XPost: comp.theory, sci.logic, sci.math   
   XPost: comp.ai.philosophy   
   From: polcott333@gmail.com   
      
   On 2/12/2026 6:29 AM, Richard Damon wrote:   
   > On 2/11/26 9:17 PM, olcott wrote:   
   >> We completely replace the foundation of Truth Conditional Semantics   
   >> with Proof Theoretic Semantics (PTS). 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 the purely linguistic PTS notion   
   >> of truth having no connections outside the inferential system.   
   >   
   > Now, since you just changed the basic operation of ALL logic, you need   
   > to re-prove what each system can do.   
   >   
   > You also need to handle the axioms that don't really have meaning under   
   > Proof Theoretic Semantics, like induction, that validate that a   
   > statement had "meaning" without proof, and provide a way to sometimes   
   > prove it.   
   >   
      
   ∀x (Provable(T, x) ⇔ Meaningful(T, x)) --- (Schroeder-Heister 2024)   
   ∀x (Provable(x) ⇒ True(x)) --- Anchored in (Prawitz, 2012)   
      
   >>   
   >> "true on the basis of meaning expressed in language" are elements of   
   >> the body of verbal knowledge. This can include basic facts of the   
   >> actual world as stipulated axioms of the verbal model of the actual   
   >> world. This bridges the divide between the analytic/synthetic   
   >> distinction.   
   >   
   > But, "the basis of meaning" in some systems specifiically ALLOW for   
   > "unprovable" things to be true.   
   >   
      
   Only with a wrong-headed notion of:   
   "true on the basis of meaning expressed in language"   
      
   > Note, "Facts" of the actual world can NOT be axioms, as they are   
   > categorically different type of things.   
   >   
      
   Facts are expressions of language that are necessarily true.   
   Without language the world is merely a continuous stream   
   of physical sensations not even a "state of affairs" exists.   
      
   >>   
   >> ∀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...   
   >>    Prawitz, D. (2012). Truth as an Epistemic Notion. Topoi, 31(1), 9–16   
   >>    https://doi.org/10.1007/s11245-011-9107-6   
   >   
   > Note, "approximate".   
      
   "appropriate" not "approximate"   
      
   >   
   > You are quoting ideas about general Philosophy, which seek a way to try   
   > to define what truth means, vs Formal Logic, which STARTS with a   
      
   incorrect   
      
   > definition of what Truth is, a definition you began by changing, and   
   > thus you need to reexamine ALL the works of logic to see what changes,   
   >   
   > As mentioned, it seems you lose mathematics, as you suddenly can't   
   > finitely express it without an axiom which is based on Truth-Conditional   
   > logic, which you reject.   
   >   
   >   
      
   *This is ALL that changes*   
   ∀x (Provable(T, x) ⇔ Meaningful(T, x)) --- (Schroeder-Heister 2024)   
   ∀x (Provable(x) ⇒ True(x)) --- Anchored in (Prawitz, 2012)   
      
   >>   
   >>    1.2 Inferentialism, intuitionism, anti-realism   
   >>    Proof-theoretic semantics is inherently inferential,   
   >>    as it is inferential activity which manifests itself   
   >>    in proofs. It thus belongs to inferentialism (a term   
   >>    coined by Brandom, see his 1994; 2000) according to   
   >>    which inferences and the rules of inference establish   
   >>    the meaning of expressions   
   >>    Schroeder-Heister, Peter, 2024 "Proof-Theoretic Semantics"   
   >>   
   >> https://plato.stanford.edu/entries/proof-theoretic-semantics/   
   >> #InfeIntuAntiReal   
   >>   
   >> When we understand that linguistic truth (just like   
   >> an ordinary dictionary) expressions of language only   
   >> get their semantic meaning from other expressions of   
   >> language then we directly understand entirely based on   
   >> the meaning of words that when no such connection exists   
   >> then no semantic meaning is derived.   
   >   
   > And, you thus get a system with no "root" of meaning, and thus no actual   
   > ability to prove things.   
   >   
      
   No you get a system that knows how to reject the   
   Liar Paradox as meaningless nonsense instead of   
   the foundation of Tarski Undefinability.   
      
   Here are the Tarski Undefinability Theorem proof steps   
   (1) x ∉ Provable if and only if p   
   (2) x ∈ True if and only if p   
   (3) x ∉ Provable if and only if x ∈ True.   
   (4) either x ∉ True or x̄ ∉ True;   
   (5) if x ∈ Provable, then x ∈ True;   
   (6) if x̄ ∈ Provable, then x̄ ∈ True;   
   (7) x ∈ True   
   (8) x ∉ Provable   
   (9) x̄ ∉ Provable   
      
   These two pages are his actual complete proof   
   https://liarparadox.org/Tarski_275_276.pdf   
      
   Within PTS Tarski's line (5) becomes an axiom   
   that rejects his line (3) thus causing his   
   whole proof to completely fail.   
      
   >>   
   >> When we understand this then we can see that   
   >> "true on the basis of meaning expressed in language"   
   >> is reliably computable for the entire body of knowledge   
   >> by finite string transformations applied to finite strings.   
   >>   
   >>   
   >   
   > Nope, Just shows that you don't understand what you are talking about.   
   >   
      
   No it shows that you don;t understand Proof Theoretic Semantics   
   deeply enough.   
      
   Understanding that this is true is all that you need.   
   ∀x (Provable(T, x) ⇔ Meaningful(T, x)) --- (Schroeder-Heister 2024)   
   ∀x (Provable(x) ⇒ True(x)) --- Anchored in (Prawitz, 2012)   
      
   > Either you definitions define that math (and related systems) is outside   
   > your logic, or it accepts that some things are not computable.   
   >   
   > The problem is either you accept math with its infinite chains of   
   > deduction, or you need an infinite number of "facts" to express "all   
   > knowledge".   
   >   
   > TRY to express all that can be known about arithmatic, even simple   
   > addition, without either rules that are allowed to be applied in   
   > unbounded number, or an infinite number of base axioms.   
   >   
   > Your problem is you mind just doesn't seem to understand the concept of   
   > the infinite system, bevause it is just too small.   
   >   
   >   
      
      
   --   
   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)