I'm not sure if I'm engaging in some sort of circular logical trap but I don't really think "this sentence is false" is all that logically problematic. But it would be helpful if someone could fix up my thinking on this if I've misstepped.

"This sentence is false" is correct that it is false but we find it problematic because we conflate correctness with truth. Undergoing a proof by contradiction, in my mind, is a kind of cross-examination of hypothetical correctness with regards to what a statement purports. But when we engage in such an indirect logical proof we are dealing with the sentence in two different ways, and then conflating them.

  1. The sentence as the sentence
  2. The sentence as a claimant of truth

The sentence as the sentence, with a specific (semantic) relationship to truth and falsity, is correct insofar as we can't prove it incorrect. Whereas the sentence as a claimant of truth by our instantiation of it through cross-examination, has a specific (logical) relationship to truth and falsity.

We can say the semantic sentence has a relationship to itself which is correct in that it says, without anything to contradict it, that it is false - as in not true, and not making a logical claim to truth.

Thus, this sentence is correct that it is false, because we cannot disprove its correctness through a proof by contradiction regarding its hypothetical truth or falsity, and therefore in a sense the paradox is simply false - semantically, and true - logically.

Apologies if I'm wasting people's time with this question. Please help me smooth over this difficulty.

closed as off-topic by Jishin Noben, Eliran, jobermark, Philip Klöcking Feb 21 at 7:37

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  • Why Russell's Paradox ? – Mauro ALLEGRANZA Feb 20 at 7:12
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    What is "correctness" ? – Mauro ALLEGRANZA Feb 20 at 7:13
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    Everyone has already explained why what you're saying is wrong, but you seem to not accept it. In order to try to make the issue clearer, you should think about the other semantic paradoxes. Is the Grelling sentence false? Is the Berry sentence false? They are not false, they are antimonic. As everyone has pointed out, your idea of "correctness" as separate from "true" is nonsense. If a sentence is correct then what it describes is true, that is the definition of "correct". Can you do the same mental gymnastics about the Grelling sentence? You should expand your question on that. – Not_Here Feb 20 at 7:43
  • @Not_Here I think I found the answer to my question in Wittgenstein. Though I was asking it poorly. projecteuclid.org/download/pdf_1/euclid.ndjfl/1093891616 – Jayden Rivers Feb 20 at 8:16
  • @JaydenRivers you could answer your own question below if you think it can be useful to others – Quentin Ruyant Feb 20 at 10:03

The OP claims the following about the liar paradox:

But when we engage in such an indirect logical proof we are dealing with the sentence in two different ways, and then conflating them:

  1. The sentence as the sentence
  2. The sentence as a claimant of truth

Beall, Glanzberg and Ripley note this about the significance of the liar paradox:

We have now seen that with some elementary assumptions about truth and logic, a logical disaster ensues.

They then explore the significance of this. One of the approaches they mention comes from Tarski which may be relevant to the OP's interests:

In a skeptical vein, Tarski himself seems to have thought the Liar shows the ordinary notion of truth to be incoherent, and in need of replacement with a more scientifically respectable one.

Furthermore, and perhaps further along the same lines the OP is investigating:

More common, and perhaps the dominant thread in the solutions to the Liar, is the idea that the basic principles governing truth are more subtle than the T-schema reflects.

There may be, as the OP suggests, a conflating of two different ways of looking at the sentence constituting the liar paradox.

Beall, Jc, Glanzberg, Michael and Ripley, David, "Liar Paradox", The Stanford Encyclopedia of Philosophy (Fall 2017 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/fall2017/entries/liar-paradox/.

  • This is very appreciated. I’m glad I persisted further instead of immediately backing down about what I intuited. – Jayden Rivers Feb 21 at 1:25
  • @Philip Klocking - I'm not sure how to comment except to do so here. I have no idea why you deleted my post rather than engage with it, I see no reason to alter it but would be interested in hearing your argument against it developed a little further, As it is I'll stick to my guns. . – PeterJ Feb 21 at 11:59
  • @PeterJ I voted to undelete your post. Since a moderator deleted it, that has only symbolic effect. I also voted to reopen this question and I up-voted your answer earlier. The problem of the Liar paradox is too quickly brushed aside. It is mainly a challenge for logic as you suggested with your phrase "logico-semantic muddles". – Frank Hubeny Feb 21 at 12:59
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    @FrankHubeny - Many thanks. I was a little confused since my answer presented a common view, be it correct or otherwise. The idea that 'this sentence' stands for the whole sentence (as suggested) just leads to an infinite regress of nested sentences. I feel you're right about the problem of conflating different ways of reading the sentence. . – PeterJ Feb 21 at 13:09

"My dog is brown." has both of those aspects, and nobody has any intention of making the statement and entailing 1 but not 2. Why should we make a special case out of this one just to avoid the inevitable? If the statement "My dog is brown." is true for a given instance of 'me' then it is a claim to truth. For other instances of the pronoun, it is still a claim to truth, just a failed one.

"This statement is false." cannot be taken as false. There is only one possible referent of the pronoun, and for that referent, if the statement were true, then it would also be false. These two interpretations simply do not exist separately, the former implies the latter. We do not make statements that are not meant to have implications. Meaning lies in usage an that is not how words are actually used.

