Jack, List. You wrote: > 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.
Edwina: No - I didn’t at all mean that there is an 'external object' that is external -to-the-universe!! I concur with Peirce that there is nothing outside of the universe. But INSIDE the universe [ if one can even acknowledge such boundaries] , I acknowledge his outline of three objects; two of them are within MY current semiosic interaction - the Dynamic and the Immediate. One of them, the External Object, is in some other semiotic interaction which has nothing immediately to do with me. Again, Peirce outlined the three types very well in his example of the weather in 8.314. Note: The reason I suggest that the External Object is in some other semiotic interaction, is my view that ALL entities in the universe are in semiotic interactions with their environment - whether it be a tree interacting with the soil, or with the caterpillars eating those leaves or.. Edwina > On Aug 23, 2025, at 6:04 AM, Jack Cody <[email protected]> wrote: > > Jon, Edwina, > > Jon, I appreciate your pushing. I did anticipate most, if not all, of your > objections. > > Formal Table.PNG > > <https://1drv.ms/i/c/d04faa036421906b/EXKXTYjIJG1Nog1W1WdXgAoBR9g5fK25IgDQ0YmhtdNj-g> > 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 > Assume L is consistent and can express meta-propositions (Gamma graphs). > By Peirce’s Axiom, ϕ exists in L. > 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). > 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. > 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 > _ _ _ _ _ _ _ _ _ _ > ► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON > PEIRCE-L to this message. PEIRCE-L posts should go to [email protected] . > ► <a href="mailto:[email protected]?subject=SIG%20peirce-l";>UNSUBSCRIBE FROM > PEIRCE-L</a> . But, if your subscribed email account is not your default > email account, then go to > https://list.iu.edu/sympa/signoff/peirce-l . > ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; and > co-managed by him and Ben Udell.
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