XPost: comp.theory, sci.math, sci.lang   
   From: polcott333@gmail.com   
      
   On 2/11/2026 4:51 AM, Mikko wrote:   
   > On 10/02/2026 15:37, olcott wrote:   
   >> On 2/10/2026 3:06 AM, Mikko wrote:   
   >>> On 09/02/2026 17:36, olcott wrote:   
   >>>> On 2/9/2026 8:57 AM, Mikko wrote:   
   >>>>> On 07/02/2026 18:43, olcott wrote:   
   >>>>>>   
   >>>>>> Conventional logic and math have been paralyzed for   
   >>>>>> many decades by trying to force-fit semantically   
   >>>>>> ill-formed expressions into the box of True or False.   
   >>>>>   
   >>>>> Logic is not paralyzed. Separating semantics from inference rules   
   >>>>> ensures that semantic problems don't affect the study of proofs   
   >>>>> and provability.   
   >>>>   
   >>>> Then you end up with screwy stuff such as the psychotic   
   >>>> https://en.wikipedia.org/wiki/Principle_of_explosion   
   >>>   
   >>> That you call it psychotic does not make it less useful. Often an   
   >>> indirect proof is simpler than a direct one, and therefore more   
   >>> convincing. But without the principle of explosion it might be   
   >>> harder or even impossible to find one, depending on what there is   
   >>> instead.   
   >>   
   >> Completely replacing the foundation of truth conditional   
   >> semantics with proof theoretic semantics then an expression   
   >> is "true on the basis of meaning expressed in language"   
   >> only to the extent that its meaning is entirely comprised   
   >> of its inferential relations to other expressions of that   
   >> language. AKA linguistic truth determined by semantic   
   >> entailment specified syntactically.   
   >>   
   >> Well-founded proof-theoretic semantics reject expressions   
   >> lacking a "well-founded justification tree" as meaningless.   
   >> ∀x (~Provable(T, x) ⇔ Meaningless(T, x))   
   >   
   > Usually it is thought that an expression can be determined to be   
   > meaningful even when it is not known whether it is provable. For   
   > example, the program fragment   
   >   
   >    if (x < 5) {   
   >      show(x);   
   >    }   
   >   
   > is quite meaningful even when one cannot prove or even know whether   
   > x at the time of execution is less than 5.   
   >   
      
   Only Proof-Theoretic Semantics   
   https://plato.stanford.edu/entries/proof-theoretic-semantics/   
      
   Can make   
   "true on the basis of meaning expressed in language"   
   reliably computable for the entire body of knowledge.   
      
   We completely replace the foundation of truth conditional   
   semantics with proof theoretic semantics. Then expressions   
   are "true on the basis of meaning expressed in language"   
   only to the extent that all their meaning comes from   
   inferential relations to other expressions of that language.   
   This is a purely linguistic PTS notion of truth with no   
   connections outside the inferential system.   
      
   Well-founded proof-theoretic semantics reject expressions   
   lacking a "well-founded justification tree" as meaningless.   
     ∀x (~Provable(T, x) ⇔ Meaningless(T, x))   
      
   ∀x (Provable(x) ⇒ True(x)) --- Anchored in Prawitz, (2012)   
      
   What is the appropriate notion of truth for sentences whose   
   meanings are understood in epistemic terms such as proof or   
   ground for an assertion? It seems that the truth of such   
   sentences has to be identified with the existence of proofs or grounds...   
      
   Prawitz, D. (2012). Truth as an Epistemic Notion. Topoi, 31(1), 9–16   
   https://doi.org/10.1007/s11245-011-9107-6   
      
      
   --   
   Copyright 2026 Olcott

   
      
   My 28 year goal has been to make 
   
   "true on the basis of meaning expressed in language"
   
   reliably computable for the entire body of knowledge.

   
      
   This required establishing a new foundation
   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)