The posts have come full circle. If any new points arise, I'll consider addressing them. — TonesInDeepFreeze
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
But people use the expression all the time in daily ordinary communications. Why reject?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
It wasn't "Did you lie?" we were talking about. It was "You lied, didn't you?" That was the original sentence. It cannot be chopped into two sentences. It is one sentence, which is both declarative and questioning form. It means, you lied, and it is true.The sentence: "Did you lie?"
is not a truth bearer thus would be rejected by a correct Truth Predicate. — PL Olcott
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
"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
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. — PL Olcott
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. — PL Olcott
Every truth or falsity must be derived from some facts in the world or the known axioms which are self evidently true. — Corvus
The paradox starts with the obscure sentence whose truth falsity value no one knows where or what it was derived from. Therefore there is no point for you progressing into the If then arguments or inferencing. That is my point. — Corvus
Again, Tarski was not trying to figure out how to deal with the liar paradox. Rather, he used the fact that there is no sentence that is true if and only if it is false to prove that there is no formula in the language of arithmetic that defines the set of true sentences of arithmetic. — TonesInDeepFreeze
Again, Tarski did not "include" such a sentence, especially an informal one. — TonesInDeepFreeze
:ok:I agree. — PL Olcott
In "This sentence is false", whether "is false" or "is true" referred to the subject of the sentence "The sentence" or the whole sentence "This sentence is false" was obscure. Would this be part of the undecidability? Or is it for something else? If for something else, then can you give a few example of the undecidability?Not when the entire notion of undecidability depends on these things. In that case we use your first quote as the basis of True(x). From this we derive False(x) ≡ True(¬x) and by this process the whole notion of undecidability utterly ceases to exist. — PL Olcott
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. — PL Olcott
In "This sentence is false", whether "is false" or "is true" referred to the subject of the sentence "The sentence" or the whole sentence "This sentence is false" was obscure. Would this be part of the undecidability? Or is it for something else? If for something else, then can you give a few example of the undecidability? — Corvus
Every truth or falsity must be derived from some facts in the world or the known axioms which are self evidently true. The paradox starts with the obscure sentence whose truth falsity value no one knows where or what it was derived from. Therefore there is no point for you progressing into the If then arguments or inferencing. That is my point. — Corvus
"This sentence is not true." can be true in the form of the sentence X is not true in grammar. Nothing wrong with that. But the content of the sentence is unclear. It doesn't say which sentence it is talking about, and "not true" in what sense. So, it is both true and unclear."This sentence is not true" is called the strengthened Liar Paradox and is its best form. — PL Olcott
This wiki document needs to be verified, the wiki says. But going back to the OP, you need to bring out some arithmetic sentences or expressions, which proves Tarski's undefinability is correct or incorrect. And then we will try them under HOL, and see if it is still valid.In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. https://en.wikipedia.org/wiki/Undecidable_problem — PL Olcott
If it is true that makes it untrue if it it false that makes it true. This proves that it is neither true nor false. — PL Olcott
Often the smartest sounding folks can be the dumbest. You have to learn to think differently from others. If you sound the same as the others on these issues, then it wouldn't prove anything apart from that you have spent all your life browsing the internet. If you think differently and come up with different ideas, you may get told that you have a problem in logic by the crowds, but you know that you are thinking with your own mind, not just parroting or agreeing and yearning to be accepted to the group of the crowds. — Corvus
"This sentence is not true." can be true in the form of the sentence X is not true in grammar. Nothing wrong with that. But the content of the sentence is unclear. It doesn't say which sentence it is talking about, and "not true" in what sense. So, it is both true and unclear. — Corvus
This wiki document needs to be verified, the wiki says. But going back to the OP, you need to bring out some arithmetic sentences or expressions, which proves Tarski's undefinability is correct or incorrect. And then we will try them under HOL, and see if it is still valid. — Corvus
If it can't be a proposition then it must be rejected by any logic system
from propositional logic on up to higher order logic. — PL Olcott
In FOL or PL, "X is not true" depends on the content of X.
In the traditional propositional logic, there is no option for that, hence it is only true in grammatical form of the sentence. Some folks insist it is still true. Likewise "What time is it now?" is true in the form of grammar. So is, "There are the Martians living in Mars." — Corvus
It seems that you are not able to tell the difference between propositional logic, predicate logic and HOL. What you were saying is confined to propositional logic. But once you are in the realm of predicate logic and upwards, the concept of truths becomes multifaceted nature. — Corvus
The sentence, "This sentence is not true." can be true, unknown, false or contradictory depending on the condition of truth. — Corvus
Every increment of HOL above FOL quantifies over expressions of the next lower order. FOL quantifies over propositions, thus propositions are the ground level of all every order of logic. — PL Olcott
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