Jon, Edwina, Jon, I appreciate your pushing. I did anticipate most, if not all, of your objections. <https://1drv.ms/i/c/d04faa036421906b/EXKXTYjIJG1Nog1W1WdXgAoBR9g5fK25IgDQ0YmhtdNj-g> [https://ukwest1-mediap.svc.ms/transform/thumbnail?provider=spo&farmid=188926&inputFormat=PNG&cs=OTE5OWJmMjAtYTEzZi00MTA3LTg1ZGMtMDIxMTQ3ODdlZjQ4fFNQTw&docid=https%3A%2F%2Fmy.microsoftpersonalcontent.com%2F_api%2Fv2.0%2Fdrives%2Fb!lwR8cgTI1E2aKZh9t-jAqqjMU1xldHhLj_BG1XNqW06K5qeznDLVQr2fcvlp5V5v%2Fitems%2F01CEFMJ5TSS5GYRSBENVG2EDKW2VTVPAAK%3Ftempauth%3Dv1e.eyJzaXRlaWQiOiI3MjdjMDQ5Ny1jODA0LTRkZDQtOWEyOS05ODdkYjdlOGMwYWEiLCJhcHBpZCI6IjkxOTliZjIwLWExM2YtNDEwNy04NWRjLTAyMTE0Nzg3ZWY0OCIsImF1ZCI6IjAwMDAwMDAzLTAwMDAtMGZmMS1jZTAwLTAwMDAwMDAwMDAwMC9teS5taWNyb3NvZnRwZXJzb25hbGNvbnRlbnQuY29tQDkxODgwNDBkLTZjNjctNGM1Yi1iMTEyLTM2YTMwNGI2NmRhZCIsImV4cCI6IjE3NTU5NjEyMDAifQ.kW8jbhMnIEBrkw6sSIR0DSauiyv8GdLpq1LHp2KOpyDFCvsUP5e43CEVkiSCTac0jGKKLaa6mvKkp6LDxEqujgUnod_QjqgSa96QMIUmD9S4wyg_W1MeOssTRB7nDr3hYUu_Lt7ve2XBuqcu7OQK0qSrqu8CMjIQDhku1zenL7IfqSoEDQD5liBx-CE8hT7QDiplf9WggZX9i2RMyISBx5Mky1lZdAqU4XwiRYXD04mr2g9EIfLy5EwVAXGZ4kcQOJNN7vNFG8aGXkNY71ISD_mp2pdHj7ns7WWzXH289ItHWDY1kMySE239s6sOSsCEtHqsfz0M4anTyPAQf0WIUq4luDJRO8NhJXn3M8FlJisf3JYkUmMnB4om2_FZcbXYGthGHqFRLiEpYuumjitU2Uz5nHRWTRc9X6M30_D61s3ZFDAlxP6bFgamXLKum9vG.jV9XyStn47rKyREjHwIL0UqnhQAVtlxP6XOG8_LpyRQ%26version%3DPublished&width=176&height=176&cb=63891539980] [https://res-1.cdn.office.net/assets/mail/file-icon/png/cloud_blue_16x16.png]Formal Table.PNG There is a way to solve the problem you mention. Whether you'll accept it or not, I'm not sure, but it works. +---------------------------------------+ | L : Peirce Beta / Gamma (choose A/B) | +---------------------------------------+ | v +---------------------------------------+ | P : predicate-expressions available | | (lexical predicates; optionally | | + composite graph-forms under A) | +---------------------------------------+ [Objection 3: "Predicates are words, not graph constructs"] [Reply 3: Define P explicitly; state Peircean (lexical) or Formal (lexical+composite) reading] | v +---------------------------------------+ | Define syntactically: | | U(P,x) := P(x) AND forall y (P(y)->y=x) | | phi := exists x forall P not U(P,x) | +---------------------------------------+ [Objection 2: "Vague philosophical remark, not about decidability"] -Note, I don't consider it vague (I pre-empted there). [Reply 2: Treat as precise indefinability principle; formalize U and phi. Under soundness and syntactic-resource considerations this yields a decidability consequence.] | v +---------------------------------------+ | Construct model M: | | D = {a, b}; for every predicate p: | | p^M(a) iff p^M(b) | | => No predicate isolates a or b | | => M |= phi | +---------------------------------------+ [Objection 4: "LOI is an indefinite individual, not a Tarskian referent"] [Reply 4: Terminology only — state LOI <-> ∃x mapping explicitly; in M the LOI corresponds to some a∈D] | v +---------------------------------------+ | Provability in L? | +---------------------------------------+ | ---------------------+--------------------- | | v v +-------------------------------+ +-------------------------------+ | If L |- phi (proof exists) | | If L |- not-phi | +-------------------------------+ +-------------------------------+ | CONTRADICTION with -S: | | CONTRADICTION with M: | | a proof that genuinely singles| | by soundness, if L proved | | out a witness would produce a | | not-phi then M |= not-phi, | | defining predicate, contradict| | contradicting M |= phi. | | forall P not U(P,x). | +-------------------------------+ +-------------------------------+ \______________________________________________/ | v +---------------------------------------+ | Conclusion: | | phi is true in M but undecidable in L | | (structural / semantic incompleteness)| +---------------------------------------+ [Objection 5: "Peirce meant universality; also strict Beta cannot scribe self-referential φ"] [Reply 5: Universality stronger than needed — an existential instance suffices. For scribing φ: either (A) adopt modest second-order object-language (recommended), or (B) treat φ as a Gamma metalanguage assertion (historically faithful). State choice.]
