"This sentence is false."

If it is false, then it is true.
If it is true, then it is false.

The If parts need reference (under what ground it is false or true) to claim it is either false or true.
There is no indication of what the reference for presuming it is false or true.
Hence the arguments are invalid. — Corvus

...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
(Gödel 1931:43-44)

Basically I am saying that self-contradictory expressions such as the epistemological antinomies that Gödel refers to are not truth bearers (neither true nor false) thus must be excluded from formal systems and never any part of any formal proof.

An expression of language that is both a question and a statement would also have
to be rejected until it is translated into one or the other.
— PL Olcott
But people use the expression all the time in daily ordinary communications. Why reject? — Corvus

Natural language cannot be accurately evaluated until it is translated into some totally precise form. An expression that is both a statement and a question cannot be properly evaluated by any Boolean True(L, x) predicate.

It must be broken down into its constituent parts. The question aspect must rejected by any Boolean True(L, x) predicate as not a truth bearer.

If we ask people is this sentence true: "What time is it?" the smartest ones will say type mismatch error. Those that have less insight will simply be confused.

Self-contradictions are false in all models.
For a given model M, every sentence in the formal language is either true in M or it is false in M. — TonesInDeepFreeze

OK now we are getting somewhere. "This sentence is not true" cannot be true because that would make it untrue and cannot be false because that would make it true. Thus it is not a bearer of truth anywhere.

Every closed WFF of the formal language of any formal system must be true or false thus the Liar Paradox is excluded from every formal system and Tarski was wrong for including it.

The posts have come full circle. If any new points arise, I'll consider addressing them. — TonesInDeepFreeze

No one anywhere on any forum ever addressed the issue that:
(a) Undecidability is fully met by self-contradictory expressions.
(b) Self-contradictory expressions cannot possibly be truth bearers.
(c) Formal systems requires that all of its expressions must truth bearers.

Everyone everywhere used the change the subject form of rebuttal or denied the verified facts stated above.

Mentioning that you believe that (a) (b) or (c) is incorrect once and then dropping it is not enough to arrive at closure. I think that our sticking point may be (a). You disbelieve (a) yet will not allow me sufficient dialogue to prove (a).

I cannot provide for progress in a conversation by repeating that I cannot provide for progress in a conversation by repeating refutations and explanations that are ignored while what has been refuted is simply reasserted. — TonesInDeepFreeze

Try and show how it makes sense to base undecidability on self-contradictory expressions or acknowledge that you do not understand that Tarski Undefinability is anchored in the Liar Paradox and we can move forward. Simply changing the subject to something else blocks all actual progress.

No important point has been ignored. It's the other way around. — TonesInDeepFreeze

The important point is that it self-contradictory expressions are still considered valid proof of undecidability and this proves that the entire notion of undecidability that ALL mathematical incompleteness depends is totally bogus. When we understand and accept that then your repeatedly going back to utterly extraneous details to this point is simply deflection.

I cannot provide for progress in a conversation by repeating refutations and explanations that are ignored while what has been refuted is simply reasserted. — TonesInDeepFreeze

You seem to simply ignore the main points that prove my case.

...We are therefore confronted with a proposition which asserts its own unprovability. 15 ...
(Gödel 1931:43-44)

When I talk about that quote you simply change the subject to something else totally ignoring my analysis of it.

Mathematical logic does not assign "fault". Fault though would be vital to assign if one were a judge in a traffic accident case. — TonesInDeepFreeze

By saying that F is incomplete when the real issue is that G is incorrect the blame
for the unprovability of G is F it misallocated.

The Godel sentence is not a contradiction and it is not nonsense. — TonesInDeepFreeze

The Gödel sentence itself cannot possibly be directly understood because all of its actual semantics are completely hidden from view. Because of this we must use these quotes to have a glimpse into his underlying reasoning:

...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
...We are therefore confronted with a proposition which asserts its own unprovability. 15 ...
(Gödel 1931:43-44)

When G asserts its own unprovability in F the proof of G in F does require a sequence of inference steps in F that prove that they themselves do not exist. That unhides the whole essence of Gödel's proof where we can see WHY G is unprovable in F not merely THAT G is unprovable in F.

