XPost: comp.theory, sci.math, sci.lang   
   XPost: comp.lang.prolog   
   From: polcott333@gmail.com   
      
   On 2/6/2026 6:23 PM, Richard Damon wrote:   
   > On 2/6/26 7:10 PM, olcott wrote:   
   >> On 2/6/2026 5:18 PM, Richard Damon wrote:   
   >>> On 2/6/26 3:00 PM, olcott wrote:   
   >>>> On 2/6/2026 12:15 PM, Richard Damon wrote:   
   >>>>> On 2/6/26 10:30 AM, olcott wrote:   
   >>>>>> On 2/6/2026 3:01 AM, Mikko wrote:   
   >>>>>>> On 05/02/2026 18:55, olcott wrote:   
   >>>>>>>   
   >>>>>>>> Changing the foundational basis to Proof Theoretic Semantics   
   >>>>>>>> Tarski Undefinability is overcome   
   >>>>>>>>   
   >>>>>>>> x ∈ Provable ⇔ x ∈ True // proof theoretic semantics   
   >>>>>>>   
   >>>>>>> A definition in terms of an undefined symbol does not really define.   
   >>>>>>>   
   >>>>>>   
   >>>>>> It is an axiom: ∀x (Provable(x) ⇒ True(x))   
   >>>>>   
   >>>>> But the axiom uses ⇒ which goes in just one direction, while you   
   >>>>> statements used ⇔ which attempts to go both ways.   
   >>>>>   
   >>>>   
   >>>> This was corrected by an expert that seems   
   >>>> to really know these things.   
   >>>>   
   >>>> This same expert agrees that with within PTS:   
   >>>> "if x is provable, then it is true."   
   >>>>   
   >>>   
   >>> Right, Provable leads to Truth. But Not Provable does not mean not   
   >>> true, or Truth require provability by the axiom.   
   >>>   
   >>> I gues you are just admitting that you are just a pathetic liar.   
   >>>   
   >>>   
   >>>>>   
   >>>>>>   
   >>>>>> There are dozens of papers needed to verify this.   
   >>>>>> It will take me quite a while to form proper citations   
   >>>>>> of these papers. It is anchored in proof theoretic semantics.   
   >>>>>> Generic PTS states that ~Provable(x) ⇔ Meaningless(x).   
   >>>>>> Model theory and truth conditional semantics are rejected.   
   >>>>>>   
   >>>>>   
   >>>>> And, I think your problem is you don't actually understand what you   
   >>>>> are reading. This shows in that you have been making the claim for   
   >>>>> years, but you are now admitting you can't ACTUALLY show why it is   
   >>>>> (yet).   
   >>>>>   
   >>>>   
   >>>> ∀x  (~Provable(x) ⇔ Meaningless(x))   
   >>>> Seems to be exactly and precisely what Proof Theoretic   
   >>>> Semantics actually says. Since the SEP article was   
   >>>> written by the guy that coined the term:   
   >>>> "Proof Theoretic Semantics"   
   >>>> It should be pretty definitive.   
   >>>   
   >>> No, which is part of your problem. Proof-Theoretic Semantics say we   
   >>> can't talk about the truth of a statement we can not prove, NOT that   
   >>> the statement can't be true without the proof, just we can't talk   
   >>> about it.   
   >>>   
   >>   
   >> Lets try to say this exactly accurately.   
   >> In PTS expressions that are unprovable are   
   >> ungrounded in semantic meaning.   
   >   
   > Right, which means you can't talk about them.   
   >   
      
   Your way of saying it is way too weak.   
   Is gibberish nonsense is more accurate.   
      
