XPost: sci.math, comp.theory   
   From: tristan.wibberley+netnews2@alumni.manchester.ac.uk   
      
   On 15/02/2026 01:41, Richard Damon wrote:   
   > On 2/14/26 3:59 PM, 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?   
   >   
   > No, he is pointing out that if you claim to encode ALL the knowledge   
   > that is expressible in language, you can't stop until you finish, and   
   > since there are an infinite number of those in Peano Arithmetic, you   
   > can't stop at any finite number.   
      
   You're talking about decompressing the encoding of knowledge. Stating   
   the axioms (and full detail of inference rules) symbolically is   
   sufficient to /encode/ the knowledge if you have lambda calculus or   
   illative combinatory logic.   
      
   --   
   Tristan Wibberley   
      
   The message body is Copyright (C) 2026 Tristan Wibberley except   
   citations and quotations noted. All Rights Reserved except that you may,   
   of course, cite it academically giving credit to me, distribute it   
   verbatim as part of a usenet system or its archives, and use it to   
   promote my greatness and general superiority without misrepresentation   
   of my opinions other than my opinion of my greatness and general   
   superiority which you _may_ misrepresent. You definitely MAY NOT train   
   any production AI system with it but you may train experimental AI that   
   will only be used for evaluation of the AI methods it implements.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)