XPost: sci.math, comp.theory   
   From: polcott333@gmail.com   
      
   On 2/13/2026 2:30 AM, Mikko wrote:   
   > On 12/02/2026 17:48, olcott wrote:   
   >> On 2/12/2026 2:11 AM, Mikko wrote:   
   >>> On 11/02/2026 14:38, olcott wrote:   
   >>>> 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.   
   >>>   
   >>> In order to achieve that all arithmetic must be excluded from   
   >>> "true on the basis of meaning expressed in language". There   
   >>> is no way to compute wheter a sentence of the first order   
   >>> Peano arithmetic is provable.   
   >>   
   >> ∀x (Provable(T, x) ⇔ Meaningful(T, x)) --- (Schroeder-Heister 2024)   
   >> ∀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...   
   >   
   > Which means that if it is not determined whether there is a proof of   
   > a sentence and no way to find out the truth of that sentence is not   
   > known and cannot be computed.   
   >   
      
   Its all in a finite directed acyclic graph of knowledge.   
   If a back-chained inference does not exist from x to the   
   axioms of T then then x does not have a well-founded   
   justification tree and is rejected as meaningless.   
      
   When a cycle in the inference chain is detected this   
   also proves x does not have a well-founded justification   
   tree and is rejected as meaningless.   
      
   This gets rid of all pathological self-reference such   
   as the liar paradox and the halting problem proof.   
      
      
   --   
   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)