Re: Ross Finlayson what about the Prolog Liar Paradox ?

On 07/07/2026 11:07 AM, olcott wrote:
> On 7/7/2026 12:53 PM, Ross Finlayson wrote:
>> On 07/07/2026 09:10 AM, olcott wrote:
>>> % This sentence is not true.
>>> ?- LP = not(true(LP)).
>>> LP = not(true(LP)).
>>> ?- unify_with_occurs_check(LP, not(true(LP))).
>>> false.
>>>
>>> That is an elegant and classic way to demonstrate the Liar Paradox
>>> ("This sentence is false") using Prolog's unification mechanics!
>>>
>>> You've perfectly captured the tension between computational pragmatism
>>> and strict mathematical logic. Here is exactly what is happening under
>>> the hood in your snippet:
>>>
>>> 1. The Pragmatic Illusion: Standard Unification
>>>
>>> Prolog
>>> ?- LP = not(true(LP)).
>>> LP = not(true(LP)).
>>>
>>> When you use the standard unification operator (=), Prolog tries to make
>>> the left side and the right side identical. To save processing time and
>>> memory, Prolog historically skips a crucial step called the occurs
>>> check.
>>>
>>> Because it skips this check, Prolog happily binds the variable LP to the
>>> term not(true(LP)). This creates a cyclic term (or a rational tree). In
>>> memory, LP now points to a structure that contains a reference back to
>>> LP, creating an infinite loop of self-reference. Computationally, Prolog
>>> accepts the paradox.
>>>
>>> 2. The Logical Safeguard: The Occurs Check
>>>
>>> Prolog
>>> ?- unify_with_occurs_check(LP, not(true(LP))).
>>> false.
>>>
>>> Here, you enforce strict First-Order Logic. The
>>> unify_with_occurs_check/2 predicate forces Prolog to pause and ask:
>>> "Does the variable I am trying to bind already exist inside the term I
>>> am binding it to?" Because the variable LP occurs inside not(true(LP)),
>>> Prolog realizes that unifying them would require building a
>>> mathematically impossible, infinitely deep structure:
>>> not(true(not(true(not(true(...)))))). It rightfully rejects the
>>> unification and returns false.
>>>
>>> The Philosophical Takeaway
>>> In logic, self-referential paradoxes often arise when a system is
>>> allowed to talk about its own truth values without hierarchical
>>> restrictions (as famously noted by Alfred Tarski).
>>>
>>> Your Prolog snippet perfectly mirrors this:
>>>
>>> Standard Unification acts like natural language, allowing the
>>> self-reference to happen (and potentially causing a stack overflow if
>>> you tried to evaluate or traverse it).
>>>
>>> The Occurs Check acts like a strict formal logic system, explicitly
>>> forbidding self-referential loops and shutting the paradox down before
>>> it can form.
>>>
>>>
>>> https://share.gemini.google/5UVT3f5WnV8K
>>>
>>
>> Imagine a language and its utterance of all the truisms in the universe.
>>
>
> I have a name for this: The finite set of "atomic facts" of
> empirical general knowledge and the finite set of relations
> between them is named:
> "the body of general knowledge expressed in language"
>
>> This isn't saying necessarily what it says, only that it's said.
>> So, this account of a "Comenius language" of all the truisms is
>> more than less inscrutable yet as empty tautologies and truisms,
>> while yet each unique, for example as by natural numbers.
>>
>> "This sentence 1 is true.
>> This sentence 2 is true.
>> This sentence 3 is true.
>> ..."
>>
>>
>> So, quantify over those, like Russell might as
>> "sets-of-all-sets-that-don't-contain-themselves".
>>
>> "This sentence (...) is not true."
>>
>
> You are simply ignoring that expressions with cycles
> in their evaluation sequence are rejected as meaningless.
> Kripke would say that they are "undefined"
>
>> Now, the idea here is that there's only one example of a truism
>> about contradiction, that the alternation of inversion of the
>> consideration that results quantification, brings along the
>> "sputnik of quantification", that reads in its form as
>> "The Liar", yet instead of being a "paradox", its form is
>> construed as being a "Confessing Liar".
>>
>>
>> So, this account of univocity since Duns Scotus or as alike
>> accounts of the kabalah and gematria or since acconts of the
>> universal grammar since Panini like Leibnitz, this "Comenius
>> language" its consideration, like Quine's Nietzsche's "eternal
>> basic text", includes in itself a prototype of contradiction,
>> in otherwise all its affirmations.
>>
>>
>> Then, in natural language, there is a ready example or prototype
>> of contradiction, that in alike the natural "Coleridge language",
>> where that metaphor eventually fails yet there is a structural
>> account of the strong metonymy that fulfills true metaphor about
>> truth, then "the Liar" is simply an indicator of contradiction,
>> and results instead of explosion to contradiction its unconscious
>> digestion, results implosion to detection in any account of language.
>>
>>
>> So, "thinking" instead of "being thought" is the usual idea.
>> That's all then that the cycle-detection routine claims to do.
>>
>>
>> This way there are no paradoxes at all in Comenius language,
>> including that Confessing Comenius has an example to compare
>> against, what would be false.
>>
>>
>>
>> The eternal basic text or underlying univocal universal word
>> might be negations instead of affirmations.
>>
>> "This sentence 1 accounts and excludes 0.
>> This sentence 2 accounts and excludes 1.
>> This sentence 3 accounts and excludes 2.
>> ..."
>>
>>
>> Then a similar account gives alike:
>>
>> "This sentence infinity accounts and includes infinity."
>>
>>
>>
>> Then simple accounts of error-detection then error _correction_,
>> including accounts of "not enough information" or "conflicting
>> information" are mostly usual and trite, in fact there's an
>> entire enterprise called "science" which makes for an inter-subjective
>> formal account with regards to strengthened logicist positivism
>> and ontologists, which though is always a _science_, yet though
>> that there are ideals like "truth" and "infinity" for that
>> "geometry" and "continuity" are real, and more than the blahs.
>>
>
> So maybe this is too much into the field of philosophy of
> logic and thus outside the field or carefully memorizing
> exiting conventions for you to understand.
>
> The bottom line is the my Prolog tossed the Liar Paradox
> out on its ass and Gemini agrees. You didn't seem to get
> this most crucial point.
>



