The ordinary distinction between truth and validity is:

A sentence is true or not per a given model.

A sentence is valid if and only if it is true in every model.

The principle of entailment goes very far back in the history of logic. It is in model theory that the principle is given mathematical exactness. The model theoretic version adheres to the general principle: A set of premises entails a conclusion if and only if there are no circumstances in which the premises are all true but the conclusion is false.

/

If a proposal for a logic does not include models than it is fundamentally different from Montague grammar/semantics.

/

"If C is any contradiction and P is any sentence, then we have C -> P, but that does not allow inferring P. Rather, we would infer P from (C -> P) & C. But since we never have C, don't have (C -> P) & C so we still don't have P."

(1) That is exactly correct. (2) Wikipedia is not a reliable source on the subject of logic. (3) I highly doubt that Wikipedia disagrees with what I wrote anyway.

/

I have not opined on the analytic-synthetic distinction.

Rather than merely bandying Richard Montague, the poster would do well to start at the beginning with symbolic logic as presented in his textbook:

Logic: Techniques Of Formal Reasoning, 2nd ed. - Kalish, Montague and Mar

Start there, with the basics of the subject before pretending to speak meaningfully about more advanced topics.

* 'entailment' and 'consequence' are usually taken as specifying the same relation. That is the relation between a sets of sentences G and a sentences P that holds when there are no models in which all the members of G are true but P is false.

* If C is any contradiction and P is any sentence, then we have C -> P, but that does not allow inferring P. Rather, we would infer P from (C -> P) & C. But since we never have C, don't have (C -> P) & C so we still don't have P.

* Montague semantics is based on compositionality as with the method of models (though with extended aspects such as types, modality, intensionality and possible world models).

As with any subject, before purporting to critique mathematical logic and model theory, one should know something about it. The number of decades one has been floundering in ignorance and confusion on the subject is not a positive index of the cogency of one's critique.

The meanings of sentences are given by the method of models. The truth or falsehood of sentences is determined by rules operating on the truth and falsehood of the components of the sentences, down to the truth or falsehood of the atomic sentences.

The principle of explosion adheres to the principle of truth preservation.

The principle of truth preservation is: All cases in which the premises are true are cases in which the conclusion is true. Put another way: There are no cases in which the premises are true but the conclusion is false.

Since there are no cases in which a contradiction is true, there are no cases in which both a contradiction is true and the conclusion is false.

Whether or not the mathematics mentioned here properly applies to the paradoxes mentioned, at least we should be clear about that mathematics. Adding to certain correct points made by the poster fishfry:

A sequence is a function whose domain is an ordinal. If the domain is finite, then the sequence is finite. If the domain is countable, then the sequence is countable. If the domain is w (read as 'omega' the set of natural numbers), then the sequence is denumerable. If the domain is uncountable then the sequence is uncountable. Sometimes we omit 0 from the domain so that there are sequences whose domain is the set of positive natural numbers.

Let t be the following sequence whose domain is the set of positive natural numbers:

t(n) = 0 if n is odd
t(n) = 1 if n is even

t is a mathematically defined sequence. It is not required to express its values per some other arithmetical formula.

t has no greatest member in its domain, there is no last value for t, and t does not converge.

Let x be any mathematical value. Let s be the following sequence whose domain is the set of positive natural numbers along with w itself (the domain of s is {n | n is a positive natural number or n = w}):

s(n) = t(n) if n is a natural number
s(n) = x if n = w

s is a mathematically defined sequence. It is not required to express its values per some other formula.

s has a greatest number in its domain, and the last value for s is x.

This is a way of saying that for any x, there is a sequence in which x is the last value.

On may reasonably propose an alternative formalized logic, but a formalized logic requires that we have a purely mechanical method by which to determine whether a given finite sequence of sentences is or is not a proof, which requires a mechanical method by which to determine whether a given sequence of symbols is or is not a sentence.

The context I am using is ordinary mathematical logic applied to classical logic:


For a given language, we have different models. A model is an interpretation of the meaning of the symbols of the language. Per a given model, every sentence receives exactly one of the two truth values. That is, per a given model, no sentence is both true and false, and every sentence is either true or it is false. Moreover, no proof has an infinite number of steps, since we cannot mechanically check an infinite number of steps.

Some sentences are true in every model (these are called 'logical truths')

Some sentences are true in some models and false in other models (these are called contingent sentences')

Some sentences are false in all models (these are called ''logical falsehoods').


