Re: Tarski Undefinability and the correctly formalized Liar Paradox

On 5/25/24 1:51 PM, olcott wrote:

On 5/25/2024 10:56 AM, Richard Damon wrote:

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
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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

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But, your final sentence no longer DEFINES what p is, it just references an undefined term, whch is an error.
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 You didn't finish reading the rest of my correction
to Tarski's formalization of the Liar Paradox.
 p if and only if p ∉ True
The above sentence says that p is logically equivalent
to itself not being a member of true sentences.
 

No that ststement says that p is true only if p is not an element of the set True.
There is no definiton of what p actually is on the statement.
if and only if doesn't "define" a symbol, but is a claim about the relationship of the truth value of the sentences. Since p hasn't been defined, the statement is meaningless.
You then went of into a reference to another article, so your discussion of Tarski appears to end until later, which I am not handling here.
What I am pointing out is that your "converstion" is just faulty, like most of your conversions/paraphrases, normally because you just don't understand what was being said.
The statement you started from, in the context defined x, and then defined p to be a reference to the statement x (but if you read carefully, they are in different logic systems).
Your paraphrase just uses p without defining it, so isn't a proper paraphrase or reinterpretation.

Note, p and x are not "identical" because x is a statement in the "Science", while p is a symbol in the metatheory.
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You don't seem to understand the difffernce between these.
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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.
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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/
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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.
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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
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Next we turn this into actual self-reference
p := p ∉ True
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Next we limit the scope to one formal system with a predicate
p := ~True(L, p)
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Next we change the name to the more recognizable name
LP := ~True(L, LP)
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<Tarski Undefinability>
    We shall show that the sentence x is actually undecidable
    and at the same time true ...(page 275)
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    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>
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*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))
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So what Tarski says is undecidable in his theory is actually
not a truth-bearer in his theory.
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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.
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