This G is unprovable in F because this G is nonsense and Gödel expressly states that this kind of nonsense "can likewise be used for a similar undecidability proof"

Again, however one characterizes the Godel sentence, it is not a contradiction. Indeed it is a true sentence of arithmetic. — TonesInDeepFreeze

This G is unprovable in F because this G is nonsense in F
That G is nonsense in F does not show that there is anything wrong with F
the issue is ALL G's fault.

When G asserts its own unprovability in F the proof of G in F does require a sequence of inference steps in F that prove that they themselves do not exist. We at the meta-math level can see that there cannot possibly be such a proof of G in F thus we know that the assertion that G is unprovable in F is true.

That unhides the whole essence of Gödel's proof where we can see WHY G is unprovable in F not merely THAT G is unprovable in F. — PL Olcott

Godel never said any such nonsense that if a system proves a contradiction then the system is incomplete. Indeed, if a system proves a contradiction then the system is complete. — TonesInDeepFreeze

You misquoted me. An epistemological antinomy <is> a self-contradictory expression

...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
(Gödel 1931:43-44)

Does say that the inability to prove a SELF-contradictory expression
"can likewise be used for a similar undecidability proof..."

No self-contradiction is provable in a consistent theory, irrespective of incompleteness. — TonesInDeepFreeze

Gödel specifically states that the inability to prove a self-contradictory expression
DOES MAKE THE FORMAL SYSTEM INCOMPLETE.

...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
(Gödel 1931:43-44)

Using stipulative definition Incomplete(F) is simply a euphemism for Incorrect(G).
Incomplete does not retain any of its conventional meaning
incomplete: not having all the necessary or appropriate parts.

Using stipulative definition this same way we could say that the inability of a formal system to prove a self-contradictory expression makes this formal system "A big fat cow".

A theory T is incomplete if and only if there is a sentence S in the language for T such that neither S nor its negation are a theorem of T. — TonesInDeepFreeze

The problem with this definition is that it proves that mathematical systems are "incomplete" when they cannot prove or refute nonsense. Self-contradictory expressions are nonsense and cannot be proven or refuted only because they are nonsense.

There is no proof of G in F.

That's the point.

Too miss that point is to utterly not know what the theorem is about. — TonesInDeepFreeze

That there is no proof of nonsense does not make any formal system incomplete unless
Incomplete(F) is a euphemism for Incorrect(G).

Regarding Tarski's undefinablity theorem, Tarski proved that in certain systems, there does not even exist such a sentence. Not only did Tarski not use such sentences as a basis, he actually proved that such sentences don't even exist in the relevant systems. To not understand that is to not understand what the theorem is even about. — TonesInDeepFreeze

When (as in Prolog) True(L, x) means Provable(L, x) and
(as in Prolog) False(L, x) means Provable(L, ~x) then Tarski Undefinability theorem utterly fails.
Self-contradictory expressions are simply rejected as not bearers of truth.

Proofs don't "hide" things. From fully declared axioms and rules of inference, we may prove Godel-Rosser. We may prove versions that do not mention semantics. And we may prove versions that mention both syntax and semantics. This is all famous and understood by reading an introductory textbook in mathematical logic. — TonesInDeepFreeze

When G asserts its own unprovability in F the proof of G in F does require a sequence of inference steps in F that prove that they themselves do not exist. We at the meta-math level can see that there cannot possibly be such a proof of G in F thus we know that the assertion that G is unprovable in F is true.

That unhides the whole essence of Gödel's proof where we can see WHY G is unprovable in F not merely THAT G is unprovable in F.

"Did you lie?" doesn't have a truth value, because it is not a declarative sentence. Indeed, interrogatory sentences do not appear as lines in proofs. — TonesInDeepFreeze

Your statement here sounds nonsense. Some questions can be for true or false. For example,"You lied, didn't you?" This means you lied, and it is true. It is also to mean you should be aware of the fact that you lied. — Corvus

Corvus was testing the boundaries of what is included and what is not included by using a rhetorical question as a pseudo statement.