If P is a sentence and t is a closed term term, then

~(P & ~P) is a theorem in every theory and it is true in every model (non contradiction)

P v ~P is a theorem in every theory and it is true in every model (excluded middle)

t = t is a theorem in every theory and it is true in every model (identity).


Moreover, we have the meta-theorem that a sentence is true in every theory if and only if it is provable in every theory.

/

With formal theories, it is required to have a mechanical method to check whether a given sequence of formulas is a proof, and for that we need a mechanical method to check whether a given sequence of symbols is a formula of a certain kind, and for that we need a mechanical method to check whether a given sequence of symbols is a formula. And we need a mechanical method to check whether a given formula is a sentence.

It would be circular if, to know whether a given formula is a sentence, we needed first to know whether a given formula is such that either it or its negation is provable. To know whether it is a sentence we would need to know whether it is provable, but to know whether it is provable, we would need whether either it or its negation is provable.

Moreover, we have the meta-theorem that there are theories such that there are sentences such that neither the sentence nor its negation are provable in the theory. This does not contradict the law of excluded middle (P v ~P), since the law of excluded middle semantically is that either P is true or ~P is true, and the law of excluded middle syntactically is that P v ~P is provable in all theories, but the law of excluded middle is not that in all theories either P is provable of ~P is provable.

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.

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'.

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.

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.

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.

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.

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.

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.

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'.

It has been fully addressed that the proof of undecidability on pages 275-276 does not use the liar sentence at any step.

/

I am not opining about a notion of a "type mismatch error" so I don't have to rebut it. I do not address the notion of "type mismatch error" without a crisp definition of it, and I wouldn't approach it without the poster at least first understanding his misconceptions about the actual proofs, which I have more than fully addressed:

(1) The liar sentence is not in any step of Godel's incompleteness proof and not in any step in Tarski's undecidability proof. Instead, the Godel sentence "I am not provable" is used, which is crucially different from the liar sentence "I am not true". The poster still does not grasp this, but instead he cites, like a disinformation bot, a Godel footnote out of context of the passage it footnotes, as in that passage Godel explicitly says that reference to the liar paradox is by analogy. The poster needs to stop talking around this.

(2) And the liar sentence itself is not a step, and especially not a premise, in Tarski's undefinability proofs, but rather the proofs start with the assumption, toward a contradiction, that the language can defined its own truth predicate, then shows that that would allow the language to form the liar sentence, which would yield a contradiction, so we conclude the denial of the assumption that the language can define its own truth predicate. The poster still refuses to understand this point that is at the very heart of undefinability. The poster needs to stop talking around this.

(3) The incompleteness proof does not at all entail that the theory itself would have to prove that the Godel sentence is not provable. The exact opposite: We prove that the theory itself does not prove that the Godel sentence is not provable. This has been explained to the poster probably at least 20 times in this forum. The poster needs to stop talking around this.

Also:

(4) In another thread, that the definition of 'antinomy' (especially in philosophy and logic) is not merely 'self-contradictory', which can be verified by looking at a number of dictionaries and articles. The poster still has not recognized this fact.

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.

First, as has been pointed out to the poster at least a dozen times, this is not a proof of undefinability. It is a proof of undecidability.

In that proof, Tarski does refer to the set of true sentences, but he does not use the liar sentence. Before the proof, Tarski specifically explains that he does not use the liar sentence, which pertains to truth, but rather a sentence regarding provability. And in the proof itself, he makes good on that by not using the liar sentence.

Notice also that the set Tr is defined in the meta-theory and not in the object theory.

And the poster repeats for the 10000th time his misconception that the theory would have to prove that the sentence is not provable. It's the opposite. The proof that the sentence is not provable does not occur in the theory but rather in the meta-theory, as the theory would be inconsistent if it proved that the sentence is not provable.

The easily verifiable fact is that in the undecidability proof, pages 275-276 of the paper, Tarski does not use the liar sentence in any step in that proof. Indeed, Tarski specifically mentions that he doesn't use the liar sentence but rather uses 'provable' rather than 'true'. This plain, incontrovertible fact has been brought to the poster's attention at least a dozen times.

And to evade that fact, the poster switches to a different theorem and proof elsewhere in the article, even though the context of that proof does not support the poster's false claim.

