XPost: comp.theory, sci.math, comp.ai.philosophy   
   XPost: comp.lang.prolog   
   From: polcott333@gmail.com   
      
   On 1/16/2026 5:21 PM, Richard Damon wrote:   
   > On 1/16/26 5:09 PM, olcott wrote:   
   >> On 1/16/2026 3:54 PM, Richard Damon wrote:   
   >>> On 1/16/26 3:51 PM, olcott wrote:   
   >>>> On 1/16/2026 2:34 PM, Richard Damon wrote:   
   >>>>> On 1/16/26 3:24 PM, olcott wrote:   
   >>>>>> On 1/16/2026 1:34 PM, Richard Damon wrote:   
   >>>>>>> On 1/16/26 2:16 PM, olcott wrote:   
   >>>>>>>> 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.   
   >>>>>>>   
   >>>>>>> I did. Where do you actually define the initial axioms of your   
   >>>>>>> syste,/   
   >>>>>>>   
   >>>>>>>>   
   >>>>>>>>> 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?   
   >>>>>>>   
   >>>>>>> So. how is your definition of the criteria to be non-well-founded   
   >>>>>>> not non-well-founded for some questions?   
   >>>>>>>   
   >>>>>>> Note, asking LLMs for a definition doesn't define it in your system.   
   >>>>>>>   
   >>>>>>>>   
   >>>>>>>> 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   
   >>>>>>>>   
   >>>>>>>>   
   >>>>>>>   
   >>>>>>> And, thus, your "definition" of non-well-founded   
   >>>>>>   
   >>>>>> Is the standard definition in truth theoretic semantics making   
   >>>>>> "true on the basis of meaning expressed in language"   
   >>>>>> reliably computable for the entire body of knowledge.   
   >>>>>>   
   >>>>>> This includes expressing all of PA in a complete system.   
   >>>>>>   
   >>>>>   
   >>>>> I think not.   
   >>>>>   
   >>>>> One problem you are going to run into is that this "entire body of   
   >>>>> knowledge" is itself not built on those semantics,   
   >>>>>   
   >>>>   
   >>>> I knew that this would be philosophically too deep   
   >>>> for you so I am using PA to build a bridge.   
   >>>>   
   >>>>> It is a problem trying to process "knowledge" based on a different   
   >>>>> logic than the logic you are trying to process it.   
   >>>>>   
   >>>>> Also, part of our knowledge is about mathematics, which, for   
   >>>>> instance will assert that the Goldbach Conjecture is one of the   
   >>>>> great puzzles of mathematics, and must either be true or false, but   
   >>>>> that FACT is incompatible with proof-theoretic semantics, as   
   >>>>> mathematics can show that some true statements do not have proofs   
   >>>>> in the system.   
   >>>>>   
   >>>>   
   >>>> You seem to keep forgetting the specified domain   
   >>>> is the body of knowledge that is   
   >>>> "true on the basis of meaning expressed in language"   
   >>>   
   >>> Which means NOTHING about the real world, only man's own   
   >>> classification of things.   
   >>>   
   >>   
   >> When viewed within proof theoretic semantics it   
   >> specifies a precisely defined and coherent set   
   >> that shows all of the details of exactly how   
   >> conventional logic diverges from correct reasoning.   
   >   
   > No, it shows how your concept of "correct reasoning" is just defective.   
   >   
      
   A sentence is meaningful only if its justification graph   
   is well‑founded. A well‑founded graph always has a terminating   
   evaluation. Truth is defined as the result of that terminating   
   evaluation. Any sentence whose justification graph is   
   non‑well‑founded has no terminating evaluation, so it is   
   not meaningful and not truth‑apt. Therefore truth is total   
   and computable over the meaningful fragment.   
      
      
   --   
   Copyright 2026 Olcott

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