"This square is not a square" is seen as a self-contradiction on its face, and its truth value is falsehood, and there is no contradiction in saying its truth value is falsehood.
"This sentence is false" also implies a self-contradiction, but it is not so easy to say its truth value is falsehood, since if its truth value is falsehood then its truth value is truth and if its truth value is truth then its truth value is falsehood. — TonesInDeepFreeze
given the LNC, a contradiction between X and Y necessitates one of the following three: a) X is valid but Y is invalid, b) Y is valid but X is invalid, or else c) neither X nor Y are valid. But given the LNC, possibility d), that of both X and Y being valid, will be excluded as impossible. — javra
neither true nor false — javra
this amount to the liar's paradox being syntactically coherent gibberish — javra
the sense you mention a 'truth predicate', we actually say a 'truth function'. On the other hand, as to truth predicates, (Tarksi) for an adequately arithmetic theory, there is no truth predicate definable in the theory.
For a language, per a model for that language, in a meta-theory (not in any object theory in the language) a function is induced that maps sentences to truth values. It's a function, so it maps a statement to only one truth value, and the domain of the function is the set of sentences, so any sentence is mapped to a truth value. — TonesInDeepFreeze
In the real world we don't use sentences as truth bearers. — Tate
I don't think we need to break from ordinary language use in assessing Russell's paradox. — Tate
I'm just pointing out that the solution you've been talking about is artificial. — Tate
Sure we do.
"Provo is in Utah" bears truth.
"Provo is not in Utah" bears falsehood. — TonesInDeepFreeze
Russell's paradox was first presented in context of formal theories. And, at least usually, the interest in Russell's paradox centers around mathematics. — TonesInDeepFreeze
don't know how you evaluate for "artificiality". However, of course, since the subject of mathematical logic is conveyed courtesy of human intellect, I guess it's "artificial" in the same sense that just about any other area of study presented by humans is "artificial". — TonesInDeepFreeze
Anyway, it's not clear to me that you understand the solution per mathematical logic. — TonesInDeepFreeze
A sentence has to be contextualized by some form of utterance to qualify as a truthbearer. — Tate
you'll need more weight than this offers to show that we can't evaluate Russell's paradox using ordinary English rules. — Tate
But given some reasonable understanding of given contexts, we do view sufficiently clear sentences as being true or false — TonesInDeepFreeze
It's artifical in the sense that we could change it if we wanted to, at least we can imagine doing so. — Tate
said that we can evaluate it by formal methods. I didn't say that we must evaluate it only by formal methods. — TonesInDeepFreeze
Yes, mathematical logic offers the freedom for anyone to present alternative formulations, definitions, methods, and paradigms. That's a good thing. — TonesInDeepFreeze
In any case, ordinary language and ordinary naive approaches not can be imagined to change but we know that they do change. — TonesInDeepFreeze
Once you add context you have more than just the sentence. You have a statement. The statement can have the property of truth. The string of words can't, not in ordinary language use. — Tate
If we can evaluate it by ordinary standards, the paradox stands. — Tate
But a solution that's subject to revision is not a strong solution. — Tate
Again, you're not seeing the point among your unnecessarily split hairs.
Sometimes informally we use 'sentence' and 'statement' synonymously. Whether or not to do that is a matter of choice in definition. We don't need to get bogged down in disputes about such choices. Meanwhile, the distinction you mention is usually made in logic as the difference between a sentence and a proposition. And there it becomes a matter of the particular development of the subject whether we say that sentences bear truth values or whether only propositions bear truth values. — TonesInDeepFreeze
In the case of "Provo is in Utah" I mean the ordinary interpretation we share of the city we know of and its location in the state we know of. — TonesInDeepFreeze
I don't propose any argument that it is not paradoxical in ordinary language. — TonesInDeepFreeze
I don't propose that they change anything. — Tate
And I don't propose any specific changes to the explication of the paradox per mathematical logic. On the other hand, no matter what you propose or do not propose, natural language changes drastically, so if change is your determinant of 'artificiality' then natural language is quite artificial too. — TonesInDeepFreeze
If you wander through the SEP articles touching on the issue you'll get up to speed pretty quickly. — Tate
You're providing a context for the sentence, so it's more than just the string of words. It's a statement. — Tate
I don't think we can imagine changing the rules of natural language the way we can imagine changing a formal system. — Tate
But then I don't see much persuasiveness in the argument that mathematical (especially mathematical logic) has its explanatory potency diminished by the fact that it always can be augmented in clear, unambiguous, and rigorous ways. — TonesInDeepFreeze
You don't know really anything about the subject of mathematical logic, yet you are persistent to somehow fault it in a quite flimsy way. I wonder why. — TonesInDeepFreeze
I don't propose any argument that it is not paradoxical in ordinary language.
— TonesInDeepFreeze
I think we're broadly in agreement. — Tate
I didn't say anything about its explanatory potency. — Tate
You don't appear to know the basics of the philosophy of truth — Tate
so we're even — Tate
AP — Tate
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