Creating an entire grammatical category just to avoid the very few cases where a paradox would result is simply dishonest. The rules of language are clear, here, and they do not allow the sentence to be given 'false' as a truth value any more than they allow it to be given 'true' as one. The sentence just proves that the Law of the Excluded Middle does not always apply.

If it were truly alone, we might simply except it. But it joins Russel's set, Barry's number, Curry's list, and five or six other equally problematic forms, each of which can be disguised in numerous ways. There is no problem, logic is simply incomplete, paradoxes are just real and do not need to be explained away.

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    This statement is false implies it is false = true. This statement is false implies it is true = true. Therefore it is true that it is false, as in it is correct that what it purports is that it is false. To my mind, these are just different spheres of meaning - one operating in a truth evaluation about truth evaluation and the other operating in truth evaluation. – Jayden Rivers Feb 20 at 11:58
  • 'This statement is false" implies it is false, by stating that it is false. And that implies that the statement is true. You can be as dogmatic as you like, the logic remains the same. By simply ignoring everyone else, you can't make a philosophical point. I am moving to close the question on the basis that there was never any question -- this is a 'I am right!' "question". – jobermark Feb 20 at 20:46
  • What if I stated 'Every sentence is true.' Would it also have a truth value that did not necessitate resolving its external references? Would it be true by virtue of declaring itself true because it purports that it is true? There is no distinction between evaluating truth and considering the referents of the sentence. The former never exists without the latter. – jobermark Feb 25 at 16:18

The liar sentence can be shown to lead to contradiction.

From the Stanford Encyclopedia of Philosophy entry on the liar paradox:

Consider a sentence named ‘FLiar’, which says of itself (i.e., says of FLiar) that it is false.

FLiar: FLiar is false.

This seems to lead to contradiction as follows. If the sentence ‘FLiar is false’ is true, then given what it says, FLiar is false. But FLiar just is the sentence ‘FLiar is false’, so we can conclude that if FLiar is true, then FLiar is false. Conversely, if FLiar is false, then the sentence ‘FLiar is false’ is true. Again, FLiar just is the sentence ‘FLiar is false’, so we can conclude that if FLiar is false, then FLiar is true. We have thus shown that FLiar is false if and only if FLiar is true. But, now, if every sentence is true or false, FLiar itself is either true or false, in which case—given our reasoning above—it is both true and false. This is a contradiction.

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    But that doesn’t counterdict what I’m saying I think. FLiar can be thought of having two parts: the FLiar label and its instantiation in the proof by contradiction as a logical claimant. It’s like mixing up the general Cat animal label with a cat you see on the street. The Cat general label has certain properties associated with it but a certain cat might do things which seemingly contradict those properties. It doesn’t mean the cat itself is changing the general properties, we’d still consider the cat a Cat. Not the best analogy, but do you see where I’m coming from? – Jayden Rivers Feb 20 at 4:09
  • Like isn’t it like saying (assumeFalse)FLiar: “FLiar: FLiar is false” and (assumeTrue)FLiar: “FLiar: FLiar is false”. Two different instantiations of claimants to truth, without dealing with the semantics of “FLiar is false”. – Jayden Rivers Feb 20 at 4:30
  • @JaydenRivers "Cat" as a general concept versus a particular cat is a type-token distinction; "cats have pointy ears" is talking about the type cat, whereas "my cat is hungry" is talking about a token cat. I could consider whether my cat rubbing against my legs means he's hungry, or whether it means he's bored; those are two different considerations, but they both are considering what's true of a token, my cat, versus a type, the cat category. Here, FLiar is a token statement. Considering the implication of it being true vs false is like considering whether my cat is hungry vs bored. – H Walters Feb 20 at 4:40
  • @HWalters I’m getting closer to understanding, thanks for your help. But it still worries me. I think my problem is still that when we are talking about your token cat, we make it out like we are talking about the type Cat, when it shouldn’t be that we can talk about the type Cat from the token cat - simply because to talk about the type Cat means talking about all cats. I’m worried that FLiar as a token is being elevated to FLiar as a type. I also don’t understand the hungry vs bored consideration because your cat could be both, neither, hungry xor bored. – Jayden Rivers Feb 20 at 4:52
  • I actually have a cat; he's 20 pounds, white with black spots, and is an occupant of my house. Most individuals of the category "cat" are none of these things; in fact, most occupants of my house aren't even cats. So it makes no sense to consider the token cat as the type cat; that's conflating type and token. Now if we treat bored/hungry independently as binary states, that's still 4 states to consider. Once we consider all 4, though, I still only have one 20 pound cat, not 4 cats=80 pounds. Likewise, considering FLiar=true then FLiar=false doesn't make FLiar two statements. – H Walters Feb 20 at 6:53

“This sentence is false.” Many years ago, I heard of a way to cure the problem of the ambiguous self-reference. As follows:

creates a negation when appended to its own quotation creates a negation when appended to its own quotation

So the quotation creates a negation, so it does not, so it does, and so forth, without end.

If the suggested revision is a satisfactory restatement of “this sentence is false”, then run it through your analysis. Does it revive the paradox that you see as cured by the correctness/truth dichotomy?

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