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| sci.logic | Logic -- math, philosophy & computationa | 262,912 messages |
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| Message 262,650 of 262,912 |
| olcott to Richard Damon |
| Re: Back in 2020 I proved that Wittgenst |
| 21 Jan 26 09:14:42 |
XPost: comp.theory, sci.math, sci.lang.semantics XPost: comp.ai.nat-lang From: polcott333@gmail.com 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. 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 That’s not breaking anything. That’s the definition of a recognizer. So what, specifically, do you think is broken? >> >>> >>>> >>>> ∀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) > 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) That is proof‑theoretic semantics. It is literally the definition of truth in a proof‑theoretic framework. >> >>>> >>>>>> >>>>>> 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. >>>>> >>>>> So, therefore you can't actually evaluate your statement. >>>>> >>>> >>>> All meta-math is defined to be outside the scope of PA. >>> >>> But we don't need "meta-math" to establish the answer. >>> >>> It is a FACT that no number will satisfy the Relationship, >> >> That relationship does not even exist outside of meta-math >> >> > > So, numbers don't exsist? > OR is it the "for all" part that doesn't exist, and thus your proof- > theoretic logic can't exist either? > > Sorry, you are just stuck trying to outlaw that which you need. PA contains arithmetic relations about numbers. It does not contain meta‑mathematical relations about PA itself. Gödel’s construction uses the latter, not the former. -- Copyright 2026 Olcott |
(c) 1994, bbs@darkrealms.ca