XPost: comp.theory, sci.math, comp.ai.philosophy   
   XPost: comp.lang.prolog   
   From: polcott333@gmail.com   
      
   On 1/17/2026 6:03 AM, Richard Damon wrote:   
   > On 1/16/26 10:42 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   
      
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