   >>   
   >> When I refer to a formal system I am referring to   
   >> Russell's atomic facts written down and placed in   
   >> a simple Type Hierarchy.   
   >   
   > Then you aren't talking about a real Formal System.   
   >   
   > This is your problem, You don't understand what a Formal system actually   
   > is.   
   >   
   > You keep on thinking it is just a form of Philosophy, which it really   
   > isn't.   
   >   
   >>   
   >>    atomic facts, which consist either of a simple   
   >>    particular exhibiting a quality, or multiple   
   >>    simple particulars standing in a relation.   
   >> https://plato.stanford.edu/entries/logical-atomism/   
   >   
   > So, how do you fit Peano Arithmatic into that system?   
   >   
   >>   
   >>> Proof-Theoretic Semantics limits our way of looking at things to what   
   >>> can be proven, and things outside of what can be proven are just   
   >>> outside the domain of discussion.   
   >>>   
   >>   
   >> "true on the basis of meaning expressed in language"   
   >> necessarily includes the entire body of knowledge   
   >> expressed in language.   
   >   
   > Try to do it.   
   >   
   > Since that body of knowledge expresses facts about mathematics and such   
   > system, it include things like the Pythagorean Theorem, which is *NOT*   
   > true just on the meaning of its words,   
      
   "true on the basis of meaning expressed in language"   
      
   NOT THE MEANING OF WORDS   
   NOT THE MEANING OF WORDS   
   NOT THE MEANING OF WORDS   
   NOT THE MEANING OF WORDS   
   NOT THE MEANING OF WORDS   
   NOT THE MEANING OF WORDS   
      
   *THE MEANING IN ALL FORMAL MATHEMATICAL AND NATURAL LANGUAGES*   
   *THE MEANING IN ALL FORMAL MATHEMATICAL AND NATURAL LANGUAGES*   
   *THE MEANING IN ALL FORMAL MATHEMATICAL AND NATURAL LANGUAGES*   
      
   I spent 25 years coming up with that you could   
   take two minutes to pay COMPLETE ATTENTION.   
      
      
   >>>   
   >>> Since the definition of arithmatic of Natural Numbers falls apart if   
   >>> you try to force this on it, all you are doing is saying that you   
   >>> logic can't handle mathematics.   
   >>>   
   >>   
   >> ∀x (Provable(PA, x)  ⇒ True(PA, x))   
   >> ∀x (Provable(PA, ~x) ⇒ False(PA, x))   
   >> ∀x (~True(PA, x) ∧ ~False(PA, x) ⇔ ~Truth_Apt(PA, x))   
   >>   
   >>   
   >   
   > Which isn't what Proof Theoretic says,   
   >   
   > As it doesn't introduce the concept of the predicate "True".   
   >   
   > It says if you CAN prove the statement in the system, then you can say   
   > the statement is true.   
   >   
   > And, if you CAN prove the converse of the statement in the system, then   
   > you can say the statement if false,   
   >   
   > And, if you CAN prove that the you can never do either of the above, you   
   > can say the statement is non-well-founded.   
   >   
      
   *Good job, you got the most important point exactly correctly*   
   Non-well-founded means not truth-apt.   
      
   > You might not be able to do any of the above, in which case you can't   
   > talk about the statement and it truth.   
   >   
      
   The key thing is that PTS rejects cases of pathological   
   self-reference as lacking a well-founded justification   
   tree thus semantically ill-formed within PTS.   
      
   > Since in mathematics, there ARE statements for which you can't do any of   
   > the above, Proof-Theoretic Semantic fall apart for it, as you start to   
   > run into the issue of not knowing if you can talk about the statements.   
   > It works better in simpler systems where there are many statements for   
   > which you can reduce it to one of the three cases you can talk about.   
   >   
   >   
      
   You are conflating mathematics within the foundation of   
   Truth Conditional Semantics with mathematics itself.   
      
   Mathematics within Proof Theoretic Semantics cannot   
   be incomplete.   
      
      
   --   
   Copyright 2026 Olcott

   
      
   My 28 year goal has been to make 
   
   "true on the basis of meaning expressed in language"
   
   reliably computable for the entire body of knowledge.

   
      
   This required establishing a new foundation
   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)