---------------------------------------------------------------------------------------------------------------------------------------- That's the simplest reply I can give you now which is provisional to your concerns as I understand them. The overall argument/proof, to me, is basically done unless a serious (I'm not saying your objections are not serious, only that I anticipated them insofar as I have had to write this a thousand or more times) objection occurs (also, I'll get to them in far more precision before I merely state that I have overcome them — this post is not that post which as I'm sure you're aware will require time). The interesting part is I tend to agree with you on universality and well-defined logical formula — these are objections I listed in the previous post for certain key reasons, not necessarily because I agree with them but only because either (a) they would be raised or (b) would have a bearing on the systematicity of the proofing. When I assume universality, I can make the same proof (in a different way) but it raises other objections regarding the scope of inquiry. My prefferred method is to go from minimal, in all respect, to whatever maximal there is. That is, I am also dealing with the "universal" aspect of Peirce's 5.5252 — it just isn't in this version as such. tl;dr I appreciate your prodding. I'll get back to you with the final product which will take your concerns into account. I've included a formal table which is prior to your latest ask for more definite terms — it will help overall in terms of situation and temporal progression (if any is interested in such). Edwina, I too am interested in object-definitions. There's no doubt, to me, that there is an obvious universe-external object (if only to the universe of discourse, but likely beyond even that). I have no idea what it is, mind. Just that I seem to think it has to exist - whatever it is. Best Jack ________________________________ From: [email protected] <[email protected]> on behalf of Jon Alan Schmidt <[email protected]> Sent: Friday, August 22, 2025 11:26 PM To: Peirce-L <[email protected]> Subject: Re: [PEIRCE-L] Peirce and Incompleteness -- Why the Parsimony of "Credit"? Jack, List: I was going to complain that your initial reply to my previous post in this thread still did not answer my direct questions, but I am now happy to say that your subsequent "relatively minimal response" helpfully clarifies what you have in mind--and why I continue to disagree. I sincerely appreciate it. JRKC: Peirce’s statement about “indescribable subjects” is a vague philosophical remark unrelated to decidability. I would say instead that Peirce's statement about "indescribable subjects" is a precise logical principle unrelated to decidability. JRKC: Let P be the set of all predicates expressible in L. This corresponds to all forms of description that words (or graph constructs) in the system can convey. The inclusion of the portion in parentheses renders this definition incorrect. In the Beta and Gamma parts of Existential Graphs (EG)--specified here as L--all predicates are general concepts, which are only represented by words, not by "graph constructs." JRKC: A subject S is a semantic referent (truth-maker) in a model M. I am not sure about this definition. In Beta/Gamma EG, a subject is an indefinite individual, which is denoted by a heavy line of identity. As you note, this corresponds to an existentially quantified variable in first-order predicate logic. JRKC: Peirce’s Axiom (From CP 5.525): There exists a proposition ϕ∈L such that: ϕ asserts: “The subject of this proposition is indescribable by P.” This formalization is incorrect. Peirce states in CP 5.525 (c. 1905) that every proposition--not just some propositions, let alone that one specific proposition--has a subject that is indescribable using words, so it must always be indicated or found instead. As he put it two decades earlier, "the subject of discourse ... can, in fact, not be described in general terms; it can only be indicated. The actual world cannot be distinguished from a world of imagination by any description. Hence the need of pronouns and indices, and the more complicated the subject the greater the need of them" (CP 3.363, EP 1:227, 1885). The line of identity in Beta/Gamma EG also fulfills that need for such indexical signs, and it is the Beta part's only axiom in addition to the blank sheet representing the inexhaustible continuum of propositions that are true within the universe of discourse, any of which may be distinctly scribed on it as a graph. In any case, I am not aware of any way to represent a self-referencing proposition such as ϕ in Beta/Gamma EG; are you? If so, how exactly would you scribe the graph of ϕ? If not, then the key premiss that ϕ exists in L is false. JRKC: Assume L is consistent and can express meta-propositions (Gamma graphs). As discussed on the List a while back, Peirce indeed provides a notation in Gamma EG for metalanguage, enabling a proposition to refer to another proposition. However, propositions are obviously signs, and thus not individuals; therefore, a heavily drawn line of identity cannot denote a proposition. Instead, Peirce initially proposes a lightly drawn line attached