XPost: comp.theory, sci.math, comp.lang.prolog   
   From: mikko.levanto@iki.fi   
      
   olcott kirjoitti 5.12.2025 klo 19.40:   
   > On 12/5/2025 3:13 AM, Mikko wrote:   
   >> olcott kirjoitti 4.12.2025 klo 16.10:   
   >>> 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.   
   >>   
   >> The usual way is to rehject them as syntactically invalid.   
      
   > Even this simplified version has the same pathological self-reference   
   > (G) F ⊢ GF ↔ ¬ProvF(┌GF┐).   
      
   There is no self reference there. F is a formal system. A formal system   
      
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