Re: Tarski Undefinability and the correctly formalized Liar Paradox

On 5/25/24 11:27 AM, olcott wrote:

x ∉ True if and only if p
where the symbol 'p' represents the whole sentence x
https://liarparadox.org/Tarski_275_276.pdf
 First we convert the clumsy indirect approximation of
self-reference by getting rid of the extraneous x we
also swap the LHS with the RHS.
p if and only if p ∉ True

But, your final sentence no longer DEFINES what p is, it just references an undefined term, whch is an error.
Note, p and x are not "identical" because x is a statement in the "Science", while p is a symbol in the metatheory.
You don't seem to understand the difffernce between these.
This is the first error in your arguement, so I won't comment further, but it demonstrates that you just don't understand what people are saying, mostly because you just don't understand the level of logic being used. You are like a first grader sitting in a Calculus course.

 ψ ↔ ϕ⟨ψ⟩ … The sentence ψ is of course not self-referential
in a strict sense, but mathematically it behaves like one.”
https://plato.stanford.edu/entries/self-reference/
 Thus Stanford acknowledges that it is formalizing self-reference
incorrectly in its article about self-reference. This seems to
be the standard convention for all papers that formalize the Liar
Paradox.
 Here is actual self-reference
x := y means x is defined to be another name for y
https://en.wikipedia.org/wiki/List_of_logic_symbols
 Next we turn this into actual self-reference
p := p ∉ True
 Next we limit the scope to one formal system with a predicate
p := ~True(L, p)
 Next we change the name to the more recognizable name
LP := ~True(L, LP)
 <Tarski Undefinability>
    We shall show that the sentence x is actually undecidable
    and at the same time true ...(page 275)
     the proof of
    the sentence x given in the meta-theory can automatically be
    carried over into the theory itself: the sentence x which is
    undecidable in the original theory becomes a decidable sentence
    in the enriched theory. (page 276)
    https://liarparadox.org/Tarski_275_276.pdf
</Tarski Undefinability>
 *When we stick with theory L we get the same results*
*thus no need for any meta-theory*
True(L, LP) is false
True(L, ~LP) is false
Truthbearer(L,x) ≡ (True(L,x) ∨ True(L,~x))
 So what Tarski says is undecidable in his theory is actually
not a truth-bearer in his theory.
 What Tarski said is provable in his meta-theory making it true
in his theory is ~True(L, LP) is true in his theory because
LP is not a truth-bearer in L.