XPost: sci.logic, sci.math, comp.theory   
   From: mikko.levanto@iki.fi   
      
   On 18/01/2026 15:45, olcott wrote:   
   > On 1/18/2026 5:54 AM, Mikko wrote:   
   >> On 16/01/2026 01:40, olcott wrote:   
      
   >>> The term *proof‑theoretic semantics* has always   
   >>> proved my point long before I ever heard of it.   
      
   >> A term does not prove anything. Only a proof proves.   
      
   Note that my comment is not conradicted nor denied below:   
      
   > Proof Theoretic Semantics with the notion of   
   > non-well-founded expressions is the same thing   
   > that I have been saying for years.   
   >   
   > True and False in PA have always been x or ~x is   
   > provable from the actual axioms of PA, otherwise   
   > x is simply not a truth bearer in PA. The only   
   > clarification that I make now explicitly adding a   
   > truth predicate to PA.   
   >   
   > ∀x ∈ PA ((True(PA, x)  ≡ (PA ⊢ x))   
   > ∀x ∈ PA ((False(PA, x) ≡ (PA ⊢ ~x))   
   > ∀x ∈ PA ((~True(PA, x) ∧ (~False(PA, x) ≡ ~TruthBearer(PA, x))   
   --   
   Mikko   
      
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