XPost: sci.math, comp.theory, comp.ai.philosophy   
   From: polcott333@gmail.com   
      
   On 1/18/2026 5:54 AM, Mikko wrote:   
   > On 16/01/2026 01:40, olcott wrote:   
   >> On 1/15/2026 5:50 AM, Richard Damon wrote:   
   >>> On 1/15/26 12:24 AM, olcott wrote:   
   >>>> On 1/14/2026 8:57 PM, Richard Damon wrote:   
   >>>>> On 1/13/26 1:43 PM, olcott wrote:   
   >>>>>> On 1/13/2026 6:10 AM, Richard Damon wrote:   
   >>>>>>> On 1/12/26 11:46 PM, olcott wrote:   
   >>>>>>>> On 1/12/2026 9:16 PM, Richard Damon wrote:   
   >>>>>>>>> On 1/12/26 4:41 PM, olcott wrote:   
   >>>>>>>>>> How The Well-Founded Semantics for General Logic Programs   
   >>>>>>>>>>   
   >>>>>>>>>> of (Van Gelder, Ross & Schlipf, 1991)   
   >>>>>>>>>> Journal of the Association for Computing Machinery,   
   >>>>>>>>>> volume 38, number 3, pp. 620{650 (1991).   
   >>>>>>>>>> https://users.soe.ucsc.edu/%7Eavg/Papers/wf.pdf   
   >>>>>>>>>>   
   >>>>>>>>>> handle the Liar Paradox when we construe   
   >>>>>>>>>> non-well-founded / undefined as not a truth-bearer?   
   >>>>>>>>>>   
   >>>>>>>>>> % This sentence is not true.   
   >>>>>>>>>> ?- LP = not(true(LP)).   
   >>>>>>>>>> LP = not(true(LP)).   
   >>>>>>>>>> ?- unify_with_occurs_check(LP, not(true(LP))).   
   >>>>>>>>>> false.   
   >>>>>>>>>>   
   >>>>>>>>>> WFS assigns undefined to self-referential paradoxes   
   >>>>>>>>>> without external support.   
   >>>>>>>>>>   
   >>>>>>>>>> When we interpret undefined as lack of truth-bearer   
   >>>>>>>>>> status the Liar sentence fails to be about anything   
   >>>>>>>>>> that can bear truth values   
   >>>>>>>>>>   
   >>>>>>>>>> The paradox dissolves - there's no contradiction   
   >>>>>>>>>> because there's no genuine proposition   
   >>>>>>>>>>   
   >>>>>>>>>> This is actually similar to how some philosophers   
   >>>>>>>>>> (like the "gap theorists") handle the Liar: sentences   
   >>>>>>>>>> that fail to achieve determinate truth conditions   
   >>>>>>>>>> simply aren't truth-bearers. WFS's undefined value   
   >>>>>>>>>> provides a formal mechanism for identifying exactly   
   >>>>>>>>>> these cases.   
   >>>>>>>>>>   
   >>>>>>>>>> A Subtle Point The occurs-check failure in Prolog is   
   >>>>>>>>>> slightly different from WFS's undefined assignment -   
   >>>>>>>>>> it's a structural constraint on term formation. But   
   >>>>>>>>>> both point to the same insight: circular, unsupported   
   >>>>>>>>>> self-reference doesn't create genuine semantic content.   
   >>>>>>>>>>   
   >>>>>>>>>>   
   >>>>>>>>>   
   >>>>>>>>>   
   >>>>>>>>> I thought you said that no one in the past handled the liar   
   >>>>>>>>> paradox?   
   >>>>>>>>>   
   >>>>>>>>   
   >>>>>>>> That is no one in the past handling the Liar Paradox.   
   >>>>>>>> That all happened today.   
   >>>>>>>   
   >>>>>>> So, today is 1991?   
   >>>>>>>   
   >>>>>>   
   >>>>>> The paper provides the basis for me to   
   >>>>>> handle the Liar Paradox today. The Paper   
   >>>>>> does not mention the Liar Paradox it   
   >>>>>> only shows how to implement Proof Theoretic   
   >>>>>> semantics in a logic programming system.   
   >>>>>>   
   >>>>>>>>   
   >>>>>>>>> I guess you are just admitting you are just a liar.   
   >>>>>>>>>   
   >>>>>>>>>   
   >>>>>>>>> Note, since Prolog's logic is not sufficient to handle PA,   
   >>>>>>>>   
   >>>>>>>> I never said it was. A formal system anchored in   
   >>>>>>>> Proof Theoretic Semantics is powerful enough.   
   >>>>>>>   
   >>>>>>> Nope. It can't handle PA.   
   >>>>>>>   
   >>>>>>   
   >>>>>> It definitely can. I already showed you the details   
   >>>>>> of how.   
   >>>>>   
   >>>>> Nope,  you PRESUME that Godel is non-sense.   
   >>>>>   
   >>>>   
   >>>> “When PA is interpreted within proof‑theoretic semantics, only   
   >>>> well‑founded inferential structures are admissible as meaningful   
   >>>> statements. Gödel’s diagonal construction produces an ungrounded,   
   >>>> self‑referential formula whose proof‑dependency graph contains a   
   >>>> cycle. Since such expressions are not truthbearers in this   
   >>>> framework, the classical incompleteness phenomenon does not arise.   
   >>>> PA itself remains sound and complete with respect to its grounded   
   >>>> proof rules.”   
   >>>   
   >>> In other words, you are just admitting to be an idiot that deosn't   
   >>> care what your words actually mean.   
   >>>   
   >>   
   >> The term *proof‑theoretic semantics* has always   
   >> proved my point long before I ever heard of it.   
   >   
   > A term does not prove anything. Only a proof proves.   
   >   
      
   Proof Theoretic Semantics with the notion of   
   non-well-founded expressions is the same thing   
   that I have been saying for years.   
      
   True and False in PA have always been x or ~x is   
   provable from the actual axioms of PA, otherwise   
   x is simply not a truth bearer in PA. The only   
   clarification that I make now explicitly adding a   
   truth predicate to PA.   
      
   ∀x ∈ PA ((True(PA, x)  ≡ (PA ⊢ x))   
   ∀x ∈ PA ((False(PA, x) ≡ (PA ⊢ ~x))   
   ∀x ∈ PA ((~True(PA, x) ∧ (~False(PA, x) ≡ ~TruthBearer(PA, x))   
      
      
      
   --   
   Copyright 2026 Olcott

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

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