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Re: A new foundation for correct reasoning

From Mikko <mikko.levanto@iki.fi>
Newsgroups comp.lang.prolog, comp.theory, sci.logic, sci.math
Subject Re: A new foundation for correct reasoning
Date 2025-12-15 11:04 +0200
Organization A noiseless patient Spider
Message-ID <10hoive$1p2m3$2@dont-email.me> (permalink)
References (24 earlier) <10hh88u$2v73q$2@dont-email.me> <10hjejt$3u3sm$1@dont-email.me> <10hju2h$4hat$1@dont-email.me> <10hm25a$tpc7$1@dont-email.me> <10hngct$1gbn9$1@dont-email.me>

Cross-posted to 4 groups.

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On 15/12/2025 01:14, olcott wrote:
> On 12/14/2025 4:05 AM, Mikko wrote:
>> On 13/12/2025 16:43, olcott wrote:
>>> On 12/13/2025 4:19 AM, Mikko wrote:
>>>> olcott kirjoitti 12.12.2025 klo 16.19:
>>>>> On 12/12/2025 2:50 AM, Mikko wrote:
>>>>>> olcott kirjoitti 11.12.2025 klo 16.17:
>>>>>>> On 12/11/2025 2:42 AM, Mikko wrote:
>>>>>>>> olcott kirjoitti 10.12.2025 klo 16.10:
>>>>>>>>> On 12/10/2025 4:04 AM, Mikko wrote:
>>>>>>>>>> olcott kirjoitti 8.12.2025 klo 21.09:
>>>>>>>>>>> On 12/8/2025 3:13 AM, Mikko wrote:
>>>>>>>>>>>> olcott kirjoitti 5.12.2025 klo 19.43:
>>>>>>>>>>>>> On 12/5/2025 3:38 AM, Mikko wrote:
>>>>>>>>>>>>>> olcott kirjoitti 4.12.2025 klo 16.06:
>>>>>>>>>>>>>>> On 12/4/2025 2:58 AM, Mikko wrote:
>>>>>>>>>>>>>>>> Tristan Wibberley kirjoitti 4.12.2025 klo 4.32:
>>>>>>>>>>>>>>>>> On 30/11/2025 09:58, Mikko wrote:
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> Note that the meanings of
>>>>>>>>>>>>>>>>>>   ?- G = not(provable(F, G)).
>>>>>>>>>>>>>>>>>> and
>>>>>>>>>>>>>>>>>>   ?- unify_with_occurs_check(G, not(provable(F, G))).
>>>>>>>>>>>>>>>>>> are different. The former assigns a value to G, the 
>>>>>>>>>>>>>>>>>> latter does not.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> For sufficiently informal definitions of "value".
>>>>>>>>>>>>>>>>> And for sufficiently wrong ones too!
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> It is sufficiently clear what "value" of a Prolog 
>>>>>>>>>>>>>>>> variable means.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> % This sentence cannot be proven in F
>>>>>>>>>>>>>>> ?- G = not(provable(F, G)).
>>>>>>>>>>>>>>> G = not(provable(F, G)).
>>>>>>>>>>>>>>> ?- unify_with_occurs_check(G, not(provable(F, G))).
>>>>>>>>>>>>>>> false.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> I would say that the above Prolog is the 100%
>>>>>>>>>>>>>>> complete formal specification of:
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> "This sentence cannot be proven in F"
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> The first query can be regarded as a question whether "G = 
>>>>>>>>>>>>>> not(provable(
>>>>>>>>>>>>>> F, G))" can be proven for some F and some G. The answer is 
>>>>>>>>>>>>>> that it can
>>>>>>>>>>>>>> for every F and for (at least) one G, which is 
>>>>>>>>>>>>>> not(provable(G)).
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> The second query can be regarded as a question whether "G 
>>>>>>>>>>>>>> = not(provable
>>>>>>>>>>>>>> (F, G))" can be proven for some F and some G that do not 
>>>>>>>>>>>>>> contain cycles.
>>>>>>>>>>>>>> The answer is that in the proof system of Prolog it cannot 
>>>>>>>>>>>>>> be.
>>>>>>>>>>>>>
>>>>>>>>>>>>> No that it flatly incorrect. The second question is this:
>>>>>>>>>>>>> Is "G = not(provable(F, G))." semantically sound?
>>>>>>>>>>>>
>>>>>>>>>>>> Where is the definition of Prolog semantics is that said?
>>>>>>>>>>>
>>>>>>>>>>> Any expression of Prolog that cannot be evaluated to
>>>>>>>>>>> a truth value because it specifies non-terminating
>>>>>>>>>>> infinite recursion is "semantically unsound" by the
>>>>>>>>>>> definition of those terms even if Prolog only specifies
>>>>>>>>>>> that cannot be evaluated to a truth value because it
>>>>>>>>>>> specifies non-terminating infinite recursion.
>>>>>>>>>>
>>>>>>>>>> Your Prolog implementation has evaluated G = not(provablel(F, G))
>>>>>>>>>> to a truth value true. When doing so it evaluated each side of =
>>>>>>>>>> to a value that is not a truth value.
>>>>>>>>>
>>>>>>>>> ?- unify_with_occurs_check(G, not(provable(F, G))).
>>>>>>>>> false.
>>>>>>>>>
