XPost: comp.theory, sci.math, sci.lang   
   From: polcott333@gmail.com   
      
   On 12/5/2025 2:48 AM, Mikko wrote:   
   > olcott kirjoitti 3.12.2025 klo 17.59:   
   >> 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)   
   >   
   > As even (a) is not answered we must interprete the above to mean   
   > that you retracted your proposal.   
   >   
      
   If you understood the above you would understand   
   that I already answered (a) in 100% complete detail.   
      
   The assumption that is false is that G is not   
   semantically incoherent.   
      
   --   
   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   
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