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Title: Logical semantic and self-referential paradoxes: The Truthteller and the Liar (draft, informal)

(major) assumption: A statement is either true or not true (law of excluded middle, classical logic)

1."This statement is true" can be or at least seems to be either true or not true, and that consistently(Truth teller sentence)

It implicitly states 2."this statement is either true or not true"

which in turn can only be true, and is therefore necessarily true because if it is not true then it is true but not vice verca.

Therefore it implicitly states about itself that it can only be true or (1)

(1) and (2) their only claims are about their own truth values but not only that they make explicitly/implicitly exactly the same claims therefore they are equivalent.

3."This statement is not true" (3) is true if and only if it is not true (Strengthened Liar sentence)

It implicitly states 4."this statement is true if and only if it is not true "

(4) is obviously strictly not true and it also implicitly claims that of itself altough inconsistently because if (4) is true then it is both true and not true, but if (4) is not true then it is only not true. Therefore it makes contradictory claims about itself, namely on the one side of the inconsistency, the contradiction of being both true and not true(explicitly), and on the other side the coherency/truth of being only not true that is implicitly the same claim as (3)

Therefore (3) and (4) make both only claims about their truth values but not only that they make explicitly/implicitly the same claims about their truth values , therefore they are equivalent.

Conclusion: the Liar sentence makes inconsistent claims about it's truth value , and the truth teller makes consistent claims about it's truth value, hence why the former gives rises to the a contradictory situation and the latter doesn't.

This becomes clear when the implicit claims are made explicit, i.e. rendering the implicit claim in the Liar sentence (3) that led to a contradictory situation (true iff only not true) explicit which is done in (4, strictly and necessarily not true) This is also the innovative aspect of this approach that gives rise to the solution to the Liar "paradox" namely it is self-contradictory and therefore meaningless or not true, and a clarification on the Truth teller sentence (1) which seems not to have prima facie an a priori proof for either being strictly true or not true (it seems to be consistent with being either true or not true) , when we render the implicit claim explicit which is done in (2,strictly and necessarily true) this clarifies that the Truth tellers sentence both its explicit and implicit claims are coherent with eachother, that's the reason why it doesn't give rise to the contradictory/"paradoxical" situation.

What yet need to be done is rigorously demonstrating the necessary explicit/implicit claim connection an example of that type of connection in classical logic is i.e given the law of excluded middle , if one claims that it is not the case that it is not true that "p" (explicit claim) then one necessarily also claims that it is true that "p" (implicit claim) and vice versa therefore both claims are equivalent I did not do here to keep the draft short and also because it seems intuitively plausible.

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    I don't see a specific question except asking for feedback. This might be closed for that reason. Regardless, welcome! – Frank Hubeny Jun 21 at 18:05
  • @FrankHubeny I am not quite sure what our standing policy is on this sort of thing. I know they do it on Math SE: somebody writes a proof and asks if there are any flaws, this is the same for an essay. It is a sort of helping with HW, but it's not like they did not show their work. – Conifold Jun 21 at 19:08
  • @Conifold The first custom close reason is explicitly for "questions" like this. – curiousdannii Jun 21 at 22:13
  • @curiousdannii No, not really. We put that there for online preachers, not for students trying to proofread HW essays. The question "did I make any beginner's mistakes?" is fairly answerable. – Conifold Jun 21 at 22:27
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I have a couple of quibbles about the question that might partially answer it if not quite make it go away.

The rule that all statements must be true or false is the Principle of Bivalence, not the LEM. The LEM is a rule for statements that obey the PB and take the form A/not-A. If they do not they cannot be decided in the dialectic but they are still statements. The LEM applies only to pairs of statements that obey Aristotle's rule for contradictory pairs (RCP), which is that one member is true and one false. It does not stipulate that statements must true or false, only that this is pre-condition when we are using dialectical logic to decide their truth of falsity.

The point is simple but widely misunderstood. The 'laws of thought' should be applied only to pairs of statements that obey the RCP. We will run into trouble if we try to decide the question 'Is the Earth square or triangular?'. This mistake is an obvious one. In metaphysics these logical mistakes become very subtle and they are easily and commonly made. Most philosophers make them.

You say - '"This statement is true" can be or at least seems to be either true or not true.'

Not to me. To me it appears to be meaningless. It is crucial to be quite sure it is meaningful, for if it is not it is not a paradox. Nobody has yet managed to convince me that it is not just a muddle of empty words. The phrase 'This statement' is not true or false, so what is the full statement saying? It says nothing, so cannot be true or false.

In order to drag me into the logical confusion that arises from awarding this statement a true or false meaning you'd have to convince me it has one.

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