions is not a way to prove anything and is what I'd call a straw-man argument. You're accusing Aristotle of allowing contradictions, but his rules prevent them when they are followed. So, contradictions arise when we break the rules of the dialectic. This is the reason for having the rules. The problem seems to arise from a very basic misundersting of logic and I cannot see what's wrong with this view. . . .

The Liar long predates Aristotle, it is due to Epimenides, so it was not made up in response. We can agree that self-referential sentences do not fall under the scope of Aristotle's semantic rules, but that is just another way of saying that their scope is limited. Neither do non-syllogistic inferences, multi-place predicates, indexicals, definite descriptions, future contingents, etc. A more productive approach is to see what minimal extensions can make sense of the Liar. There are two types of options: alter the rules for the truth predicate a la Tarski or Kripke, or alter logic a la Priest.

@Conifold - Thanks. We'd better stop here. if you're saying that the Liar is not a dialectical contradiction then we're not disagreeing in a serious way. I was looking for a proposition (or pair) that is a paradox or contradiction in Aristotle's dialectic, but nobody seems able to find one. Even Priest never cites one. The issue seems to have no significance in metaphysics, or not until someone proves Aristotle's system is flawed. Thanks for your comments. .

I think the part about Aristotle's dialectic would have clarified the OP significantly, it is quite a different question. The twin paradox, say, is not a paradox of Newtonian mechanics, etc. You may want to look at Crivelli's Aristotle on the Liar. Semantics is rather remote from metaphysics generally, it is about interpreting what we think and say, not about what is.

i think think the point about Aristotle would clarify the entire issue significantly, which was sort of my point. All good. Thanks for reality-check.