XPost: comp.theory, sci.logic, sci.math   
   XPost: comp.lang.prolog   
   From: polcott333@gmail.com   
      
   On 1/16/2026 12:52 PM, Richard Damon wrote:   
   > On 1/16/26 12:47 PM, olcott wrote:   
   >> The system uses proof-theoretic semantics, where the   
   >> meaning of a statement is determined entirely by its   
   >> inferential role within a theory. A theory T consists   
   >> of a finite set of basic statements together with   
   >> everything that can be derived from them using the   
   >> inference rules. The statements derivable in this   
   >> way are the theorems of T. A statement is true in   
   >> T exactly when T proves it. A statement is false   
   >> in T exactly when T proves its negation. Some   
   >> statements are neither true nor false in T. These   
   >> are the non-well-founded statements: statements   
   >> whose inferential justification cannot be grounded   
   >> in a finite, well-founded proof structure. This includes   
   >> self-referential constructions such as Gödel-type sentences.   
   >>   
   >> *Proof Theoretic Semantics Blocks Pathological Self-Reference*   
   >> https://philpapers.org/archive/OLCPTS.pdf   
   >>   
   >   
   > WHAT system?   
   >   
   > WHAT can you do in it?   
   >   
   > Can you actually prove that, or is it just more of your lies.   
   >   
      
   You have to actually read the paper.   
      
   > Your problem is that you system is based on a criteria that matches your   
   > own definition of non-well-founded.   
   >   
      
   What does not well-founded mean in proof-theoretic semantics?   
      
   In proof‑theoretic semantics, a statement is not well‑founded when its   
   justification cannot be grounded in a finite, well‑structured chain of   
   inferential steps. It lacks a terminating, well‑ordered proof tree that   
   would normally establish its truth or falsity. This often happens with   
   self‑referential or circular statements whose “proofs” loop back on   
   themselves rather than bottoming out in basic axioms or introduction   
   rules. // Copilot   
      
   In proof-theoretic semantics, saying that something is “not   
   well-founded” means that the structure used to define or justify   
   meanings does not rest on a base case that is independent of itself.   
   Instead, it involves circular or infinitely descending dependencies   
   among rules or proofs. // ChatGPT   
      
   In proof-theoretic semantics, not well-founded typically refers to   
   derivations or proof structures that contain infinite descending chains   
   or circular dependencies, violating the well-foundedness property.   
   In classical proof theory, well-founded derivations have a clear   
   hierarchical structure where every inference rule application depends   
   only on "smaller" or "simpler" premises, eventually bottoming out in   
   axioms or basic rules. This ensures that proofs are finitely   
   constructible and verifiable. // Claude AI   
      
   A set of introduction rules (definitional clauses) for an atom P is   
   called well-founded if every chain of successive "definitions"   
   (unfoldings) eventually terminates — i.e., there is no infinite   
   descending chain of definitional dependencies.   
   Intuitively:   
   The meaning of P is ultimately grounded in basic facts or in logical   
   structure after finitely many unfoldings. // Grok   
      
      
      
   > It seems that for many of the system you want to talk about, it is non-   
   > well-founded if statements are in fact non-well-founded because you   
   > can't KNOW if a proof exists (but isn't known yet) of the statement or   
   > its negation.   
   >   
   > This collapse your whole system into a ball of meaningless unless you   
   > restrict it to "toy" level where you can prove if a proof can exist.   
      
      
   --   
   Copyright 2026 Olcott

   
      
   My 28 year goal has been to make 
   
   "true on the basis of meaning expressed in language"
   
   reliably computable.

   
      
   This required establishing a new foundation
   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)