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Let's consider the famous liar paradox's statement:

This statement is false

Now, in classical logic, principle of bivalence could be stated as "All statements can either be assigned a value of true or false", if we assume principle of bivalence to be true, it would mean that the liar's paradox statement must be true or false, but assigning it any value out of the both would lead to a contradiction.

So this would imply that principle of bivalence is not true, i.e "Some statements cannot be assigned either a value of true or false", and this is the only straight-forward I have thought to resolve it, I have read about how fuzzy logic tries to resolve it, but just because principle of bivalence turns out to be not true, we cannot necessarily categorize such statements as both true or false (i.e a value of 0.5). I would want to know if the members of the community have/know about a solid way to resolve it, or rather find anything wrong in my resolution.

Siddharth Chakravarty
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    Both the [IEP](https://iep.utm.edu/liar-paradox/) and [SEP](https://plato.stanford.edu/entries/liar-paradox/) articles on the topic have the information you're asking about. Note that bivalence is equivalent to LEM not automatically overall but under relatively precise conditions. "This sentence is not true," still generates a "revenge" paradox, as would, "This sentence is false or meaningless," etc. – Kristian Berry Jun 10 '23 at 17:17
  • This contribution to the thinking about the paradox is very good. – Mark Andrews Jun 10 '23 at 19:40
  • LEM cannot be stated as "all statements can either be assigned a value of true or false", this is called bivalence. One can reject bivalence and admit truth value gaps, but still affirm LEM, as in [supervaluationism](https://plato.stanford.edu/entries/sorites-paradox/#SupeRela), which is used to resolve the sorites paradox. Truth value gaps, with or without LEM, are used to resolve the Liar as well, e.g. [by Kripke](https://iep.utm.edu/liar-paradox/#SH3c). But all such resolutions are unsatisfactory in one way or another. – Conifold Jun 11 '23 at 00:05
  • @Conifold and Kristian Berry, Thank you for pointing out that bivalence is more precise than LEM, however I wasn't sure which one to write because I have seen many people used them interchangeably, but as you guys suggested, I will edit the question. – Siddharth Chakravarty Jun 11 '23 at 03:42
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    You've rediscovered the principle of incompleteness. [Yanofsky 2003](https://arxiv.org/abs/math/0305282v1) is probably the best entrypoint if you want a comprehensive understanding of what's going on. Further, [Bauer 2016](https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016-01556-4/S0273-0979-2016-01556-4.pdf) is likely interesting if you'd like to know more about logic without the Law of Excluded Middle. – Corbin Jun 11 '23 at 04:59
  • @Corbin I will check out the papers, thank you. – Siddharth Chakravarty Jun 11 '23 at 16:26
  • You are unable to solve the problem because the solution do not exist in the very small set you are looking in. Your assumption "All statements can be classified as either true or false" limits you to a very small set of statements. There are statements that cannot be classified as true or false. Consider this statement "Your boss is taller than you". How will you classify it within your framework of true/false if you dont have a boss? Right way to think is "the property: my-boss do not exist. therefore any property of my-boss do not exist. my-boss dont have a height" Is it true that the numbe – Atif Jun 12 '23 at 14:49
  • @Atif that is what my conclusion is, have you even read the post carefully? I already answered this to you in your answer, which you deleted, and I don't know why. – Siddharth Chakravarty Jun 13 '23 at 11:42
  • @SiddharthChakravarty, https://philosophy.stackexchange.com/a/98298/65403, this is a similar question and I have put this answer there. This can help you. – Agnibho Dutta Jun 13 '23 at 12:19
  • Aaah! A zen moment worth relating: Of course, of course, he is not human. – Agent Smith Jul 15 '23 at 11:43
  • @AgentSmith I am not sure what you mean. – Siddharth Chakravarty Jul 15 '23 at 17:59

2 Answers2

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Suppose that, "This sentence is false," is neither true nor false. Let's use the correspondence, coherence, and truthmaker theories of truth to specify the original statement:

  1. This sentence corresponds to an anti-fact.
  2. This sentence anti-coheres with the general set of true sentences.
  3. Something makes this sentence false.

Now, what is falsity, then?

  1. If this sentence corresponds to an anti-fact, then this sentence doesn't correspond to a fact.
  2. If this sentence anti-coheres with the general truth set, then this sentence doesn't cohere with the general truth set.
  3. If something makes this sentence false, then nothing makes this sentence true.

Or we could say:

  1. This sentence anti-corresponds to a fact.
  2. This sentence coheres with the antiset of truth (the set of anti-truth).
  3. This sentence is made anti-true.