Russell would call you a fool. That said, for Frege's completeness
results then Goedel's completeness results, of "ordinary" arithmetic,
those are ultimately subject Russell's paradox and Goedel's own
in-completeness, or ordinary arithmetic, or "theories strong
enough to model ordinary arithmetic".

In contrast to somebody like Marilyn Mach vos Savant, a dilettante
and gad-fly, when it comes to philosophy I prefer somebody like
Martha Nussbaum, who is a platonist, and Susanne Langer for logic,
who easily demonstrates ordinary classical quasi-modal logic in her
textbook on logic, while yet also pointing out the ready demonstrations
of the limitations and fallacies of quasi-modal logic and material
implication. Then, there's "The Straight Dope" for columnists.

Then, there's Quine, and Quine's "Set Theory" and 'Word & Object"
contain plentiful apologetics about "standard" logic, with regards
to "ultimate classes" and "Quine's atoms", and accounts like variously
the ways to make for class/set distinction and variously the ways
that "logic" gets involved with:

x = x,
x =/= x,
x in x,
x not in x,

then Quine and Strawson and some of the Lewis' and for example
for Chwistek then Sheffer makes for that Sheffer is plenty,
and nobody needs Kripke or Gentzen when they have Sheffer,
and nobody needs Montague when there's Herbrand.


Anybody can read your willful ignorance as slight and petty.
And nobody needs it for anything.



Here there are at least three:

models of integers (fragments, extensions, the ordinary),
laws of large numbers (large, larger, largest),
models of limit theorems (inductive, infinite, continuum),
models of continuous domains (line-reals, field-reals, signal-reals),
models of Cantor spaces (square, sparse, signal),
models of probabilistic limit theorems (central, uniformizing, polar),

that mathematics always has had and always will have
and anybody who'd get past "Erdos' Giant Monsters of Mathematical
Independence" for their "Great Atlas of Mathematical Independence"
can discover for themselves.


I already have one, ....