More generally, Godel's and Tarski's proofs do not have the defects claimed in this thread (and claimed by the same poster several other times in this forum). That can be verified by reading an introductory textbook on mathematical logic in which the groundwork and proofs of Godel-Rosser incompleteness and Tarski undefinability are provided. — TonesInDeepFreeze

Tarski's proof is directly anchored in the actual Liar Paradox itself.
Liar Paradox basis of proof: https://liarparadox.org/Tarski_247_248.pdf
The actual proof itself: https://liarparadox.org/Tarski_247_248.pdf

Most people can understand that: "This sentence is not true" cannot possibly
be true or false thus is not a truth bearer. Tarski did not seem to understand that
or he would not have used it as the basis of his proof.

Contrary to a claim made in this thread (and made by the same poster several other times in this forum), it is not the case the Godel sentence requires that there is a sequence of inference steps that prove that they don't exist (as has been explained several other times in this forum). — TonesInDeepFreeze

...We are therefore confronted with a proposition which asserts its own unprovability. 15 ...
(Gödel 1931:43-44)

The above is a direct quote from the proof and the system referred to by the above quote certainly does require a sequence of inference steps that prove that they themselves do not exist. The actual proof itself hides all of its underlying semantics behind the purely symbolic manipulation of mathematical operations. Diagonalization shows THAT G is unprovable in F and hides WHY G is unprovable in F.

I am not sure if you are allowed to modify the given example sentence under the process of breakdown. If it is allowed, then virtually all questions can be truth bearers. — Corvus

Sentences with ambiguity must be disambiguated before that can be properly analyzed.
For example asking a man that has never been married: Have you stopped beating you wife?
Would be rejected as semantically incorrect on the basis of the false presupposition.

An expression of language that is both a question and a statement would also have
to be rejected until it is translated into one or the other. The sentence: "Did you lie?"
is not a truth bearer thus would be rejected by a correct Truth Predicate.

Your statement here sounds nonsense. Some questions can be for true or false. For example, "You lied, didn't you?" This means you lied, and it is true. It is also to mean you should be aware of the fact that you lied. — Corvus

That is a great counter-example. It seems to me that is actually a declarative sentence that is phrased as a rhetorical question. It is comprised of two distinctive parts: The statement: (a) "You lied." and the question: (b) Did you lie? When we break it down to its constituent parts (b) is still not a truth bearer.

The sentences in question say one way or another - and the article makes clear that exactly how they speak can be important - that they are not true, or not provable. And the analysis shows that whatever else might be true, it is self-evident and provable that they are true. Which is to say that they are, according to your exact definition, truth-bearers, which in turn makes all of your claims absurd. — tim wood

Gödel's Incompleteness use diagonalization that show G is not provable in F yet totally hides why it is unprovable. When reasoning is entirely hidden behind mathematical operations there is no basis for rebuttal.

...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
...We are therefore confronted with a proposition which asserts its own unprovability. 15 ...
(Gödel 1931:43-44)

Exposes the actual reasoning that is hidden behind the mathematical operations.
I just recently figured out that the proof of any "proposition which asserts its own unprovability"
requires a sequence of inference steps that prove that they themselves do not exist.

That would be the same as converting René Descartes famous: "I think therefore I am"
into "I think therefore thoughts do not exist", ridiculous nonsense.

Russell's Paradox was solved by simply correcting the incoherent definition of a set.
We can solve Tarski Undefinability this same way: Truth_Bearer(F, x) ≡ ((F ⊢ x) ∨ (F ⊢ ¬x))

↪PL Olcott You get right to the point so will I. There are a whole lot of so-called "contradictory" sentences that are true. — tim wood

Please provide one example. Contradiction is a very well-known and certain measure of falsity in classical logic, symbolic logic, predicate logic.

↪PL Olcott You get right to the point so will I. There are a whole lot of so-called "contradictory" sentences that are true. — tim wood

That is the same as saying there are a whole lot of integers that are greater than five and less than three, utterly ridiculous nonsense. That logicians make sure to never pay any attention at all to philosophy of logic makes them complete ignoramuses. There there are no mistakes within their false assumptions DOES NOT MEAN THERE ARE NO MISTAKES.

That actual self-contradictory sentences (not the deliberate double-talk of so-called) cannot possibly be true or false should have been universally accepted thousands of years ago shortly after the Liar Paradox was created.

