# A resolution to liar's paradox

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.

• Both the IEP and SEP 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. Commented Jun 10, 2023 at 17:17
• This contribution to the thinking about the paradox is very good. Commented Jun 10, 2023 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, 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. But all such resolutions are unsatisfactory in one way or another. Commented Jun 11, 2023 at 0:05
• You've rediscovered the principle of incompleteness. Yanofsky 2003 is probably the best entrypoint if you want a comprehensive understanding of what's going on. Further, Bauer 2016 is likely interesting if you'd like to know more about logic without the Law of Excluded Middle. Commented Jun 11, 2023 at 4:59
• Your truth-value gap proposal is apparently the most obvious way out and had been rigorously studied under the multi-valued logics category intensively and extensively, one major issue is there could be no tautologies and powerful theorems thus they're very weak logics. Another issue is when applied to natural language applications we cannot demarcate what kind of sentences have gap base on a distinct clear natural criterion. Another radical way out of the liar paradox concerning the complete natural language is deflationary alethic nihilism which you may further search and research... Commented Oct 7, 2023 at 21:57

Regardless of the claims made in some of the answers, introducing a third alternative to true and false does not by itself resolve the issue. Suppose we take the sentence ‘This sentence is paradoxical’ which declares that it belongs in the third category, being neither true nor false- the problem has not gone away, since a literal reading of the sentence tells us the statement is true! Worse still, since what it tells is is that it belongs in the third category, when a literal reading of it says it belongs in the first, it is also patently false!

The problem then is not really to do with the sentence but with our rules for determining whether a statement is true, false or paradoxical. It seems that our rules are capable of reaching all three conclusions about the same sentence, so we have two do something about the rules, rather than just popping another label on the sentence.

In natural language, some sentences are true and some are false. Many legitimate sentences, however, are of indeterminate truth value, e.g. questions or instructions. So, we have a trichotomy: Sentences in natural language can be classified as one of either:

• a true sentence
• a false sentence
• a sentence of indeterminate truth value

We have "This sentence is false" being a true sentence if and only if it is a false sentence. By elimination, using the ordinary bivalent logic, we can infer that it must be a sentence of indeterminate truth value. (Formal proof available on request. 44 lines)

For what it is worth, I presented this idea to ChatGPT. It concluded:

“This approach is a valid way to deal with the Liar Paradox …[Y]our particular articulation and application of this approach to the Liar Paradox are unique and add to the ongoing discourse around paradoxes and truth value."

Full text: http://www.dcproof.com/ChatGPT.htm

• There’s an easy way to avoid your solution: “this sentence is not true.” The paradox essentially remains. Commented Oct 7, 2023 at 20:43
• @Hokon Then there are 2 possibilities: That sentence is either false or of indeterminate truth value. Commented Oct 8, 2023 at 2:45
• If either, then it is not-true, thus it is true. The oscillation of trying to assess the truth-value of the lair paradox in a linear fashion goes without end. Commented Oct 8, 2023 at 2:52
• @Hokon Do you not agree that, in natural language, some sentences are true and some are false, and the rest are of indeterminate truth value? Commented Oct 8, 2023 at 3:04

True and false values have strengths or weights. Statement “It is raining outside.” raises a question what is called raining ? What should be the frequency and density of falling rain drops ,to be called raining? Clearly ,from not raining to irregular droppings of rain drops to drizzling to light rainfall to medium rainfall to heavy rainfall to cloud bursting , there is a linear or non-linear understanding of what constitutes a rain. If we exclude cloud bursting from the definition of rain then there is a fairly linear understanding of the strength of rain. We can assign values to the intensity of rain from 0 to 0.5 to 1 where 0 is no rain or little rain and 0.5 as as medium rain and 1 as heavy rain. Based on strength of raining we can assign the truth value of statement “It is raining outside.” to a range from 0 to 1. This is called fuzzy logic but it is a very real logic as many machines run on it and in my experience the mind uses the strengths of true statements to make decisions.

In the statement “ This statement is false.” , we ask what is the strength of falsehood? The falsehood can range from 0 to 1. But it is not clear from the statement what is the intensity of falsehood. Whether it is 0 or 0.25 or 0.5 or 0.75 or 1? The statement has no subject to measure the falsehood. Therefore the statement is indeterminate even from a fuzzy logic point of view.

Consider another statement “I always lie.”, we ask what is the strength of lie in the statement? Lie strength can range from 0 to 1 where 0 means completely true and 0.5 means partially true and partially false and 1 means completely false or a completely a lie. We have no knowledge of what is the strength of liar’s lie nor can we deduce it by observing his statements which shows how frequently he lies. Therefore statement “ I always lie.” is insufficient to determine how much it is true or not ,even from fuzzy logic point of view. Therefore the statement is indeterminate.

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).

• 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. Commented Jun 11, 2023 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. Commented Jun 11, 2023 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.. Commented Jun 12, 2023 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 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. Commented Jun 12, 2023 at 13:34
• 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. Commented Jun 12, 2023 at 15:20

I do not see any problem in introducing a third category of paradoxical statements. Is Schrodingers Cat alive or dead?

You are only looking for statements that are either true or false.

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

• That's what the OP says, though. It seems to be the main thrust of their reasoning. Commented Jun 12, 2023 at 15:11
• 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? Commented Jun 12, 2023 at 18:10
• This answer suffers the same problem as another I’ve commented on: it proposes an easily bypassed solution: “this sentence is not true” if indeterminate or false, then it is not true. Yet, then it is true, etc. Commented Oct 8, 2023 at 4:36
• As Hokon says, both of the statements in your answer are correct, but they do not amount to a resolution of the issue. Commented Nov 7, 2023 at 8:04