Thakyou. I stand corrected on the technicalities of Cohen's work. But as an example of 'problems' with classical logic I still claim validity.
I have no idea where you are hoping to go with my alleged 'confusion' between syntax and semantics etc. As far as I'm concerned the contexts in which you want to differentiate between those terms is nothing to do with the context of my anti-classical logic position. — fresco

I only read your post and commented on your remarks regarding CH. I didn't take a position on logic. Sorry for any confusion.

The point about syntax and semantics is that in terms of syntax, we can neither prove nor disprove CH within ZFC. But we can exhibit a model, or interpretation of ZFC, in which CH is true (‎Gödel 1940) and another model in which it's false (Cohen 1963). In any given model of ZFC, CH has a definite truth value. It's either true or false. That's semantics. But syntactically, we have no proof.

As I say if you are making a larger point, I didn't address it.

Russell's Paradox was dismissed by Wittgenstein as being 'aberrant language'. — fresco

This is an appeal to authority fallacy; and a poor appeal at that, for no matter what language we decide to use to refer to the ground of existence, that is, the origin, container, and final destination of all words, concepts, objects, and motions, it remains ontological and metaphysical.

Its a bit ironic that the 'authority' to which you object was the one who identified the 'language on holiday' you produce in your objection.. Only you know what you are talking about when you speak of your quasi-religious 'final destination'.. If you know the apocryphal tale of "turtles all the way down" it seems to apply to you.

The linguist John R. Ross also associates James with the phrase:

The following anecdote is told of William James. [...] After a lecture on cosmology and the structure of the solar system, James was accosted by a little old lady.
"Your theory that the sun is the centre of the solar system, and the earth is a ball which rotates around it has a very convincing ring to it, Mr. James, but it's wrong. I've got a better theory," said the little old lady.
"And what is that, madam?" inquired James politely.
"That we live on a crust of earth which is on the back of a giant turtle."
Not wishing to demolish this absurd little theory by bringing to bear the masses of scientific evidence he had at his command, James decided to gently dissuade his opponent by making her see some of the inadequacies of her position.
"If your theory is correct, madam," he asked, "what does this turtle stand on?"
"You're a very clever man, Mr. James, and that's a very good question," replied the little old lady, "but I have an answer to it. And it's this: The first turtle stands on the back of a second, far larger, turtle, who stands directly under him."
"But what does this second turtle stand on?" persisted James patiently.
To this, the little old lady crowed triumphantly,
"It's no use, Mr. James — it's turtles all the way down."

— J. R. Ross, Constraints on Variables in Syntax 1967[10]

"CH has a definite truth value. It's either true or false. That's semantics. But syntactically, we have no proof".
I'm not clear what you mean by 'syntax' here. The 'semantic point' is that the phrase 'definite truth value' automatically invokes the semantic context of classical binary logic.

....on further consideration, I assume you mean 'rules governing what constitutes a valid form of answer'. On that assumption we are touching on 'Zen Koan' territory which forces the pupil to consider the assumptions regarding the structure of 'the question'.. In that case my identification the inapplicability of the rules behind the assumptions of classical logic could be regarded as a 'syntactic' point

"CH has a definite truth value. It's either true or false. — fresco

That is a Platonic claim. It can be strongly argued against. I"m not taking a position one way or another but only pointing out that your claim is arguable.

Consider a variant of the game of chess in which pawns may be promoted to queens or rooks but not knights or bishops. That is not a very radical change in the rules. There are in fact many variants of chess.

Now we come upon two expert chess players arguing over which version is true. But we can see that there is no truth of the matter at all. Chess and variant-chess are formal games. We make up the rules arbitrarily. The only requirement is that the rules are sensible enough so that the game is playable; and that enough people find it fun and enjoyable to play. There is no requirement with truth.

To a formalist, math is the same. It's a meaningless game played with marks on paper according to rules.

https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics)

To a formalist, CH has no definite truth value. We can play the game with CH or with its negation. And when it comes to CH it's a very interesting situation. All of the new axioms which set theorists have studied in order to get a handle on CH are CH-agnostic. You throw in a new large cardinal axiom, for example, and there's a CH and a not-CH version.

Now if someday some physicist determines that ZFC is instantiated in the physical world, then CH would become a research project and would have a definite truth value.

Till that day, if it ever comes, we can only ask if CH is true in the "correct true model of set theory out there somewhere." And the very existence of such a world is a Platonist dream. Gödel himself, as I've mentioned, was a Platonist. His incompleteness theorems to him mean that there is a realm of mathematical truth that's not accessible to the axiomatic method of symbol manipulation.

All that is by way of saying that when you say there is a definite truth value to CH, you might as well ask how pawns may "really" be promoted. The question is a category error. There is no truth in formal games.

That's semantics. But syntactically, we have no proof".
I'm not clear what you mean by 'syntax' here. The 'semantic point' is that the phrase 'definite truth value' automatically invokes the semantic context of classical binary logic. — fresco

Our syntax consists of:

* An alphabet of symbols;
* The usual rules by which we can form well-formed logical and mathematical formulas;
* The inference rules of first-order predicate logic, by which we can start from a set of wffs called the axioms, and derive other wffs called the theorems. Note by the way that an axiom is a theorem, since every axiom A has a one-line proof, namely A.
* The axioms of ZFC.

I should mention that the rules for wffs and the rules of inference are computable. You could write a program (ie Turing machine) to recognize a valid wff and a valid inference.

It is a fact that there is no proof in ZFC of CH nor its negation. That's syntax.

Semantics is an interpretation. Some universe of set theory, called a model, in which CH or not-CH are a matter of observable fact. The question is whether there is a Platonic "correct" model of set theory that settles the issue of CH. A lot of smart people haven't found one yet.

....on further consideration, I assume you mean 'rules governing what constitutes a valid form of answer'. On that assumption we are touching on 'Zen Koan' territory which forces the pupil to consider the assumptions regarding the structure of 'the question'.. In that case my identification the inapplicability of the rules behind the assumptions of classical logic could be regarded as a 'syntactic' point — fresco

No, nothing so woo-woo. A simple matter that syntax, the formal rules of deriving theorems from axioms, does not settle the question of CH when starting from ZFC. One can then find interpretations of the symbols in which CH is objectively true; and interpretations in which it's objectively false. And nobody knows an interpretation of set theory that's so obviously "the right one" that we're willing to call it the official model and thereby determine the truth value of CH.

I hope this wasn't too wordy and addressed some of your concerns. Syntax = derivations, semantics = interpretations.

?
The original quote about 'definite truth value' was yours not mine.

My reconsideration of 'syntax was based on my understanding of 'syntax' as the linguistic one of 'rules governing combination of components', sometimes called 'grammar'.
Thus
Does the dog bite the man?... has the same syntactic structure as... Does the man bite the dog?
The common syntax implies a yes/no answer, but the particular answer is based on semantics.

The original quote about 'definite truth value' was yours not mine. — fresco

You have a cat? Mine sometimes walks on my keyboard and writes half the stuff I post here.

"CH has a definite truth value. It's either true or false. — fresco

So you didn't write that? Ok.

No. I quoted you !
Maybe you quoted someone above.

Maybe you quoted someone above. — fresco

My cat did it.