Self contradictory expressions - assuming that is an expression which entails a contradiction or are otherwise equivalent to A&~A - are truth bearers. They have the capacity to be true or false - as in, it would mean something for the statement to be true, and mean something for the statement to be false. They just happen to be false.

Self contradictory expressions - assuming that is an expression which entails a contradiction or are otherwise equivalent to A&~A - are truth bearers. They have the capacity to be true or false - as in, it would mean something for the statement to be true, and mean something for the statement to be false. They just happen to be false. — fdrake

OK then try to explain how this sentence is a truth bearer: "This sentence is not true."

Go look up Prior's approach to the liar and see what you think.

Go look up Prior's approach to the liar and see what you think. — fdrake

Therefore Since This statement is False is , in other words, This Statement is true and this statement is false, this no more than p=(-p), which is a contradiction, which is from ealier just not the case.
https://thephilosophyforum.com/discussion/8134/liar-paradox-the-three-laws-of-logic-are-intact

ON A FAMILY OF PARADOXES
https://pages.nyu.edu/dorr/hempel/PriorTranslation.pdf

I just read the overview that was posted here. Pretty succinct.

The meaning of 'antinomy' or 'epistemological antinomy' is not just 'self-contradictory'.

The meaning of 'antinomy' or 'epistemological antinomy' is not just 'self-contradictory'. — TonesInDeepFreeze

My only source for the meaning was the compositional meaning of the two separate terms.

relating to the study of the nature, origin, and limits of human knowledge
https://www.merriam-webster.com/dictionary/epistemological

A self-contradictory phrase such as "There is no absolute truth" can be considered an antinomy
https://en.wikipedia.org/wiki/Antinomy

Whatever the case about compositional meaning, 'epistemology antinomy' does not mean merely 'self-contradictory'.

Whatever the case about compositional meaning, 'epistemology antinomy' does not mean merely 'self-contradictory'. — TonesInDeepFreeze

antinomy seems to mean that by itself.

An antinomy is not just any self-contradiction.

An antinomy is not just any self-contradiction. — TonesInDeepFreeze

I have never seen any use of the term {epistemological antinomy} that did not mean self-contradictory.

One can look in a dictionary or in books and articles to see that there is more to being an antinomy than merely being a self-contradiction, especially in the subjects of philosophy or logic.

One can look in a dictionary or in books and articles to see that there is more to being an antinomy than merely being a self-contradiction, especially in the subjects of philosophy or logic. — TonesInDeepFreeze

In any case we can simply cut-to-the-chase and carefully examine every single subtle
detail of Tarski's use of the "antinomy of the liar" in his Undefinability proof.

One thing that we do definitely do know about
Tarski's use of the "antinomy of the liar" in his Undefinability proof
is that he did not recognize it and reject it as a type mismatch error
for every formal system of bivalent logic.

It would then be possible to
reconstruct the antinomy of the liar in the metalanguage
by forming in the language itself a sentence x such that the
sentence of the metalanguage which is correlated with x
asserts that x is not a true sentence.
https://liarparadox.org/Tarski_247_248.pdf — Tarski

Tarski's Undefinability Theorem Proof
https://liarparadox.org/Tarski_275_276.pdf

Regarding the definition of 'antinomy', the point stands that, contrary to the poster's misconception, the definition is not merely 'self-contradiction'.

Regarding Tarski, the poster just quotes again and again and again out of context and ignores the context explained to him dozens of times. He will continue to do that. At a certain point, replies are futile.

Regarding Tarski, the poster just quotes again and again and again out of context and ignores the context explained to him dozens of times. He will continue to do that. At a certain point, replies are futile. — TonesInDeepFreeze

I am not a very good communicator, maybe this is more clear

Tarski's Liar Paradox from page 248
It would then be possible to reconstruct the antinomy of the liar
in the metalanguage, by forming in the language itself a sentence
x such that the sentence of the metalanguage which is correlated
with x asserts that x is not a true sentence.
https://liarparadox.org/Tarski_247_248.pdf

Formalized as:
x ∉ True if and only if p
where the symbol 'p' represents the whole sentence x
https://liarparadox.org/Tarski_275_276.pdf

adapted to become this
x ∉ Pr if and only if p // line 1 of the proof

Here is the (first three steps of the) Tarski Undefinability Theorem proof
(1) x ∉ Provable if and only if p // assumption (see above)
(2) x ∈ True if and only if p // Tarski's convention T
(3) x ∉ Provable if and only if x ∈ True. // (1) and (2) combined

x e T if and only if p

is not the liar sentence.

