XPost: comp.theory, sci.math, comp.ai.philosophy   
   XPost: comp.lang.prolog   
   From: news.x.richarddamon@xoxy.net   
      
   On 1/16/26 10:44 PM, olcott wrote:   
   > 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   
      
   [continued in next message]   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)