XPost: comp.theory, sci.logic, sci.math   
   XPost: comp.lang.prolog   
   From: polcott333@gmail.com   
      
   On 1/16/2026 9:24 PM, Richard Damon wrote:   
   > On 1/16/26 8:27 PM, olcott wrote:   
   >> 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   
      
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