The poster just reposts over and over and over his same dogmatic and already rebutted out of context claims. At a certain point, replies are futile. The poster is out of reach of rational discussion.

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.

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.

In sum:

(1) In the Tarski proof of undecidability lately discussed here, Tarski did not use the liar sentence, but rather he used a different formulation involving provability rather than truth. And Tarski himself not only did not claim that the liar sentence has a truth value, but in the undefinability proofs he showed that in the relevant languages, it cannot even be formulated.

(2) In the Godel in the incompleteness paper, he did not claim that the liar sentence has a truth value, and he did not use the liar sentence in the actual incompleteness proof.

It could only be a mistake if the footnote didn't pertain to the passages that begin with "The analogy [...]".

So if one concedes, by actually reading the paper, that the footnote pertains to those passages, then one doesn't have grounds to claim there is a mistake.

This stands:

https://thephilosophyforum.com/discussion/comment/907058

The proof uses the Godel sentence that is analogous to the liar sentence but is very different in crucial ways from the liar sentence. The proof itself does not have any mention of the liar sentence. The proof does not need the liar sentence. It would not even make sense for the proof to use the liar sentence. No one who has studied this subject thinks that the proof uses the liar sentence. The proof of just incompleteness doesn't even have to mention the notion of truth or semantics and can be formulated purely in finitisitc arithmetic and regarding only syntax.

Against all those facts, the only person who would claim that Godel is mistaken in this regard is someone obsessed with insisting that Godel is wrong no matter what, no matter how factually and logically wrong it is required to maintain that insistence.

He said, even as you admit that he swapped, that the proof uses 'provable' instead of 'true'.

That is not the liar paradox.

I guess the poster won't concede that footnote 14 is to the passages that begin by saying that the antinomies are analogous to the Godel argument.

Rational discussion is barely possible with a person who doen't have the honesty to recognize the connection between a footnote and what it footnotes.

In another thread, I showed exactly that Tarski did not use the liar sentence in the proof discussed there.

For the 100th time, Tarski himself said that instead of "true" he used "provable. It's in the exact text of the paper.

(1) The footnote pertains to the sense of using the antinomy analogously to a certain argument. That argument does not itself use the antinomy, but rather Godel's remarks show that the antinomy is analogous to that argument.

(2) That argument is then developed in full detail in a full proof. That proof does not use the liar sentence in any premise, line or conclusion. And the argument doesn't need to use the liar sentence in any premise, line or conclusion. Moreover, the liar sentence could not be used, since the liar sentence cannot even be formulated in the system under consideration.

(3) Textbooks that prove incompleteness do so without using the liar sentence as a premise, line or conclusion.

(4) Tarski even proved that the liar sentence cannot be formulated in such systems.

In all that context, it is seen that Godel did not use the liar sentence in the proof and did not say he did. What he said is that such antinomies can be use for such proofs, which in all this context, can only be understood not as used in the proofs formally but as an analogy that is adapted to the proofs. That adaptation is using 'provable' rather than 'true'. And that is also exactly what Tarski mentions explicitly in one of his proofs of incompleteness. Anyone with a sincere interest in understanding this subject sees this clearly, as opposed to someone who is interested only in taking a footnote out of context (to the extent of ignoring the very passage to which it is a footnote) for the purpose of insisting on a claim to be right no matter how ridiculous.

The context in which they are not mistaken is the context in which he wrote them.

The poster seems to have a problem: Posting the quote from the footnote over and over again, as if the passage that it is a footnote to does not exist, even as the poster included my own quote of that passage.

An antinomy is not just any self-contradiction.

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

Godel is not wrong. What is wrong is brazenly, dishonestly attributing to him out of context, and then doing it yet again even after the context was reiterated yet again.

It's right there in the paper. Footnote 14 pertains to the passages that begin, "The analogy of this argument [...]"

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

I am looking at the only Godel-approved translation right now. Page 598 of 'From Frege To Godel'.

"The ANALOGY of this argument [...]" [emphasis added ] which is the context of the footnote quoted.

Looking at the specific argument that Godel mentions as "this argument" we see that the liar sentence is not in that argument. Rather, Godel mentions that the liar sentence has an ANALOGOUS role.

So, again, I point out that the poster is quoting while intentionally omitting the crucial context.

Again, the wings of birds are analogous to the wings of airplanes, but they are not the same. The liar sentence "I am not true" is analogous to the Godel sentence "I am not provable", but they are not the same.

