Re: How does the philosophical foundation of analytical truth defeat the Tarski Undefinability Theorem?

On 4/16/24 10:11 AM, olcott wrote:

There is expressions of language that are true on the basis of their semantic meaning (analytic) and there are expressions of language that are true on the basis of direct observation by the sense organs (empirical).

Which you seem to fundamentally not understand. "Semantic Meaning" isn't just "The Meaning of the words", as that definition doesn't handle things like Pythagorean's Theorem.

 Analytic truth is essentially entirely comprised of relations between
expressions of language.

Right, within the bound of the system. This means that a statement can be Analytically True, based on a INFINITE sequence of relationships, but such a statement can not be PROVEN, as a PROOF, by its definition, is a FINITE set of steps from known true statements to the statement.
For instance, a statement about no number existing that meets a given criteria might be True, if no such number exists, and this is established by the infinite testing of EVERY possible number (which are infinite in number), but unless some "shortcut" can be found, like an induction, such an infinite testing is not a proof of the statement.

 Kurt Gödel in his 1944 Russell's mathematical logic gave the following
definition of the "theory of simple types" in a footnote:
 By the theory of simple types I mean the doctrine which says that the
objects of thought (or, in another interpretation, the symbolic
expressions) are divided into types, namely: individuals, properties of
individuals, relations between individuals, properties of such
relations, etc. (with a similar hierarchy for extensions), and that
sentences of the form: " a has the property φ ", " b bears the relation
R to c ", etc. are meaningless, if a, b, c, R, φ are not of types
fitting together.
https://en.wikipedia.org/wiki/History_of_type_theory#G%C3%B6del_1944
 When we try to define a truth predicate on the basis of simple type
theory we have a set of expressions of language that are stipulated
to be true that define the semantic meaning of each type these are
like the axioms of a formal system or the Facts in Prolog.

And, as Tarski showed, if such a predicate exists in a system with sufficient logical power, that system can be shown to be inconsistant.

 Then we have expressions of language that can be derived from
expressions built from these defined types these are derived by
applying truth preserving operations, like Prolog Rules.

You like pointing out "Prolog" logic systems, but seem to fail to note that Prolog doesn't rise to the level of logic capability described in the theorems that you are discussing.
This seems to be becaue you yourself don't understand logic more complicated than the simple logic that Prolog is capable of, so you think it actually is everything.

 https://liarparadox.org/Tarski_247_248.pdf
https://liarparadox.org/Tarski_275_276.pdf
 In the same sort of way that ZFC screened out Russell's
Paradox a correct Boolean Truth(L, x) predicate can screen out the
epistemological antinomy basis of Tarki's Undefinability Theorem.
Truth_Bearer(F, x) ≡ ((F ⊢ x) ∨ (F ⊢ ¬x))

Which means that, as he shows the statement: (Where True(F,S) is the proposed Truth Predicate)
Truth_Bearer(F, True(F, S)) isn't always true, as for some statements, True(F, S) can't generate a consistant truth value, and thus "True(F,S)" fails to meet the definition of a PREDICATE.

 https://en.wikipedia.org/wiki/Truth-bearer
https://plato.stanford.edu/entries/truthmakers/