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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.logic,comp.theory
Subject: Re: True on the basis of meaning --- Good job Richard ! ---Socratic
 method MTT
Date: Tue, 28 May 2024 23:38:41 -0400
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On 5/28/24 10:39 PM, olcott wrote:
> On 5/28/2024 9:04 PM, Richard Damon wrote:
>> On 5/28/24 10:59 AM, olcott wrote:
>>> On 5/28/2024 1:59 AM, Mikko wrote:
>>>> On 2024-05-27 14:34:14 +0000, olcott said:
>>>>
>>>>> ?- LP = not(true(LP)).
>>>>> LP = not(true(LP)).
>>>>>
>>>>> ?- unify_with_occurs_check(LP, not(true(LP))).
>>>>> false.
>>>>>
>>>>> In other words Prolog has detected a cycle in the directed graph of 
>>>>> the
>>>>> evaluation sequence of the structure of the Liar Paradox. Experts seem
>>>>> to think that Prolog is taking "not" and "true" as meaningless and is
>>>>> only evaluating the structure of the expression.
>>>>
>>>> The words "not" and "true" of Prolog are meaningful in some contexts
>>>> but not above. The word "true" is meaningful only when it has no 
>>>> arguments.
>>>>
>>>
>>> That Prolog construes any expression having the same structure as the
>>> Liar Paradox as having a cycle in the directed graph of its evaluation
>>> sequence already completely proves my point. In other words Prolog
>>> is saying that there is something wrong with the expression and it must
>>> be rejected.
>>
>> But Prolog doesn't support powerful enough logic to handle the system 
>> like Tarski and Godel are talking about.
>>
>> The fact that Prolog just rejects it shows that.
>>
> 
> Your ignorance is no excuse.

What ignorance?

The fact that I understand the limitation of Prolog and what forms of 
logic it can do, which seems to be beyond your understanding?

Claiming that Prolog rejects a statement because it doesn't fit its 
grammer is meaningless for more complicated logics that don't have that 
same grammer restricition.

IF you want to limit the logic you use to what Prolog can handle, go 
ahead, the rest of the world likes its mathematics.

> 
>>>
>>>> You could try
>>>> ?- LP = not(true(LP), true(LP).
>>>>
>>>> or
>>>> ?- LP = not(true(LP), not(true(LP)).
>>>>
>>>> The predicate unify_with_occurs_check checks whether the resulting
>>>> sructure is acyclic because that is its purpose. Whether a simple
>>>
>>> Yes exactly. If I knew that Prolog did this then I would not have
>>> created Minimal Type Theory that does this same thing. That I did
>>> create MTT that does do this same thing makes my understanding much
>>> deeper.
>>>
>>>> unification like LP = not(true(LP)) does same is implementation
>>>> dependent as Prolog rules permit but do not require that. In a
>>>> typical implementation a simple unification does not check for
>>>> cycles.
>>>>
>>>
>>> ISO Prolog implementations have the built-in predicate
>>> unify_with_occurs_check/2 for sound unification
>>> https://en.wikipedia.org/wiki/Occurs_check#Sound_unification
>>>
>>> Alternatively such expressions crash or remain stuck in infinite loops.
>>>
>>>
>>>> Anyway, none of this is relevant to the topic of this thread or
>>>> topics of sci.logic.
>>>>
>>>
>>>     ...14 Every epistemological antinomy can likewise be used for
>>>     a similar undecidability proof...(Gödel 1931:40)
>>>
>>> Gödel, Kurt 1931.
>>> On Formally Undecidable Propositions of Principia Mathematica And 
>>> Related Systems
>>>
>>> https://monoskop.org/images/9/93/Kurt_G%C3%B6del_On_Formally_Undecidable_Propositions_of_Principia_Mathematica_and_Related_Systems_1992.pdf
>>>
>>>     It would
>>>     then be possible to reconstruct the *antinomy of the liar* in the
>>>     metalanguage, by forming in the language itself a sentence x
>>>     such that the sentence of the metalanguage which is correlated
>>>     with x asserts that x is not a true sentence.
>>>
>>> CONCEPT OF TRUTH IN FORMALIZED LANGUAGES, Tarski
>>> https://liarparadox.org/Tarski_247_248.pdf
>>>
>>> The Liar Paradox and other such {epistemological antinomies} must be
>>> rejected as type mismatch errors for any system of bivalent logic thus
>>> cannot be correctly used for any undecidability or undefinability proof.
>>>
>>
>> But you just don't don't understand what was done in those proofs.
>>
>> Neither of them assumed the Liar's paradox had a truth value. Only 
>> statements formed from VALID logical sequences in the field.
>>
>> Please try to show what step in Godel's or Tarski's proof where they 
>> made a logical error (not just came up with a statement you think 
>> can't be valid).
>>
> 
> Tarski's Liar Paradox from page 248
>     It would then be possible to reconstruct the antinomy of the liar
>     in the metalanguage, by forming in the language itself a sentence
>     x such that the sentence of the metalanguage which is correlated
>     with x asserts that x is not a true sentence.
>     https://liarparadox.org/Tarski_247_248.pdf

Right, He has SHOWN that the logic system, when given the assumption of 
the existance of the Truth Predicate, can construct the liar as a 
truth-bearing statement.

That means the logic system must be inconsistant.

The ONLY added assumption to the system, that was initially required to 
be conssistant, was the existance of the Truth Predicate, so that 
assumption must be incorrect.

If you try your idea that the statement somehow gets intercepted as a 
type error, then all you have done is shown that you can, by your 
detecting the type error, is prove the new system with the truth 
predicate is inconsistant, so we STILL get to the assumption of a truth 
predicate can't be correct, and your traps must reject THAT statement as 
the source of the problem.

> 
> Formalized as:
> x ∉ True if and only if p
> where the symbol 'p' represents the whole sentence x
> https://liarparadox.org/Tarski_275_276.pdf
> 
> adapted to become the first line of his proof
> x ∉ Pr if and only if p
> 

But that isn't the first line of the full proof, but a consequence from 
his previous logic.

A system that can prove a non-truth-bearer (or a 'syntax error') is an 
inconsistent system.