XPost: comp.theory, sci.logic, sci.math   
   From: polcott333@gmail.com   
      
   On 12/3/2025 4:41 AM, Mikko wrote:   
   > olcott kirjoitti 2.12.2025 klo 16.00:   
   >> On 12/2/2025 2:53 AM, Mikko wrote:   
   >>> olcott kirjoitti 1.12.2025 klo 19.15:   
   >>>> On 12/1/2025 5:02 AM, Mikko wrote:   
   >>>>> olcott kirjoitti 29.11.2025 klo 23.59:   
   >>>>>   
   >>>>> G := (F ⊬ G) // G says of itself that it is unprovable in F   
   >>>>>   
   >>>>> With a reasonable type system that is a type error:   
   >>>>> - the symbol ⊬ requires a sentence on the right side   
   >>>>> - the value of the ⊬ operation is a truth value   
   >>>>> - the symbol := requires the same type on both sides   
   >>>>> - thus G must be both a sentence and a truth value   
   >>>>>   
   >>>>> But G cannot be both. A sentence has a truth value but it isn't one.   
   >>>>>   
   >>>>   
   >>>> % 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.   
   >>>>   
   >>>> It is an expression of language having no truth value   
   >>>> because it is not a logic sentence.   
   >>>>   
   >>>> https://en.wikipedia.org/wiki/Sentence_(mathematical_logic)   
   >>>   
   >>> Yes, that is the exxential difference between the two G's.   
   >>> The expession F ⊬ G has a truth value because it is either   
   >>> true or false   
   >>   
   >> I propose that is a false assumption.   
   >   
   > If you want to propose anygthng like that you should   
   > (a) specify what is the assumption you want to propose as false   
   > (b) why should that assumption be considered false   
   > (c) what assumption would be true or at least less obviously false   
   >   
      
   ?- G = not(provable(F, G)).   
   G = not(provable(F, G)).   
   ?- unify_with_occurs_check(G, not(provable(F, G))).   
   false.   
      
   G is neither True nor False its resolution remains stuck   
   in an infinite loop.   
      
   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.   
   END:(Clocksin & Mellish 2003:254)   
      
      
   --   
   Copyright 2025 Olcott   
      
   My 28 year goal has been to make   
   "true on the basis of meaning" computable.   
      
   This required establishing a new foundation   
   for correct reasoning.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)