XPost: comp.theory, sci.math, sci.lang   
   From: mikko.levanto@iki.fi   
      
   olcott kirjoitti 5.12.2025 klo 18.41:   
   > 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.   
      
   Apparently "that" in your "I propopose that is a false assumption"   
   refers to my "yes" response to your previous posting. But that   
   response does not oresent any assumption.   
      
   As everyone can see, you did not indentify the assumption.   
      
   > The assumption that is false is that G is not   
   > semantically incoherent.   
      
   That assumption is not present in any plase that the word "that"   
   could refer to.   
      
   --   
   Mikko   
      
   --- SoupGate-Win32 v1.05   
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