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).

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

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.

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.

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))

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

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

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/
>