XPost: comp.theory, sci.logic, sci.math   
   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.   
   >   
   >>   
   >> We do not get the psychotic nonsense that   
   >> (A & ~A) Proves that Donald Trump is Jesus the Christ.   
   >   
   > Which only happens in incoherent systems like yours.   
   >   
   >>   
   >> the principle of explosion is the law according to   
   >> which any statement can be proven from a contradiction.   
   >   
   > No, it says that if a systems says that a contradiction can be proven   
   > true, then you can prove anything you want in the system.   
   >   
      
   I quoted the words that it said sheep dip !!!   
      
   > Remember, a PROOF must be based on true statements. Thus to prove   
      
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