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 / twitter.com/JonAlanSchmidt On Fri, Aug 22, 2025 at 9:21 AM Jack Cody <[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]> 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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