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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic,comp.theory
Subject: Re: True on the basis of meaning --- Good job Richard ! ---Socratic
 method MTT
Date: Tue, 28 May 2024 09:59:30 -0500
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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.

> 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.

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