XPost: comp.theory, sci.math   
   From: polcott333@gmail.com   
      
   On 2/16/2026 5:25 AM, Mikko wrote:   
   > On 15/02/2026 15:02, polcott wrote:   
   >> On 2/15/2026 3:18 AM, Mikko wrote:   
   >>> On 14/02/2026 17:31, polcott wrote:   
   >>>> On 2/14/2026 3:14 AM, Mikko wrote:   
   >>>>> On 13/02/2026 15:32, olcott wrote:   
   >>>>>> 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.   
   >>>>>   
   >>>>> No, it is not. The set of provable statements of the first order Peano   
   >>>>> arithmetic is infinite so it cannot be in a finite graph.   
   >>>>   
   >>>> The specialized nature of my work has exceeded the technical   
   >>>> knowledge of people here and most everywhere else.   
   >>>   
   >>> That an inifinite sent cannot be in a finite graph may exceed your   
   >>> technical knowledge but certainly doesn't everyone else's.   
   >   
   >> ∀x((x > 10) ⇒ (x > 0))   
   >> Does not mean to test every x.   
   >   
   > Irrelevant. That is only one sentence, not infinitely many.   
   >   
   >> ∀x ∈ PA (True(PA, x) ↔ PA ⊢ x)   
   >> Does not mean to test every x in PA   
   >   
   > No, it merely declares that there are two symbols for one predicate   
   > (which, if interpreted accordint to the usual meaning of either symbol,   
   > is uncomputable).   
   >   
      
   What do you think that this means: PA ⊢ x ?   
      
   > But that is irrelevant, too. The set of provable sentences is infinite   
   > so it cannot be in a finite graph.   
   >   
      
   So if I say that all cats are animals this   
   does not count as true until after you check   
   every cat?   
      
   --   
   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)