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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic,comp.theory
Subject: Re: Tarski Undefinability and the correctly formalized Liar Paradox
Date: Sat, 25 May 2024 12:51:12 -0500
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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
>>
>> 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.
> 

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.

> 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.
>>
> 

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