I think you have said that a truth-bearer is a proposition that is true, or if false then its negation is a truth-bearer. Yes? Or if no, then what, exactly, do you say a truth-bearer is? — tim wood

Very close. A proposition / logic sentence is defined to always be a {truth-bearer}. This means that it is either true or its negation is true.

A truth-bearer is any expression of language that can have the semantic value of true or false. That after 2000 years people do not understand that the Liar Paradox (and other self-contradictory expressions) are not {truth-bearers} seems as ridiculous as a PhD math professor that simply "does not believe in" numbers.

Tarski's Undefinability theorem still stands even though its basis is that a Truth predicate cannot correctly determine that the actual Liar Paradox is true or false. To me this is like all of the bakers in the world trying again and again to bake an angel food cake using only house bricks for ingredients and never having any idea that this can't possibly work.

So there are two distinct mechanisms to determine the truth value of a language expressions, yes? — EricH

Yes. There is expressions of language that are true on the basis of their semantic meaning and there are expressions of language that are true on the basis of direct observation by the sense organs. The first kind boils down to relations between expressions of language.

I think everyone gets it as something that is defined in a particular way. But having defined it, you then misapply it where it doesn't apply, leading you to make foolish claims. — tim wood

That almost everyone including the greatest experts in the field do not fully understand that self-contradictory expressions are not {truth-bearers} does seem ridiculously stupid to me. To me it seems like the same thing as a PhD mathematics professor that disagrees with first grade arithmetic because they simply "do not believe in" numbers.

The article is long, comprehensive, and in parts very interesting - I won't pretend to have read it all. But to you, PL, I commend it as necessary for your understanding. As for your quote above, it simply establishes - as a truth-maker - that someone is totally incompetent. — tim wood

A {truth-maker} is a more difficult subject than a {truth-bearer} It took me 20 years of primary and secondary research to fully understand exactly what a {truth-maker} is for expressions of language that are true on the basis of their semantic meaning. It turns out that the final analysis of this is very simple.

On the other hand a {truth-bearer} is much simpler than this:

A truth-bearer is an entity that is said to be either true or false and nothing else.
https://en.wikipedia.org/wiki/Truth-bearer

In other words a {truth-bearer} is any expression of language that can possibly have a semantic value of true or false. This generally includes declarative sentences and generally excludes questions.

I have focused 20 years of primary and secondary research on the relationship between epistemological antinomies and expressions of language that are true on the basis of their semantic meaning.

At this point it does seem very very stupid that people cannot understand that self-contradictory expressions are not {truth-bearers}.

As I said before, will say again. The whole confusion with the paradox and undefinability have been originated from the single narrow perspective seeing the problems in propositional logic, which only allows a proposition must be either True or False.

If you think about the real world situations and objects, there are cases where things are neutral i.e. neither true nor false such as Number 0. And there are the real world cases where things are both True and False, read on QM or some Metaphysical topics. — Corvus

The term {true bearer} is very widely know throughout philosophy. That declarative sentences can be {truth bearers} and questions cannot be {truth bearers} is known by everyone that knows what {truth bearers} are. It takes very little additional understanding to know that {self-contradictory expressions} are not {truth bearers}.

That Tarski and Gödel did not understand something as simple as this makes them totally incompetent.

Really? In which book or article did he do that? I have his Mathematical Logic, Method of Logic, Elementary Logic and The Significance of New Logic, total 4 books. But cannot recall seeing it. — Corvus

Two Dogmas of Empiricism Willard Van Orman Quine
https://www.theologie.uzh.ch/dam/jcr:ffffffff-fbd6-1538-0000-000070cf64bc/Quine51.pdf

Bachelor is a rather simple term. There are many other words in English which are more abstract to define. — Corvus

Yet he wrote a whole paper on his failure to understand that the term {bachelor} is simply assigned the semantic meaning of {unmarried + male + adult}.

is there another mechanism/method to determine the truth value of expressions of language? — EricH

Yes there is a TV in my living room right now is true on the basis of eyesight.

I think Quine did understand what bachelor meant. But his point was that a word can mean different things, the meanings of words can change through time and culture, and for a word to convey clear meanings, it needs the context in the expressions in grammatically correct sentence reflecting the reality situations. — Corvus

No that it not it. He used the term {synonymous} 98 times. He did not understand that the term {bachelor} is simply assigned the semantic meaning of {unmarried + male + adult}.

