XPost: comp.theory, sci.math, comp.lang.prolog   
   From: polcott333@gmail.com   
      
   On 12/7/2025 5:02 AM, Mikko wrote:   
   > olcott kirjoitti 6.12.2025 klo 14.50:   
   >> On 12/6/2025 3:30 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?   
   >>>   
   >>> When "G = not(provable(F, G))." is used as a Prolog goal the   
   >>> applied semantics is what Prolog lauguage definition specifies.   
   >>> Does "semantically sound" mean something in that context?   
   >>>   
   >>> At least your Prolog interpretation finds it meaningful. It determines   
   >>> that the excution of that goal succeeds and assigns a value G but not   
   >>> to F.   
   >>   
   >> Is this sentence true or false:   
   >> "This sentence is not true"   
   >> It is not semantically sound.   
   >   
   > Formal systems we usually use have no expression for "this".   
   >   
      
   Yes that is the why I created Minimal Type Theory.   
   There is no reason why encoding "this" should   
   require dozens of different formulas.   
   LP := ~True(LP)   // says: "this sentence says itself is not true"   
   G := ~Provable(G) // says: "this sentence says itself is not provable".   
      
      
      
   --   
   Copyright 2025 Olcott

   
      
   My 28 year goal has been to make 
   
   "true on the basis of meaning" computable.

   
      
   This required establishing a new foundation
   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)