XPost: sci.math, comp.theory   
   From: mikko.levanto@iki.fi   
      
   On 14/02/2026 22:59, 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.   
   >   
   > So it looks like you are saying that no one can count   
   > until after they first count to infinity?   
      
   So it looks like you can't read.   
      
   --   
   Mikko   
      
   --- SoupGate-Win32 v1.05   
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