>>>>>>>>> Proves that
>>>>>>>>> G = not(provable(F, G)).
>>>>>>>>> would remain stuck in infinite recursion.
>>>>>>>>>
>>>>>>>>> unify_with_occurs_check() examines the directed
>>>>>>>>> graph of the evaluation sequence of an expression.
>>>>>>>>> When it detects a cycle that indicates that an
>>>>>>>>> expression would remain stuck in recursive
>>>>>>>>> evaluation never to be resolved to a truth value.
>>>>>>>>>
>>>>>>>>> BEGIN:(Clocksin & Mellish 2003:254)
>>>>>>>>> Finally, a note about how Prolog matching sometimes differs
>>>>>>>>> from the unification used in Resolution. Most Prolog systems
>>>>>>>>> will allow you to satisfy goals like:
>>>>>>>>>
>>>>>>>>> equal(X, X).
>>>>>>>>> ?- equal(foo(Y), Y).
>>>>>>>>>
>>>>>>>>> that is, they will allow you to match a term against an
>>>>>>>>> uninstantiated subterm of itself. In this example, foo(Y)
>>>>>>>>> is matched against Y, which appears within it. As a result,
>>>>>>>>> Y will stand for foo(Y), which is foo(foo(Y)) (because of
>>>>>>>>> what Y stands for), which is foo(foo(foo(Y))), and so on.
>>>>>>>>> So Y ends up standing for some kind of infinite structure.
>>>>>>>>>
>>>>>>>>> Note that, whereas they may allow you to construct something
>>>>>>>>> like this, most Prolog systems will not be able to write it
>>>>>>>>> out at the end. According to the formal definition of
>>>>>>>>> Unification, this kind of “infinite term” should never come
>>>>>>>>> to exist. Thus Prolog systems that allow a term to match an
>>>>>>>>> uninstantiated subterm of itself do not act correctly as
>>>>>>>>> Resolution theorem provers. In order to make them do so, we
>>>>>>>>> would have to add a check that a variable cannot be
>>>>>>>>> instantiated to something containing itself. Such a check,
>>>>>>>>> an occurs check, would be straightforward to implement, but
>>>>>>>>> would slow down the execution of Prolog programs considerably.
>>>>>>>>> Since it would only affect very few programs, most implementors
>>>>>>>>> have simply left it out 1.
>>>>>>>>>
>>>>>>>>> 1 The Prolog standard states that the result is undefined if
>>>>>>>>> a Prolog system attempts to match a term against an 
>>>>>>>>> uninstantiated subterm of itself, which means that programs 
>>>>>>>>> which cause this to
>>>>>>>>> happen will not be portable. A portable program should ensure 
>>>>>>>>> that wherever an occurs check might be applicable the built-in 
>>>>>>>>> predicate
>>>>>>>>> unify_with_occurs_check/2 is used explicitly instead of the normal
>>>>>>>>> unification operation of the Prolog implementation. As its name 
>>>>>>>>> suggests, this predicate acts like =/2 except that it fails if an
>>>>>>>>> occurs check detects an illegal attempt to instantiate a variable.
>>>>>>>>> END:(Clocksin & Mellish 2003:254)
>>>>>>>>>
>>>>>>>>> Clocksin, W.F. and Mellish, C.S. 2003. Programming in Prolog
>>>>>>>>> Using the ISO Standard Fifth Edition, 254. Berlin Heidelberg:
>>>>>>>>> Springer-Verlag.
>>>>>>>>
>>>>>>>> Thank you for the confirmation of my explanation of your error.
>>>>>>>
>>>>>>>  >> Y will stand for foo(Y), which is foo(foo(Y)) (because of
>>>>>>>  >> what Y stands for), which is foo(foo(foo(Y))), and so on.
>>>>>>> As I say non-terminating, thus never resolves to a truth value.
>>>>>>
>>>>>> As according to Prolog rules foo(Y) isn't a truth value for any Y
>>>>>> the above is obviously just an attempt to deive with a distraction.
>>>>>
>>>>> That was a quote from the most definitive source
>>>>> for the Prolog Language.
>>>>
>>>> As I already said, that source agrees with what I said above.
>>>>
>>>>> Prolog only has Facts and Rules thus the only
>>>>> derivation is to a truth value.
>>>>
>>>
>>> You just don't seem to understand:
>>> ?- G = not(provable(F, G)).
>>> G = not(provable(F, G)).
>>> ?- unify_with_occurs_check(G, not(provable(F, G))).
>>> false.
>>>
>>> The first statement creates a cyclic term, also called
>>> a rational tree. The second executes logically sound
>>> unification and thus fails.
>>> https://www.swi-prolog.org/pldoc/man?predicate=unify_with_occurs_check/2
>>
>> Saying the same as I said does not support a claim of non-understanding.
> 
> It finally resolves the Liar Paradox
> as not a truth bearer or proposition.

