XPost: comp.theory, sci.math, sci.lang   
   From: mikko.levanto@iki.fi   
      
   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 that G is no provable in F, and the same truth   
   value is given to G in the expression G := (F ⊬ G). The   
   Prolog term not(provable(F, G)) does not have a truth value.   
   After G = not(provable(F, G)) the value of G is that data   
   structure, so it has no truth value, unlike the G in   
   G := (F ⊬ G).   
      
   --   
   Mikko   
      
   --- SoupGate-Win32 v1.05   
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