And again, if one actually reads the proof, then one will see that the liar sentence is not used anywhere in the proof.

Moreover, Godel could not use it as a line in any step of the proof, because the liar sentence cannot even be formulated in such systems that are the subject of the proof, which is what Tarski proved.

We can read the many posts in which the poster claimed that Godel used the liar sentence (i.e. the epistemological antinomy) in the proof.

And the quote of Godel just mentioned drops the context that Godel explicitly wrote this as a matter of "ANALOGY".

The wings of birds are analogous to the wings of airplanes, but they are not the same. The liar sentence "I am not true" is analogous to the Godel sentence "I am not provable", but they are not the same.

And the Tarski quote regards not the theorem that the poster previously cited (pages 275-276). Jumping around, changing contexts like that, is incoherent. But it goes right along with what I wrote. IF[/b] we had a truth predicate then we would have the liar paradox.

Saying again that Godel used the liar sentence in the incompleteness/undecidability proofs is to yet again ignore the plain hard fact that he did not.

Saying that Tarski derived the liar paradox is to yet again ignore the plain hard fact that he did not. Not only did Tarski not claim that the liar sentence is a truth bearer in the relevant formal systems, but even more fundamentally he showed the the liar sentence cannot even be formed in those formal systems. It really helps to actually study the subject of undefinability and incompleteness rather than to brazenly misrepresent it, as a seemingly perpetual strawman, as saying the opposite of what it actually says.

Tarski did not use the liar sentence as a premise, and he did not derive the liar sentence. Rather, he showed that a certain assumption would provide that the liar sentence could be formed in the interpreted language ((1) not even as a theorem, but merely as a sentence having the liar property, especially since there is no system of theorems involved but only an interpreted language, thus this is not a "derivation" of the liar sentence in the sense of a theorem, but only a proof that it could be formed as a sentence; moreover the derivation of the existence of the sentence is not a conclusion in the argument, but only a conditional result based on the reductio ad absurdum assumption) but if the liar sentence could be formed then it would be both true and false in the given interpretation, which contradicts that no sentence can be both true and false in an interpretation, thus we conclude the denial of the original assumption.

It has been pointed out at least half a dozen times in other threads: Godel is referring to using the general idea of such paradoxes to spring the idea for his proof, but the actual proof does not use the liar paradox, as instead Godel constructs a sentence that says "I am unprovable" and not one that says "I am false". This can be verified by reading the actual proof.

And again, for about the tenth time: Tarski does not use the liar sentence as a premise in his proofs. Rather, for undefinability, he makes a reductio ad absurdum assumption that there is a truth predicate, from which he shows that that assumption provides a liar sentence that is a contradiction, thus refuting the reductio ad absurdum assumption. And in another proof, for incompleteness (undecidability actually), he explicitly says that he uses the predicate of provability not the predicate of truth. And one can verify in the actual proofs that the liar sentence is never a premise.

To put this in most stark form:

Assume P.
Derive Q.
Show that Q is contradictory.
Conclude ~P.

P is "a truth predicate can be formed"
Q is "the liar sentence can be formed"

The point is not that we adopt any liar sentence, but the opposite: that the liar sentence cannot be formed therefore there a truth predicate cannot be formed.

Tarski very much stresses that we do NOT deploy the liar sentence, since the liar sentence cannot even be formed in these kinds of languages.

To say that Tarski deploys the liar sentence in his proofs is to brazenly reverse what he wrote.

One more time: One can look at the actual proof steps for incompleteness and undefinability to see that the liar sentence is not used as a premise.

Godel didn't hide steps regarding Godel numbering or diaganolization. Rather they are treated in exact detail.

"x is true if and only p" is not, according to Tarski or anyone who has reasonably studied this subject, the liar paradox nor the liar sentence. Moreover, as has been explained several times to the poster, Tarski does no use the liar sentence as a premise in any proof. Rather, Tarski assumes, toward a contradiction, that in the interpreted language there is a truth predicate for that language, and then shows that that assumption would allow the formation of the liar sentence and its contradiction, therefore that the assumption is contradictory and there is no such truth predicate.

It was explained exactly why your versions are not paradoxes.

Clear versions are available on the Internet. Moreover, I stated clear versions in this thread. Here again, in greatest generality:

Someone shaves all and only those who do not shave themselves.