What the above hopefully brings out is that, "This sentence is false," and, "This sentence is not true," are almost identical, and a proposed solution to the falsity-framed version is only as good, eventually, as a proposed solution to an untruth-framed version. To wit (we'll leave evaluating the coherence/truthmaker versions as an exercise for the reader):

  1. This sentence doesn't correspond to a fact.
  2. If (10) is not true, then (10) doesn't correspond to a fact.
  3. Any sentence that is what it says it is, corresponds to a fact.
  4. Therefore, if (10) doesn't correspond to a fact, then (10) corresponds to a fact.
  5. Therefore, (10) does and does not correspond to a fact. QED

And so on. Now, the weakest premise is (12), or at least it is open to an interpretation such that we might say, "If something is fully X and fully not X, then there is no difference between being X and being not X/not being X." In other words, here, for, "This sentence is not true," there is no difference between being true and not being true. This can be seen by plugging the sentence into the truth biconditional:

  1. (TB) "S is X," is true if and only if S is X. E.g., "Kittens are cute," is true if and only if kittens are cute.
  2. "This sentence is not true," is true if and only if that sentence is not true. (Note: there is a sort of "indexical degeneracy" here in that we cannot repeat the left-hand "This" on the right-hand side of the biconditional or else we form a degenerately nested sequence of right-hand sides.)
  3. "This sentence is true," is not true if and only if that sentence is not true.
  4. "This sentence is true," is true if and only if, "This sentence is not true," is not true.
  5. Therefore, "This sentence is true," is true if and only if, "This sentence is not true," is true.
  6. "This sentence is true," eventually means the same thing as, "This sentence is not true."
  7. Therefore, the liar and honest sentences are equivalent as to their truth-conditional semantics.
  8. Therefore, for these two sentences, there is no distinction between truth and untruth.
  9. If there is no difference between A and B, then saying, "X is both A and B," is the same as to say, "X is both A and A," or, "X is both B and B," which is redundant.
  10. There is no difference between the liar sentence's being true and the liar sentence's being untrue.
  11. "X is both A and A," is not actually a contradiction.
  12. Therefore, the liar sentence's being true and untrue is not a contradiction. QED

Incidentally, fuzzy logics or other logics with partial values of truth (not exactly the same thing as partial truth values, but we'll not go over that topic here) do not claim that a sentence whose truth value is 1/2 is "both true and false" just like that. They might say, "Such a sentence is partly true and partly false," but this is not so as to conform to either bivalence or a truth-predicate application of the LEM (fuzzy logic is normally about as far from bivalent as can be, though using "just true" and "just false" as endpoints in the sequence of possible truth values is perhaps a higher-level sort of bivalence).

Kristian Berry
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  • The main flaw that I see is, for example, it asserts that "This sentence is false" corresponds to an anti-fact or anti-corresponds to a fact, but these statements are based on the assumption that the sentence has a well-defined truth-value, which is precisely what the liar paradox challenges. The argument you presented only works, if you strive to say that the statement necessarily has truth-value and somehow, you make it correspond to the value of truth. I could agree fuzzy logic works for vague statements like "He is a tall boy" where assigning a value of truth makes sense there. – Siddharth Chakravarty Jun 11 '23 at 16:24
  • @SiddharthChakravarty they're not based on that assumption at all, the conclusion of the whole argument is that the use of the word "truth" for, "This sentence is untruth," is very unlike the attribution of a distinctively meaningful truth value. – Kristian Berry Jun 11 '23 at 19:25
  • While you claim that the use of the truth is not the same as 'truth is usually understood' for the liar sentence, the properties of correspondence and anti-correspondence you assign to the sentence still involve the usual notion of truth or falsity. Thus, the conclusion sounds gibberish when you mix up both the usages, also the self-referential nature of the liar paradox remains unaddressed, and your analysis does not sufficiently resolve this inherent contradiction.. – Siddharth Chakravarty Jun 12 '23 at 12:24
  • @SiddharthChakravarty I'm not sure you're familiar with the topics at hand. I'd recommend reading about [the revision theory of truth](https://plato.stanford.edu/entries/truth-revision/) and the other SEP/IEP articles I linked in my comment on the OP. As it is, I wasn't trying to absolutely resolve the contradiction but was highlighting how switching from "false" to "not true" isn't a sustainable solution, either. – Kristian Berry Jun 12 '23 at 13:34
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    before I argue something more, and not to be biased or be wrong in any way, I would need time to go through the links you mentioned so that I can finally be ensured that we are on the same page. Thank you, as of now, I will reply back soon after giving the whole thing a thought properly again after going through your references otherwise we would just end up in a useless long discussion. – Siddharth Chakravarty Jun 12 '23 at 15:20
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You are only looking for statements that are either true or false.

The statement "This statement is false" is neither true nor false.

GlassFullOrEmpty
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  • That's what the OP says, though. It seems to be the main thrust of their reasoning. – Kristian Berry Jun 12 '23 at 15:11
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    Maybe I'm just being paranoid but I think it's odd that two answers to the OP, two answers that don't even understand the OP (or the topic more broadly), have received several upvotes. Almost like people are signing in under different usernames to inflate their upvotes? – Kristian Berry Jun 12 '23 at 18:10