XPost: comp.theory, sci.logic, sci.math   
   XPost: comp.lang.prolog   
   From: polcott333@gmail.com   
      
   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.   
      
   We do not get the psychotic nonsense that   
   (A & ~A) Proves that Donald Trump is Jesus the Christ.   
      
   the principle of explosion is the law according to   
   which any statement can be proven from a contradiction.   
      
   https://en.wikipedia.org/wiki/Principle_of_explosion   
      
   *Proof Theoretic Semantics Blocks Pathological Self-Reference*   
   https://philpapers.org/archive/OLCPTS.pdf   
   Furthermore all undecidability and incompleteness is blocked.   
      
   > So, it can't talk about things like Global Warming, or f the Earth is   
   > Round.   
   >   
   >>   
   >>> Thus, your system colapses in a contradiction that the statement   
   >>> might be not-well-founded, but that classification might be not-well-   
   >>> founded, and that determination may be not-well-founded, and so on,   
   >>> so your attempt to define you system runs into a possibly infinite   
   >>> loop of asking if we can even talk about the statement.   
   >>>   
   >>   
   >> My paper already explains all of the details of that.   
   >>   
   >> Proof Theoretic Semantics Blocks Pathological Self-Reference   
      
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