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Informally, the statement T: T: "this statement is not both true and not true" a) T is not true if and only if it is both true and not true b) T is true if and only if it is not both true and not true

we can agree that: 1. saying of T that it is not true is equivalent to saying that it is both true and not true

  1. Saying of T that it is true is equivalent to saying that it is not both true and not true .

but T says about itself that it is not both true and not true, which is equivalent to T saying that it is either true or not true (in classical logic). which is equivalent to T saying that it is EITHER not both true and not true OR both true and not true. Therefore T is equivalent to "this statement is true and not not true" (truth teller sentence) but then L: "this statement is both true and not true" is equivalent to "this statement is not true" (Liar sentence)

T is necessarily true , therefore truth teller is necessarily true

L is necessarily not true , therefore liar sentence is necessarily not true. Therefore there is no paradox. Whats even more all this works with a formalized Liar sentence too ( e.g. ∃x(Qx & ~Tx), where Q and T are predicates which are satisfied by names of sentences. More specifically, T is the one-place, global truth predicate satisfied by all and only the names [that is, numerals for the Gödel numbers] of the true sentences, and Q is a one-place predicate that is satisfied only by the name of ∃x(Qx & ~Tx).)

Why is this solution not yet popular, it seems so obvious to me?

  • Not clear... You say that T is "statement T is not (TRUE and not TRUE)" that (by De Morgan) amounts to : "statement T is (either not TRUE or TRUE)" which is not paradoxical. – Mauro ALLEGRANZA Aug 26 at 13:06
  • Not every self-refernce leads to paradox : the gist of Liar's Paradox is that self-reference may produce paradoxes. – Mauro ALLEGRANZA Aug 26 at 13:07
  • "Therefore T is equivalent to 'this statement is true and not not true'" It's not clear to me how this follows. – Eliran Aug 26 at 13:20
  • Because saying of T that it is not true is equivalent to saying of T that it is both true and not true , therefore saying of T that it is not both true and not true is equivalent to saying of T that it is not both (true) and (both true and not true) , therefore it is equivalent to saying of T that it is true and not both true and not true ,therefore it is equivalent to saying of T that is true and not not true. well T says about itself that it is not both true and not true , via these equivalences it says "this statement is true and not not true" – Noname Aug 26 at 17:01
  • Your statement L is simply false, as you point out - it is not true that "this statement is both true and not true". But that's not the liar paradox, which tries to assign a truth value to "this statement is not true". Statement L and the liar paradox statement aren't equivalent, so it's not clear how you attempt to resolve the paradox by making inferences about L (in which there is no paradox). – Nuclear Wang Aug 26 at 17:42
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Your identification of the classical liar

L: "L is false"

and the modified liar

M: "M is false and M is true"

is flawed.

It is (I would argue, and I think you would agree) true that for each sentence P we should* have the equivalence "P iff [(P) and (P is true)]." However, we have to be careful about scope here, specifically with regards to what happens inside and outside of the original sentence.

That is, for an arbitrary sentence P let [P] be the sentence "P is true." Then our equivalence does tell us that L is equivalent to the conjunction

C: "(L: L is false) and [L],"

but you want to bring the conjunction inside the name binding to get (after appropriate renaming) an equivalence between our original L and the new sentence

L': "L' is false and [L']."

This may seem benign at first, but it needs to be carefully justified; and exactly the fact that L' is unproblematic while L is quite problematic should indicate that there is something serious going on here.

So at this point you really need to fall back on some precise theory of logic-with-truth-predicates. Indeed, we really should have done that earlier, since right off the bat substitution in the context of naming and self-reference pose interesting syntactic issues which need to be addressed before things can even be appropriately formulated.

(Now that's not to say that there's nothing interesting here. In some sense you're isolating a "deflation operator" here, where we replace a sentence "P: [...]" with its deflated version "P': ... and P' is true)" (where "..." is the result of replacing each "P" in [...] with "P'"). You might now try to claim that only "fully deflated" sentences, that is, sentences which are "easily provably equivalent" to their own deflations, are actually meaningful, and this does indeed forbid the classical liar. But this is a serious semantic commitment, requiring justification, and one which I think many people would actually disagree with (myself included).)


*I say "should" because mixing classical logic and truth predicates is a fundamentally fraught task - indeed, this is sort of the point of the liar paradox in the first place. But certainly this equivalence ought to hold in any reasonable system!

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Idk if my argument below corresponds to yours but let's say

A. B is true. B. A is false.

If you interpret these disquotationwise, they "turn into" (A. A is false) and (B. B is true), which looks like something you wrote? Good question and proposal, at any rate.

  • But in this way it is a simple variant of the usual Liar's Paradox. So what ? – Mauro ALLEGRANZA Aug 26 at 14:15
  • The idea, or at least my take on it, is that this shows the liar and the honest sentences to somehow be equivalent, which supports Tarski's levels-of-truth image, where "true" and "false" inside the L and H sentences are different enough from "true" and "false" outside them, that combining them doesn't yield the required contradiction. – Kristian Berry Aug 26 at 15:59

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