XPost: comp.theory, sci.math
From: polcott333@gmail.com
On 12/4/2025 3:07 AM, Mikko wrote:
> olcott kirjoitti 3.12.2025 klo 18.11:
>> On 12/3/2025 4:53 AM, Mikko wrote:
>>> olcott kirjoitti 26.11.2025 klo 17.13:
>>>> On 11/26/2025 3:05 AM, Mikko wrote:
>>>>> olcott kirjoitti 26.11.2025 klo 5.24:
>>>>>> On 11/25/2025 8:43 PM, Python wrote:
>>>>>>> Le 26/11/2025 à 03:41, olcott a écrit :
>>>>>>>> On 11/25/2025 8:36 PM, André G. Isaak wrote:
>>>>>>>>> On 2025-11-25 19:30, olcott wrote:
>>>>>>>>>> On 11/25/2025 8:12 PM, André G. Isaak wrote:
>>>>>>>>>>> On 2025-11-25 19:08, olcott wrote:
>>>>>>>>>>>> On 11/25/2025 8:00 PM, André G. Isaak wrote:
>>>>>>>>>>>>> On 2025-11-25 18:43, olcott wrote:
>>>>>>>>>>>>>> On 11/25/2025 7:29 PM, André G. Isaak wrote:
>>>>>>>>>>>>>>> On 2025-11-25 17:52, olcott wrote:
>>>>>>>>>>>>>>>> On 11/25/2025 6:47 PM, Kaz Kylheku wrote:
>>>>>>>>>>>>>>>>> On 2025-11-25, olcott wrote:
>>>>>>>>>>>>>>>>>> Gödel incompleteness can only exist in systems that
>>>>>>>>>>>>>>>>>> divide
>>>>>>>>>>>>>>>>>> their syntax from their semantics ...
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> And, so, just confuse syntax for semantics, and all is
>>>>>>>>>>>>>>>>> fixed!
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Things such as Montague Grammar are outside of your
>>>>>>>>>>>>>>>> current knowledge. It is called Montague Grammar
>>>>>>>>>>>>>>>> because it encodes natural language semantics as pure
>>>>>>>>>>>>>>>> syntax.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> You're terribly confused here. Montague Grammar is called
>>>>>>>>>>>>>>> 'Montague Grammar' because it is due to Richard Montague.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Montague Grammar presents a theory of natural language
>>>>>>>>>>>>>>> (specifically English) semantics expressed in terms of
>>>>>>>>>>>>>>> logic. Formulae in his system have a syntax. They also
>>>>>>>>>>>>>>> have a semantics. The two are very much distinct.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Montague Grammar is the syntax of English semantics
>>>>>>>>>>>>>
>>>>>>>>>>>>> I can't even make sense of that. It's a *theory* of English
>>>>>>>>>>>>> semantics.
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> *Here is a concrete example*
>>>>>>>>>>>> The predicate Bachelor(x) is stipulated to mean ~Married(x)
>>>>>>>>>>>> where the predicate Married(x) is defined in terms of billions
>>>>>>>>>>>> of other things such as all of the details of Human(x).
>>>>>>>>>>>
>>>>>>>>>>> A concrete example of what? That's certainly not an example
>>>>>>>>>>> of 'the syntax of English semantics'. That's simply a
>>>>>>>>>>> stipulation involving two predicates.
>>>>>>>>>>>
>>>>>>>>>>> André
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> It is one concrete example of how a knowledge ontology
>>>>>>>>>> of trillions of predicates can define the finite set
>>>>>>>>>> of atomic facts of the world.
>>>>>>>>>
>>>>>>>>> But the topic under discussion was the relationship between
>>>>>>>>> syntax and semantics in Montague Grammar, not how knowledge
>>>>>>>>> ontologies are represented. So this isn't an example in anyway
>>>>>>>>> relevant to the discussion.
>>>>>>>>>
>>>>>>>>>> *Actually read this, this time*
>>>>>>>>>> Kurt Gödel in his 1944 Russell's mathematical logic gave the
>>>>>>>>>> following definition of the "theory of simple types" in a
>>>>>>>>>> footnote:
>>>>>>>>>>
>>>>>>>>>> By the theory of simple types I mean the doctrine which says
>>>>>>>>>> that the objects of thought (or, in another interpretation,
>>>>>>>>>> the symbolic expressions) are divided into types, namely:
>>>>>>>>>> individuals, properties of individuals, relations between
>>>>>>>>>> individuals, properties of such relations
>>>>>>>>>>
>>>>>>>>>> That is the basic infrastructure for defining all *objects of
>>>>>>>>>> thought*
>>>>>>>>>> can be defined in terms of other *objects of thought*
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> I know full well what a theory of types is. It has nothing to
>>>>>>>>> do with the relationship between syntax and semantics.
>>>>>>>>>
>>>>>>>>> André
>>>>>>>>>
>>>>>>>>
>>>>>>>> That particular theory of types lays out the infrastructure
>>>>>>>> of how all *objects of thought* can be defined in terms
>>>>>>>> of other *objects of thought* such that the entire body
>>>>>>>> of knowledge that can be expressed in language can be encoded
>>>>>>>> into a single coherent formal system.
>>>>>>>
>>>>>>> Typing “objects of thought” doesn’t make all truths provable —
it
>>>>>>> only prevents ill-formed expressions.
>>>>>>> If your system looks complete, it’s because you threw away every
>>>>>>> sentence that would have made it incomplete.
>>>>>>
>>>>>> When ALL *objects of thought* are defined
>>>>>> in terms of other *objects of thought* then
>>>>>> their truth and their proof is simply walking
>>>>>> the knowledge tree.
>>>>>
>>>>> When ALL subjects of thoughts are defined
>>>>> in terms of other subjects of thoughts then
>>>>> there are no subjects of thoughts.
>>>>
>>>> I am merely elaborating the structure of the
>>>> knowledge ontology inheritance hierarchy
>>>> tree of knowledge.
>>>
>>> When ALL subjects of thoughts are defined in terms of other subjects
>>> of thoughts the system of ALL subjects of thoughts is either empty
>>> or not a hierarchy. There is no hierarchy where every member is under
>>> another member.
>>
>> *I have always been referring to the entire body of general knowledge*
>
> Your condition that ALL objects of thought can be defined in terms of
> other objects of thought is false about every non-empyt collection of
> objects of thjought, inluding the entire body of general knowledge,
> unless your system allows circular definitions that actually don't
> define.
>
Yes circular definitions can be defined syntactically
and are rejected as semantically unsound.
% This sentence is not true.
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
In Olcott's Minimal Type Theory
LP := ~True(LP)
that expands to: ~True(~True(~True(~True(~True(LP)))))
% 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.
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
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