XPost: comp.theory, sci.math, sci.lang.semantics   
   XPost: comp.ai.nat-lang   
   From: news.x.richarddamon@xoxy.net   
      
   On 1/22/26 11:43 AM, olcott wrote:   
   > 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))   
   >>>>>>>>   
      
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