XPost: comp.theory, sci.math, sci.lang   
   From: mikko.levanto@iki.fi   
      
   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.   
      
   --   
   Mikko   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)