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| sci.logic | Logic -- math, philosophy & computationa | 262,912 messages |
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| Message 262,870 of 262,912 |
| olcott to Richard Damon |
| Re: Making the body of knowledge computa |
| 12 Feb 26 10:06:11 |
XPost: comp.theory, sci.math, sci.math.symbolic XPost: comp.lang.prolog, comp.software-eng From: polcott333@gmail.com On 2/12/2026 6:29 AM, Richard Damon wrote: > On 2/11/26 8:08 AM, olcott wrote: >> On 2/11/2026 6:56 AM, Richard Damon wrote: >>> On 2/10/26 11:59 PM, olcott wrote: >>>> We completely replace the foundation of truth conditional >>>> semantics with proof theoretic semantics. Then expressions >>>> are "true on the basis of meaning expressed in language" >>>> only to the extent that all their meaning comes from >>>> inferential relations to other expressions of that language. >>>> This is a purely linguistic PTS notion of truth with no >>>> connections outside the inferential system. >>>> >>>> Well-founded proof-theoretic semantics reject expressions >>>> lacking a "well-founded justification tree" as meaningless. >>>> ∀x (~Provable(T, x) ⇔ Meaningless(T, x)) >>>> >>> >>> The problem is that you new system can't handle mathematics. >>> >>> The problem, as has been pointed out, is that mathematics, by the >>> axiom of induction, accepts as true statements that can be >>> established by an infinite number of steps as true, and shows a >>> method to solve SOME of them. >>> >>> Also, "Halting" is a well-founded property of ALL machines, as they >>> MUST either Halt or not, and HALTING is always provable, so those >>> machines that do not halt, must be non-halting. >>> >>> Your "logic" essentially denies the property of the excluded middle >>> for systems that have infinite members, but some statements are >>> inherently of the class of the excluded middle. >>> >>> As I have said, TRY to show how your PTS can establish the >>> mathematics of the Natural Numbers. >>> >>> Try to even fully define ADDITION without the need for allowing >>> unbounded steps. >>> >> >> ∀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))) > > So, where is "Addition" in that? > > How do you determine ~True(PA, x)? in your proof-Theoretic semantics? > 0=1 equal(successor(0), successor(successor(0))==FALSE >> >> "What is the appropriate notion of truth for sentences whose >> meanings are understood in epistemic terms such as proof or >> ground for an assertion? It seems that the truth of such >> sentences has to be identified with the existence of proofs or >> grounds..." https://doi.org/10.1007/s11245-011-9107-6 > > A question in General Philosophy, not Formal Logic. > >> >> Spend 20 hours carefully studying this and get back to me. >> https://plato.stanford.edu/entries/proof-theoretic-semantics/ > > Which is a paper on PHILOSOPHY, not Formal Logic. > Logic and math choose a notion of truth from philosophy and the choose the wrong one. ∀x (Provable(T, x) ⇔ Meaningful(T, x)) --- (Schroeder-Heister 2024) ∀x (Provable(x) ⇒ True(x)) --- Anchored in (Prawitz, 2012) > Note, Formal Logic BEGINS with its definiton of Truth, which is based on > the, possibly infinite, application of its logical rules. > As you yourself kept harping on some times true(x) exists outside of the domain of knowledge. > To change that to a Proof-Theoretical Semantics basis changes the > results of the system. In particular, any system that generates an > infinite domain (like mathemtics) becomes problematic. > Only in cases where a truth value requires infinite inference steps. >> >> It makes "true on the basis of meaning expressed in language" >> reliably computable for the entire body of knowledge. >> > > Nope. > > IT makes your definition of "true" just a lie. > ∀x (Provable(T, x) ⇔ Meaningful(T, x)) --- (Schroeder-Heister 2024) ∀x (Provable(x) ⇒ True(x)) --- Anchored in (Prawitz, 2012) That you are woefully ignorant of PTS does not entail that I am incorrect. What you call an intentional falsehood has always been your own ignorance. > How can you "compute" if a number exist that satisfies the relationship > that Godel developes in his proof? > PTS has no notion of satisfies. > Why does that relationship, which is just built from the fundamental > operation of mathematics in PA not have "meaning"? > When any expression of language lacks a semantic connection to the expressions that define it this expression remains undefined. > Where is the line between that relationship, and the statement that we > can assert that 1 + 1 = 2? > successor(0) + successor(0) = successor(successor(0)) > Proof-Thoeretic Semantics in sets with infinite members just severly > limits what you can do in that field. > > And that is why real Proof-Theoretic Semantics doesn't assert that what > we haven't proven is meaningless, just that we don't yet know the > meaning, because it knows that all we can know is that we haven't yet > proven something, and not that it can't be proven, unless we can > actually find a proof of that (like proving its opposite). > It could be meaningless or it could be unknown either way is is outside of the domain of knowledge. > Your system becomes a lie, because while you assert you are using just > Proof-Theoretical Semantics, you need to actully have Truth-Conditional > logic to determine those semantics, as there ARE things not provablable > unprovable, and thus your tri-valued system (true, false, non-well- > founded) can't have values exstablished by proof-thoeretic logic. > Like I already said carefully study the article written by the guy that coined the term: Proof-Theoretic Semantics https://plato.stanford.edu/entries/proof-theoretic-semantics/ -- Copyright 2026 Olcott |
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