XPost: comp.theory, sci.math, sci.lang.semantics   
   XPost: comp.ai.nat-lang   
   From: polcott333@gmail.com   
      
   On 1/22/2026 6:42 AM, Richard Damon wrote:   
   > On 1/21/26 10:14 AM, olcott wrote:   
   >> On 1/21/2026 6:38 AM, Richard Damon wrote:   
   >>> On 1/20/26 11:49 PM, olcott wrote:   
   >>>> On 1/20/2026 10:00 PM, Richard Damon wrote:   
   >>>>> On 1/20/26 1:13 PM, olcott wrote:   
   >>>>>> On 1/19/2026 11:29 PM, Richard Damon wrote:   
   >>>>>>> On 1/19/26 12:56 PM, olcott wrote:   
   >>>>>>>> Back in 2020 I proved that Wittgenstein was correct   
   >>>>>>>> all along. His key essence of grounding truth in   
   >>>>>>>> well-founded proof theoretic semantics did not exist   
   >>>>>>>> at the time that he made these remarks. Because of   
   >>>>>>>> this his remarks were misunderstood to be based   
   >>>>>>>> on ignorance instead of the profound insight that   
   >>>>>>>> they really were.   
   >>>>>>>>   
   >>>>>>>   
   >>>>>>> Nope.   
   >>>>>>>   
   >>>>>>>> According to Wittgenstein:   
   >>>>>>>> 'True in Russell's system' means, as was said: proved   
   >>>>>>>> in Russell's system; and 'false in Russell's system'   
   >>>>>>>> means: the opposite has been proved in Russell's system.   
   >>>>>>>> (Wittgenstein 1983,118-119)   
   >>>>>>>   
   >>>>>>> Which is only ONE interpretation, (and not a correct one).   
   >>>>>>>   
   >>>>>>   
   >>>>>> All we need to do to make PA complete   
   >>>>>> is replace model theoretic semantics   
   >>>>>> with wellfounded proof theoretic sematics   
   >>>>>> and ground true in OA the way Haskell   
   >>>>>> Curry defines it entirely on the basis   
   >>>>>> of the axioms of PA,   
   >>>>>   
   >>>>> Nope, doesn't work.   
   >>>>>   
   >>>>> THe system breaks as it can't consistantly determine the truth   
   >>>>> value of some statements.   
   >>>>   
   >>>> Just to make it simpler for you to understand think   
   >>>> of it as a truth and falsity recognizer that gets   
   >>>> stuck in an infinite loop on anything else.   
   >>>> So PA is complete for its domain.   
   >>>   
   >>> Nope, as your idea to make it complete breaks everything.   
   >>>   
   >>   
   >> You keep asserting that it “breaks everything,”   
   >> but you haven’t identified a single axiom of   
   >> PA, rule of inference, or valid derivation that fails.   
   >   
   > What fails, is your definition of truth.   
   >   
   >>   
   >> The recognizer does exactly what it’s supposed to:   
   >> – returns true when PA proves ϕ   
   >> – returns false when PA proves ¬ϕ   
   >> – diverges on anything PA cannot settle   
   >   
   > But your "not-well-founded" isn't a REcOGNIZER, it is a PREDICATE, which   
   > ALWAYS needs to return a value.   
   >   
   >>   
   >> That’s not breaking anything.   
   >> That’s the definition of a recognizer.   
   >>   
   >> So what, specifically, do you think is broken?   
   >   
   > You definition of "Truth", which can't have a value by your logic.   
   >   
   >>   
   >>>>   
   >>>>>   
   >>>>>>   
   >>>>>> ∀x ∈ PA ((True(PA, x)  ≡ (PA ⊢ x))   
   >>>>>> ∀x ∈ PA ((False(PA, x) ≡ (PA ⊢ ~x))   
   >>>>>> ∀x ∈ PA (~WellFounded(PA, x) ≡ (~True(PA, x) ∧ (~False(PA, x))   
   >>>>>> Then PA becomes complete.   
   >>>>>   
   >>>>> And, in proof-theoretic semantics, this is just not-well-founded as   
   >>>>> there are statements that you can not determine if any of these are   
   >>>>> applicable or not.   
   >>>>>>   
   >>>>>> This is very similar to my work 8 years ago   
   >>>>>> where the axioms are construed as BaseFacts.   
