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I noticed something about the liar paradox, when the liar sentences are taken as questions. Let’s start with the liar index, “This sentence is false.” Allow this to be questioned: “Is this sentence false?” Now, if this is a yes-or-no question, then saying, “No,” amounts to reinterpreting the sentence as, “No, this sentence is not false; it’s true.” In other words, the liar index is logically interchangeable with the honest index, as such (more on this in a little).

Next, take the recursive liar, “L: L is false.” As a question this would be, “L: L is false?” However, L in that case would be a question, and questions are not truth-apt as such. The recursive liar lacks an erotetic value.

Another example would be the liar loop, “The next sentence is true; the previous sentence is false.” But if either of the sentences is taken as a question, the loop halts: “The next sentence is true; is the previous sentence false?” doesn’t work since “is the previous sentence false?” is a question, so “The next sentence” does not refer to something that is truth-apt, now.

Let’s also consider the liar imperative. This comes from the epistemic-imperative theory of erotetic logic, where a question is converted into an epistemic imperative. Here, “L: Is L false?” = “L: Let me know whether L is false.” But L in this event is an imperative: again, a kind of sentence that is neither true nor the opposite of true (false), but just “not true” in the abstract. (I think there's a related analysis of, "Don't comply with this imperative," which is complied with if and only if it is not complied with, vs., "Comply with the imperative, 'Do x'" being reducible just to, "Do x" in the same way that, "It is truth that X," reduces to the straight assertion that X. But I can't remember how to "do it" (solve the problem) right now. It shows up in Hofstadter's GEB as, "I wish that this wish would not be granted," for whatever that's worth.)

I wrote an essay on the topic once where I covered a few more examples IIRC and they all go pretty much the same way. For example, trying to work out the liar disquotation ("This sentence is false," is true if and only if this sentence is false) can be used to exactly describe how saying "no" to the liar index converts it into the honest one (i.e. how (false:not true):no "goes to" true:true; it's a kind of subtle double-negation elimination that ends up with: "This sentence is false," is false if and only if, "This sentence is true," is true).

Tarski’s model of levels of truth-predication therefore holds as a two-place relation, between questions and answers in general. That is, since the liar sentences are “unquestionable,” if we accepted them (believed them), we would be using them as unquestionable axioms. The problem, then, is that the liar sentences are not answers to questions, so they have no erotetic form of truth, which would be the alternative truth-predicate of a t1/t2/...-predicate model.

NOTE: I am not arguing that the liar sentences are “meaningless.” The liar index is certainly not meaningless: it has an entirely regular use in natural language, as in saying the sentence while pointing at some other false sentence to which “This” actually refers, e.g. pointing at a written token of, “2 + 2 = 22,” and saying, “This sentence is false.” (Moreover, the existence of this natural usage is what allows us to commit to an equivalent "This" in the interior and exterior of the liar disquotation, instead of saying, "'This sentence is false,' is true if and only if that sentence is false." This haecceitic quasi-copying is allowable on the ground that the liar indexicals have to be adjudged only as sentence-tokens: the sentence-type is truth-inapt, since it does not actually self-refer enough to be evaluated as such.)

My last question, then, is: is this the solution to the liar paradox?

  • The liar paradox already has a solution; the words "this sentence" are nebulous. If you said "He said something false," then that could be a meaningful statement, but telling us that "this sentence" is false is really telling us nothing. – David Blomstrom Aug 17 at 12:48
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    @KristianBerry "is this the solution to the liar paradox" Which part of your question are you referring to by "this"?! – Speakpigeon Aug 17 at 14:34
  • Argument: liar sentences are not answers to well-formed questions. So if we believed/used them, e.g. as evidence against the axiom of noncontradiction, we would use claims that couldn't be doubted, which seems out of the spirit of philosophy. Or: Tarski's multi-level truth-predicate idea IS correct, yet it can be simplified (I think his has possibly infinite levels?) to the difference between truth by itself and as true answers to questions. Liar sentences are disquotationally interchangeable with honest sentences, too, it seems. – Kristian Berry Aug 17 at 22:37
  • The liar paradox was an illustration of a problem in set theory, as such any of the solutions to that problem is also a solution the liar paradox e.g. Russell's "type theory". Also for interest: philosophy.stackexchange.com/q/54384/33787 – christo183 Aug 18 at 4:55
  • Declaring that Liar-like sentences are not well-formed is not a solution, a solution would be to give precise and algorithmic formation rules that allow common uses of self-reference but rule out undesirable consequences. Since this is likely impossible to everyone's satisfaction, there is no solution, see IEP Liar Paradox. – Conifold Aug 18 at 5:01

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