Rather, it is the general truth scheme.

For formal languages, the particulars of p are filled in per the interpretation of the language.

/

Anyone can look at pages 275-276 to see that no step in that proof uses the liar sentence.

Anyone can look at pages 275-276 to see that no step in that proof uses the liar sentence. — TonesInDeepFreeze

You must carefully study pages 247, 248, 275, and 276.
I have spent hundreds of hours on those four pages over the last several years.

From this basis it is easy to see that every single detail that I said in the prior post
is exactly and precisely a verified fact.

This deserves an award for being one of the most goofball comments ever posted:

"I have spent hundreds of hours on those four pages over the last several years. From this basis it is easy to see that every single detail that I said in the prior post is exactly and precisely a verified fact."

/

The proof on pages 247-249 is of undefinability in a quite technical context. It is more complicated than the proof on pages 275-276 that is of undecidability and is fairly simple.

If the liar sentence appears as a line in the undecidability proof then one could point exactly to the line.

For the undefinability proof, the poster still doesn't grasp that the liar sentence is not a premise but rather Tarski shows that if a truth predicate were definable in the language then the liar sentence could be formed, which would be contradictory.

The poster continues to not recognize that even by common dictionary definition 'antinomy' does not mean merely 'self-contradictory'.

If the liar sentence appears as a line in the undecidability proof then one could point exactly to the line. — TonesInDeepFreeze

Tarski's Liar Paradox from page 248
It would then be possible to reconstruct the antinomy of the liar
in the metalanguage, by forming in the language itself a sentence
x such that the sentence of the metalanguage which is correlated
with x asserts that x is not a true sentence.
https://liarparadox.org/Tarski_247_248.pdf

Formalized as:
x ∉ True if and only if p
where the symbol 'p' represents the whole sentence x
https://liarparadox.org/Tarski_275_276.pdf

adapted to become this
x ∉ Pr if and only if p // line 1 of the proof — PL Olcott

The formalized liar Paradox is adapted to become line 1 of the proof.

Line 1 is:

~x e Pr if and only if p.

That is to say, "x is not provable if and only if p."

As has been pointed out over and over, that is not the liar sentence as 'not provable' is crucially different from "not true".

The poster persists post after post after post to ignore that central point.

A formalization of the liar sentence "This sentence is false" would be along these lines:

"The sentence with Godel-number n is false" while "The sentence with Godel-number n is false" has Godel-number n.

It is exactly a point of Tarski that there is no such sentence in the relevant interpreted languages.

However, "This sentence is unprovable" is analogous with the liar sentence "This sentence is false", but, again, it is only analogous and they are crucially different.

A formalization of "This sentence is not provable" would be along these lines:

"The sentence with Godel-number n is not provable" while "The sentence with Godel-number n" has Godel-number n.

Godel proved that that sentence is formalizable in the relevant languages. And Tarski proved that the analogous sentence with 'false' instead of 'unprovable' is not formalizable in the relevant languages.

It is not needed to spend hundreds of hours on just four pages (!) lifted out of context from a dauntingly, densely complex paper written in outdated notation to understand these points, as millions of people who ever got a good grade in undergraduate mathematical logic would understand them.

'x e T if and only if p' is not the liar sentence. — TonesInDeepFreeze

You are not paying close enough attention

x ∉ True if and only if p
x {is not a member of} True

x ∈ True if and only if p
x {is a member of} True

I did err in my previous post by overlooking the negation sign. And I should have emphasized again, for the 100th time, that 'unprovable' and 'untrue' are profoundly different.

I edited that post completely now.

The poster stresses that line 1 is an adaptation of the liar sentence. But he still cannot grasp that they are crucially different even though analogous.

An intelligent person should understand that X may be an adaptation of Y but be profoundly different from Y.

The proof on pages 275-276 is a proof of the undecidability of certain sound systems (a system is sound if and only if all its theorems are true).

It is a plain hard fact that the liar sentence does not occur in any line of the proof.

Let '~e' stand for 'is not an element of'.