   >>>>>> It was pure proof theoretic even way back then.   
   >>>>>>   
   >>>>>> The ultimate foundation of [a priori] Truth   
   >>>>>> Olcott Feb 17, 2018, 12:42:55 AM   
   >>>>>> https://groups.google.com/g/sci.logic/c/dbk5vsDzZbQ/m/4ajW9R08CQAJ   
   >>>>>   
   >>>>> At least that accepted that there were statement that it couldn't   
   >>>>> handle as they were neiteher true or false.   
   >>>>>   
   >>>>> With your addition, we get that there are statements that can be   
   >>>>> none of True, False, or ~WellFounded.   
   >>>>>   
   >>>>   
   >>>> This was the earliest documented work that   
   >>>> can be classified as well-founded proof theoretic semantics.   
   >>>> My actual work is documented to go back to 1998.   
   >>>   
   >>   
   >> An BaseFact is an expression X of (natural or formal)   
   >> language L that has been assigned the semantic property   
   >> of True. (Similar to a math Axiom).   
   >>   
   >> A Collection T of BaseFacts of language L forms the   
   >> ultimate foundation of the notion of Truth in language L.   
   >>   
   >> To verify that an expression X of language L is True or   
   >> False only requires a syntactic logical consequence   
   >> inference chain (formal proof) from one or more elements   
   >> of T to X or ~X.   
   >>   
   >> True(L, X) ↔ ∃Γ ⊆ BaseFact(L) Provable(Γ, X)   
   >> False(L, X) ↔ ∃Γ ⊆ BaseFact(L) Provable(Γ, ~X)   
   >   
   > And what it the provable truth value of Godel's G statement?   
   >   
   > It can't be True, since it turns out to not be provable.   
   >   
   > It can't be False, as no number exists to make it false.   
   >   
   > It can't be Proven Not-Well-Founded, as proving that it can't be false,   
   > establishes that no such number exists, which makes it true in the system.   
   >   
   > Thus, your definition of "Truth" as being True/False/Not-Well-Founded is   
   > just self-contradictory.   
   >   
   > All you are doing is your normal back-pedeling and dupliciously changing   
   > you claim that actually negates your other position.   
   >   
   >>   
   >>> But it isn't well-founded, as it isn't actualy based on proof.   
   >>>   
   >>   
   >> True(L, X) means: there exists a proof of X from the base facts   
   >>   
   >> False(L, X) means: there exists a proof of ¬X from the base facts   
   >>   
   >> Everything else → the recognizer diverges (no proof either way)   
   >   
   > In other words, your "Proff-Theoretic" system is actually Truth-   
   > Conditional, and thus you can't use it.   
   >   
   >>   
   >> That is proof‑theoretic semantics.   
   >  > > It is literally the definition of truth in a proof‑theoretic   
   > framework.   
   >   
   > Which means proof-theoretic needs truth-conditional to be accepted by   
   > your logic.   
   >   
   > Proof-Theoretic can work if it says that it just can't handle some   
   > statements like G.   
   >   
   > Which is an admission of its own limitations.   
   >   
   > Proof-Theoretic ADMITS it is incomplete in PA, as there are statements   
   > it can not determine if they are true, false, or neither in the system,   
   > because a "proof" on not being true or false actually establishes the   
   > statement as true (or for other statments, that they are false).   
   >   
   >   
   >>   
   >>>>   
   >>>>>>   
   >>>>>>>>   
   >>>>>>>> Formalized by Olcott as:   
   >>>>>>>>   
   >>>>>>>> ∀F ∈ Formal_Systems ∀𝒞 ∈ WFF(F) (((F⊢𝒞)) ↔ True(F,   
   𝒞))   
   >>>>>>>> ∀F ∈ Formal_Systems ∀𝒞 ∈ WFF(F) (((F⊬𝒞)) ↔   
   ¬True(F, 𝒞))   
   >>>>>>>> ∀F ∈ Formal_Systems ∀𝒞 ∈ WFF(F) (((F⊢¬𝒞)) ↔   
   False(F, 𝒞))   
   >>>>>>>   
   >>>>>>> Which can be not-well-founded, as determining *IF* a statement is   
   >>>>>>> proveable or not provable might not be provable, or even knowable.   
   >>>>>>>   
      
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