to words expressing predicates and enclosing the referenced proposition in an oval (RLT 151, 1898), which he later changes to a dotted line (CP 4.471, LF 2/1:165-6, 1903). Moreover, just as existence is not a predicate that can be attributed to things, truth is not a predicate that can be attributed to propositions in any part of EG. Instead, every unenclosed line of identity scribed on the sheet is asserted to exist within the universe of discourse, and every graph scribed on the sheet is asserted to be true within the universe of discourse--including all the theorems that can be derived directly from the blank, which are true in any universe of discourse (tautologies). Taken together, these observations again lead me to wonder if our differences are rooted in the longstanding (and likely unresolvable) debate between nominalism and scholastic realism. I acknowledge that you are still working on what will hopefully be an even more perspicuous explanation, so please do not feel obligated to say anything further until you are ready to share that. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt<http://www.LinkedIn.com/in/JonAlanSchmidt> / twitter.com/JonAlanSchmidt<http://twitter.com/JonAlanSchmidt> On Fri, Aug 22, 2025 at 9:21 AM Jack Cody <[email protected]<mailto:[email protected]>> wrote: Hi Jon, List So i've gone through your appeal for me to clarify terms more precisely. As this was slightly tangential to what I was working on at the time, I've spent only a day or so to handle the general claims and I can present a relatively minimal response (minimalist claims). I've styled it as minimal claim (1) and then series of objections (2) (all of which objections, asking for more nuance, are fair). I hope that helps. Once again, I have two theses at once here: minimal and maximal. The minimal is what I present below (and it can be revised if one wishes — but the gist is given in the claim). If any problems with terminology or extension, happy to take into account and reformat — there' s a complete methodology scetion, longer than this respones, which is not really present here. ________________________________ Claim Peirce’s CP 5.525 contains a meta-proposition that foreshadows Gödel/Tarski incompleteness. ________________________________ Objection (Summary) “Incompleteness” is a technical property of formal systems (undecidability of theorems). Peirce’s statement about “indescribable subjects” is a vague philosophical remark unrelated to decidability. ________________________________ Refutation The objection conflates symptom (undecidability) with cause (semantic indefinability). Peirce’s principle is a structural claim about all representation, which—when formalized—implies incompleteness. Here is a minimal proof within Peirce’s framework, now including methodology. ________________________________ 1. Definitions (Within Peirce’s Graph Logic) * Let L be a formal system (Peirce’s Beta/Gamma graphs). * Let P be the set of all predicates expressible in L. This corresponds to all forms of description that words (or graph constructs) in the system can convey. * A subject S is a semantic referent (truth-maker) in a model M. * S is describable in L iff some predicate ψ∈P uniquely picks out S in M. * S is indescribable iff no such ψ exists. ________________________________ 2. Peirce’s Axiom (From CP 5.525) There exists a proposition ϕ∈L such that: * ϕ asserts: “The subject of this proposition is indescribable by P.” * Formally: ϕ≡∃S(∀ψ∈P:¬[ψ uniquely describes S]) ________________________________ 3. Methodology / Derivation of +S and -S Step 1: Predicate space → subject S * P represents the full expressive capacity of the formal system (all words or graph predicates). * Construct the meta-proposition ϕ that quantifies over all predicates in P: ϕ≡∃S∀ψ∈P¬UniqueM(ψ,S) * This step derives S as the existential witness of the proposition. * The line of identity in Beta/Gamma graphs allows existential instantiation: it marks a semantic referent whose existence is asserted by the graph. ________________________________ Step 2: +S / -S * +S (existence): The line of identity guarantees a semantic referent exists in the model M. * -S (indescribability): By construction of ϕ, no predicate in P can uniquely describe S. This is a syntactic limit: the system lacks the expressive power to fully name its own truth-maker. Formally: M⊨+SandM⊨−S ________________________________ Step 3: Line of identity ≈ existential quantifier * In Peirce’s Beta/Gamma graphs, a line of identity asserts that “some thing exists” without naming it. * Formally, this corresponds to ∃S in predicate logic. * The line anchors a semantic referent in the model, allowing one to talk about its properties (or lack thereof) relative to the predicates in P. Hence, in the minimal response-proof: Line of identity≡existential quantifier ∃S This is why the construction of ϕ is both ontologically grounded (+S) and syntactically constrained (-S). ________________________________ 4. Proof of Incompleteness 1. Assume L is consistent and can express meta-propositions (Gamma graphs). 