(1) x ~e Pr iff p
"x is unprovable if and only if p"
[this is not the liar sentence]

(2) x e Tr iff p
"x is true if and only if p"
[this is not the liar sentence]

(3) x ~e Pr iff x e Tr
"x is unprovable if and only if x is true"
[this is not the liar sentence]

(4) x ~e Tr or not-x ~e Tr
"x is untrue or not-x is untrue"
[this is not the liar sentence]

(5) if x e Pr then x e Tr
"if x is provable then x is true" (from soundness)
[this is not the liar sentence]

(6) if not-x e Pr, then not-x e Tr
"if not-x is provable then not-x is true" (from soundness)
[this is not the liar sentence]

(7) x e Tr
"x is true"
[this is not the liar sentence]

(8) x ~e Pr
"x is unprovable"
[this is not the liar sentence]

(9) not-x ~e Pr
"not-x is unprovable"
[this is not the liar sentence]

I did err in my previous post by overlooking the negation sign. And I should have emphasized again, for the 100th time, that 'unprovable' and 'untrue' are profoundly different. — TonesInDeepFreeze

Only when you fail to understand that True(L,x) requires a sequence of truth preserving operations from basic facts that are other expressions of language stipulated to be true.

That you have false assumptions does not make me incorrect.

I have no false assumptions in this context. But one of the many ridiculously dogmatic and ignorant false assumptions of the poster is that "This sentence is not provable" is the liar sentence.

To evade that, the poster resorts to insisting that we accept his own framework of the notions of languages and truth. But we are not obligated to do that.

Better yet, the poster's argument is circular:

The poster insists that we adopt his personal framework and not the mathematical logic of Godel and Tarski because Godel and Tarski are wrong because they use the liar sentence in their proofs. But they do not use the liar sentence in their proofs. So the poster insists that they actually do use the liar sentence in their proofs if we view the situation in poster's personal framework rather than the given context of the mathematical logic of Godel and Tarski.

Put this way:

The poster insists that Godel and Tarski are mistaken.

But how so?

The poster insists that it's because they use the liar sentence.

But they don't.

So the poster says they really do if we take them in the poster's personal framework.

But why should we take them in the poster's personal framework? (As well as, changing the framework doesn't change that they don't use the liar sentence.)

The poster replies because that framework avoids the mistakes of Godel and Tarski.

And so full circle.

I have no false assumptions in this context. But one of the many false assumptions of the poster is that "This sentence is not provable" is the liar sentence. — TonesInDeepFreeze

"This sentence is not provable"
is not provable because that would require a sequence
of inference steps that prove that they themselves do not exist.

Now that the poster cannot support his dogmatically ignorant and confused claim (that the liar sentence appears in Tarski's proof of undecidabilty) that he has been insisting on for 1000 posts, he evades by moving to a different claim that he's been insisting on for 1000 posts. And that claim is also wildly incorrect as has been explained to the poster over and over and over.

Again, for the 1000th time: Showing that the Godel sentence is unprovable does not require proving the Godel sentence in the system itself. In other words, showing that the Godel sentence is unprovable does not require showing in the system itself that the Godel sentence is unprovable. Indeed, we prove in the meta-theory that if the system is consistent then the Godel sentence is not provable in the system itself.

Now that the poster cannot support his dogmatically ignorant and confused claim — TonesInDeepFreeze

The sentence <IS> derived from his Liar Paradox
and has the same sort of paradoxical result.

I am taking your last post as an indication that you are really not interested
in an honest dialogue.

Meanwhile, just for sake of keeping the eye on the ball:

Godel proves incompleteness/undecidability, and the liar sentence is not in that proof.

Tarski also proves incompleteness/undecidability and the liar sentence is not in the proof.

And Tarski proves that for languages of a certain kind, within the languages themselves there is no definable truth predicate for sentences of those languages. To do that, he shows that if there were such a definition, then the liar sentence could be formed, but the formation of a liar sentence would yield a contradiction.

An example of intellectually dishonest posting is a poster evading 1000 times that being inspired by an analogy between ideas does not entail that the one idea is the same as the other.

The liar sentence was known as an informal paradox. That gave Godel the idea of an analogous but crucially different idea that could be formalized and is not paradoxical. And Tarski proved that the liar sentence cannot be formalized in certain relevant languages.

It it is hard to fathom that a poster cannot or will not understand that things that are analogous can still be very different from one another.

/

By the way, still the poster won't admit that even by dictionary definition 'antinomy' does not mean merely 'self-contradictory'.

To the point about paradox:

'paradox' may be defined in different ways. I'll use this definiens: a contradiction derived from seemingly acceptable premises.

The liar sentence yields a paradox per this contradiction:

The liar sentence is true and the liar sentence is not true.

Incompleteness yields:

The Godel sentence is true and the Godel sentence is not provable.

That is not a contradiction even though it may be surprising that it is the case.