TonesInDeepFreeze

The incompleteness theorem does not rely on the liar sentence.

For a formal language, per a given interpretation, every sentence has a truth value.

That is not the Godel sentence. We do prove it is a sentence in the language of the theory at hand. And we don't the sentence in the theory.

We can formulate the liar paradox without saying "is a statement".

Some statements do mention truth values. The incompleteness theorem is not at all disqualified by the liar sentence. And it's not a conjecture.

Knowing something about logic and the context helps to understand why the liar paradox is of interest. Good then that no one is forcing you to spend t...

Do you disagree with the point that inference rules may themselves be a mathematical object? The symbol '->' may be a primitive or defined from primit...

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It's not used only in logic and mathematics. In everyday discourse, people write "If ___, then" commonly. The source you cited mentioned mentions "If ...

It's not clear to me what you're claiming. Example?

My point is that there may be many views as to what mathematical objects are or are not, including realism, fictionalism, nominalism... But that, in t...

I should add that the above does not opine that those things are platonic things. Moreover, there is not a particular sense in which I am saying they ...

In context of modern logic, 'decidable' means either (1) the sentence or its negation is a theorem, or (2) There is an algorithm to decide whether the...

They are grammatically correct in English. Why would you claim otherwise? "If ____, then ___" is ordinary grammatical English. "I am dying now" said w...

Theologician, redeem thyself.

What do you mean by "apply"? And do you mean there are cases in which no law applies? Or do you mean that, for any law, there are cases in which that ...

"If X then Y" is incorrect because you think "If you go, then I will go" is not grammatical? Why would an ordinary sentence form be incorrect? Every t...

Usually, mathematical logic is studied by means of classical logic. Indeed, mathematical logic is formulated by classical set theory. The theorems of ...

What does that mean?

If relevance is required between the antecedent and consequent for meaningfulness, then we don't know whether a given conditional is meaningful until ...

What do you mean by "cannot be broken"? Do you mean "cannot break without being in error"?

"In English, on the other hand, we only say, "If P then Q," when we believe that the presence of P indicates the presence of Q." I speak English, and ...

Ah, I see the problem, and I carelessly extended it. I'm dumping this: "If A then B" if and only if "Every instance in which A is true is an instance ...

Anytime you want jazz album recommendations, just ring. Jazz is one thing I know a lot about, unlike logic.

Whoever first said, "if ¬(A?B) is true and B is false, A is true", the point is that it is unnecessarily cluttered by "and B is false".

Good, so we've taken care of your problem. Negation is not at issue.

'A -> B' is symbolic. In context of ordinary symbolic logic, it is unambiguous. What is ambiguous is everyday discourse. And, of course, many ordinary...

Works idiomatically. And I edited anyway for even greater sharpness: "There is rain but there is no wetness". is idiomatically the equivalent with: "T...

"Every instance in which A is true is an instance in which B is true" equivalent with: "There is no instance in which A is true and B is false." If A ...

It is not strictly speaking a sentence, but idiomatically it is understood that it means "There is rain but there is no wetness".

It's been pointed out to you at least twice that B doesn't matter: ~(A -> B) -> A Then don't read it that way. It is suggested to read it as: It is no...

First, that is not idiomatic. I've never heard someone say "There is X without there is Y". Second, it could mean at least a few different things. Thi...

I don't think that "It is not the case that" is usually ambiguous. (It is not the case that "it is not the case that" is usually ambiguous.) "If A the...

It's only my guess as to what he might mean. I've never heard of second order recursion or what it might be, though it seems like something that might...

I don't opine as to what that other poster has in mind. But: Rules themselves may be mathematical objects. Languages, axioms, rules, systems, theories...

The albums 'Conversations With Myself' and 'Further Conversations With Myself'.

So did Bill Evans.

He might have meant something parallel to the distinction between first order induction and second order induction that he seemed to be mentioning, so...

What do you mean by "A does not imply B"? Do you mean?: "It is not the case that A implies B" i.e., ~(A -> B) which is true in any interpretation in w...

"A does not imply B". In English that is ambiguous. It could mean: There are instances in which A is true but B is false. It is not the case that A en...

Don't understand that quote. But comments that might be on target: (1) "given complete induction. Unfortunately Peano's axiom of induction is not full...

Better deep in knowledge and shallow in misunderstanding. Better deep in love and shallow in hate. Then you're discussing with the wrong person. You m...

Ah, that is not a notation I would have thought means "the axiom of infinity negated". I would have thought it means "ZF without the axiom of infinity...

StackExchange also has a bad discussion design. And often some confused discussions, But at least as far as math and logic, I have found it to be far ...

Even worse than Wikipedia, which much too often is, at best, slop. Quora is close to the absolute lowest grade of discussion. It is a gutter of misinf...

" as an introduction to a topic Wikipedia is very good." I'll fix that: as an introduction to a topic Wikipedia is very good lousy.

@"Tarskian" And your claim about ZF\I is incorrect. ZF\I is not bi-interpretable with PA. Rather, it is (ZF\I)+~I that is bi-interpretable with PA. (A...

@"Tarskian" Back to this matter: Whether there are uncountably many truths or whether there are unexpressed truths depends on what is meant by 'a trut...