Hence a woman can be a bachelor, so could a man married many times. I am sure there are surnames called "Bachelor", hence some married old folk could be a Bachelor, Mr Bachelor, or if for a woman, Ms Bachelor. They are all B(b)achelors. — Corvus

A person with a 50 million IQ that cannot understand that (one the the sense meanings of) the term {bachelor} is assigned the semantic meaning of {unmarried + male + adult} is ridiculously stupid about this one point.

It can be implemented in C or Java in modified form with abstraction and generalisation. It cannot be implemented because you are seeing it in the propositional logic rather than predicate or first-order logic. — Corvus

Gödel Incompleteness can only be implemented in systems that implement Boolean True(L, x) incorrectly. It cannot exist in systems where True(L, x) means that x is provable from L and False(L, x) means ~x is provable from L and for everything else x is simply untrue in L.

This same reasoning also conquers Tarski Undefinability.
When Tarski anchors his undefinability in the Liar Paradox:
x = "This sentence is not true" we get
Truth-Bearer(L, x) ≡ ∃x ∈ L ((L ⊢ x) ∨ (L ⊢ ¬x)) // x is not a Truth-Bearer

But your example "cows don't eat house bricks" is neither a fact nor common sense. It is just an irrelevant daft statement, which is based on senseless reasoning. — Corvus

It is a concrete example of an expression of language that is true on the basis of it meaning. Quine objected to true on the basis of meaning trying to get away with saying there is no such thing as meaning. The stupid nitwit could not even begin to understand that bachelors are not married.

The good people telling the truth don't have the slightest clue of how to effectively deal with this.
— PL Olcott
The Germans do, apparently. — tim wood

The Germans currently know how to counter-propaganda?
One of the things that they did that has worked is limiting free speech.

It does seem to me that counter-factual lies should come with civil liability.
To keep free-speech open you are allowed to assert any opinion as long
as you qualify it as only an opinion.

When you assert a counter-factual lie as a verified fact you can lose up to
all of your assets no matter how much you have. This would be especially
helpful for the fossil fuel industry's hiring of liars.

You're not correcting anything. But you are making a mistake plain and simple. Let's try a test. I'm a Holocaust denier (not, actually). How does your program handle that? — tim wood

It will not focus its attention on every lie, only the ones that can have near term catastrophic consequences. I can't understand how 45% of the electorate can believe that election fraud changed the outcome of the 2020 presidential election when there has been literally no evidence of this.

The system that I propose would know 10,000-fold more about counter-propaganda than any human mind can possibly hold. The bad people that are pushing Nazi propaganda have had many decades perfecting their craft. The good people telling the truth don't have the slightest clue of how to effectively deal with this.

Facts are true
— PL Olcott
*sigh* Anyone else want to take this on? — tim wood

The reason for my persistence with correcting the notion of truth is that incorrect notions are resulting in the extinction of humanity through climate change and the end of Democracy through Nazi propaganda.

If we had a properly formalized notion of truth then it would be possible for a computer program to argue against the hired liar climate change deniers and Nazi propagandists so effectively that this people would look like complete fools even to themselves.

Here you seem to be saying that we can determine the set of facts from a well constructed dictionary. — EricH

We can combine (1) and (2) and say that every expression of language that is true on the basis of its meaning is either a fact or derived from a fact. Facts are true even if everyone disagrees or no one knows.

Writing down every single detail of all of the general knowledge of the actual world would probably need a book at least a million miles tall. Since this is only 59 petabytes this is probably not nearly enough.

My point being that there is a whole barge-load of assumptions you're making, apparently without being aware you're making them. The Boston Red Sox won the 2003 World Series: if three out of three agree with that, does that make it a fact? — tim wood

I am making zero assumptions what-so-ever. Facts are expressions of language that are true
EVEN IF NO ONE KNOWS THIS OR EVERYONE IN THE WHOLE UNIVERSE DISAGREES.

All facts are propositions that are historical in nature. A kind of hearsay, if you will. And this includes just about everything that can be said about the world — tim wood

Actual facts are expressions of language that correctly model the actual world even if everyone in the universe disagrees or no one in the universe knows them.