2. By Peirce’s Axiom, ϕ exists in L. 3. Let M be a model where ϕ is true. Then: * +S: There exists a subject S (line of identity → existential witness). * -S: S is indescribable (no ψ∈P uniquely describes it). 4. Consider provability: * If L could prove ϕ, it would describe S (contradiction with -S). * If L could disprove ϕ, it would assert S is describable, which is false in M. 5. Therefore, ϕ is true in M but undecidable in L. ________________________________ 5. Conclusion * ϕ is a true but undecidable proposition, showing incompleteness. * Cause: semantic indefinability of S (+S exists, -S is indescribable) implies syntactic undecidability. * The line of identity functions as the existential quantifier that allows this reasoning to occur formally. ________________________________ 6. Why This Refutes the Objection Objection Response “Incompleteness is about decidability, Peirce is vague” Semantic indefinability (+S / -S) directly causes undecidability in the system. Universality Only existential: some propositions are undecidable, not all. Semantic vs. syntactic conflation +S is ontological; -S is syntactic relative to P. “Not formal / outside the system” All resources (lines of identity, predicate-space P, meta-propositions) are within L. Numeric diagonalization unnecessary Meta-level quantification suffices; no Gödel numbering needed. Robust to variant models Any model satisfying the assumptions yields a similar witness S with +S / -S. Best, Jack On Wed, Aug 20, 2025 at 6:24 PM Jack Cody <[email protected]<mailto:[email protected]>> wrote: Hi Jon, That publication is a placeholder. It has absolutely zero value beyond the general concept — thus you can see how messy it is but the kernel compoments exist there. I.e., if it is cited in twenty years, I can demonstrate proof of concept (though I have internal publishings which can do that also). I would direct your attention to Tarski's general threoem and then ask you to study Godel's and try to understand, within that context, why Peirce's 5.525 is highly relevant. It can be understood far more broadly than even Tarski understands and there are problems with Tarski's own logic (in terms of realizing the full solution). He doesn't understand the "meta" aspect even as he invokes it, for instance. I will be more than clear, anyway, about all these things in due time. Briefly? All representational systems, and all possible, as I can eventually show, are incomplete (no matter how one wishes to define it or describe it — as Tarski or Godel, wisely of course, set minimal limits — but then miss entirely, though Tarski comes closer, to the general fact that all representational systems are incomplete). By the time I'm ready to respond to you, it will be very clear with much more minimal assumptions and so forth. I have no idea how you came across that (google?) because it's not something I put out there for any reason other than to "placehold". Like registering a website name I want to use ten years down line if you catch my drift. I wouldn't cite it in that formal presentation ever — genuinely. The only work it does is link some key concepts which within the context of an actual long-essay makes more than perfect sense. Yes, I appreciate your thoughts on Peirce. I think we've been over that. More on me than you to re-contextualize that and see where it goes. I don't disregard your opinions/thoughts — in fact, they work well as that against/with which I myself formalize certain responses. But readiness is not there. https://zenodo.org/records/14777823 That's the only other use I ever made of that website. Again, a place-holder (not a final product). But not even relevant to this discussion — I just post it here so you might understand that "proof" (if I even used that term — I cannot recall) is used loosely in that publication. It's an archetypal logical outline which, there, is ironcially very much incomplete. Not to be really taken as anything other than the placeholder it is. I suppose the original post to this thread is similar. By the time the full-length essay is done I'm certain you'll understand what I mean when I link 5.525 directly to incompleteness theorems (though if you play around with the internal logic and look more to Tarski than Godel — his logical statements — you might understand some of it already). Best, Jack
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