In other words you admit you were lying about me.

-- 
Mikko

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Re: A new foundation for correct reasoning olcott <polcott333@gmail.com> - 2025-12-05 11:43 -0600
  Re: A new foundation for correct reasoning Mikko <mikko.levanto@iki.fi> - 2025-12-06 11:30 +0200
    Re: A new foundation for correct reasoning olcott <polcott333@gmail.com> - 2025-12-06 06:50 -0600
      Re: A new foundation for correct reasoning Mikko <mikko.levanto@iki.fi> - 2025-12-07 13:02 +0200
        Re: A new foundation for correct reasoning olcott <polcott333@gmail.com> - 2025-12-08 13:49 -0600
  Re: A new foundation for correct reasoning Mikko <mikko.levanto@iki.fi> - 2025-12-08 11:13 +0200
    Re: A new foundation for correct reasoning olcott <polcott333@gmail.com> - 2025-12-08 13:09 -0600
      Re: A new foundation for correct reasoning Mikko <mikko.levanto@iki.fi> - 2025-12-10 12:04 +0200
        Re: A new foundation for correct reasoning olcott <polcott333@gmail.com> - 2025-12-10 08:10 -0600
          Re: A new foundation for correct reasoning Python <python@cccp.invalid> - 2025-12-10 15:01 +0000
          Re: A new foundation for correct reasoning Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2025-12-10 18:10 +0000
            Re: A new foundation for correct reasoning olcott <polcott333@gmail.com> - 2025-12-10 14:01 -0600
          Re: A new foundation for correct reasoning Mikko <mikko.levanto@iki.fi> - 2025-12-11 10:42 +0200
            Re: A new foundation for correct reasoning olcott <polcott333@gmail.com> - 2025-12-11 08:17 -0600
              Re: A new foundation for correct reasoning Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2025-12-11 23:28 +0000
                Re: A new foundation for correct reasoning olcott <polcott333@gmail.com> - 2025-12-11 17:49 -0600
                Re: A new foundation for correct reasoning Richard Damon <Richard@Damon-Family.org> - 2025-12-11 19:52 -0500
              Re: A new foundation for correct reasoning Mikko <mikko.levanto@iki.fi> - 2025-12-12 10:50 +0200
                Re: A new foundation for correct reasoning olcott <polcott333@gmail.com> - 2025-12-12 08:19 -0600
                Re: A new foundation for correct reasoning Richard Damon <Richard@Damon-Family.org> - 2025-12-12 09:24 -0500
                Re: A new foundation for correct reasoning Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2025-12-14 19:03 +0000
                Re: A new foundation for correct reasoning Mikko <mikko.levanto@iki.fi> - 2025-12-13 12:19 +0200
                Re: A new foundation for correct reasoning olcott <polcott333@gmail.com> - 2025-12-13 08:43 -0600
                Re: A new foundation for correct reasoning Richard Damon <Richard@Damon-Family.org> - 2025-12-13 13:36 -0500
                Re: A new foundation for correct reasoning Mikko <mikko.levanto@iki.fi> - 2025-12-14 12:05 +0200
                Re: A new foundation for correct reasoning olcott <polcott333@gmail.com> - 2025-12-14 17:14 -0600
                Re: A new foundation for correct reasoning Richard Damon <Richard@Damon-Family.org> - 2025-12-14 19:13 -0500
                Re: A new foundation for correct reasoning olcott <polcott333@gmail.com> - 2025-12-14 18:46 -0600
                Re: A new foundation for correct reasoning Richard Damon <Richard@Damon-Family.org> - 2025-12-14 19:53 -0500
                Re: A new foundation for correct reasoning olcott <polcott333@gmail.com> - 2025-12-14 19:08 -0600
                Re: A new foundation for correct reasoning Richard Damon <Richard@Damon-Family.org> - 2025-12-14 20:46 -0500
                Re: A new foundation for correct reasoning olcott <polcott333@gmail.com> - 2025-12-14 20:05 -0600
                Re: A new foundation for correct reasoning Richard Damon <Richard@Damon-Family.org> - 2025-12-14 21:23 -0500
                Re: A new foundation for correct reasoning olcott <polcott333@gmail.com> - 2025-12-14 20:09 -0600
                Re: A new foundation for correct reasoning Richard Damon <Richard@Damon-Family.org> - 2025-12-14 21:27 -0500
                Re: A new foundation for correct reasoning olcott <polcott333@gmail.com> - 2025-12-14 21:22 -0600
                Re: A new foundation for correct reasoning Richard Damon <Richard@Damon-Family.org> - 2025-12-15 07:33 -0500
                Re: A new foundation for correct reasoning Mikko <mikko.levanto@iki.fi> - 2025-12-15 11:04 +0200
                Re: A new foundation for correct reasoning olcott <polcott333@gmail.com> - 2025-12-15 08:03 -0600
                Re: A new foundation for correct reasoning Mikko <mikko.levanto@iki.fi> - 2025-12-16 11:44 +0200
                Re: A new foundation for correct reasoning Mikko <mikko.levanto@iki.fi> - 2025-12-16 11:48 +0200

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