Jack, List: After so much back-and-forth, it turns out that what your proof actually demonstrates is not controversial, surprising, or even especially interesting.
JRKC: Demonstrating that Beta cannot decide this existential claim is a powerful, formal consequence of the broader Peircean principle. It shows that the system is inherently incapable of settling a question that arises directly from its own design philosophy. On the contrary, the system inherently settles the question *by virtue of* its own design philosophy. Why would we need or expect the *logical principle* that Peirce states in CP 5.525--every proposition has at least one subject that is indescribable using words, and thus must be indicated or found instead--to be a theorem of *any *formal system, including Beta/Gamma EG? It is a fundamental aspect of logic *itself*, including natural languages as well as formal systems, and using it as the basis for proving the incompleteness of the latter entirely misses the point that Peirce was making in that passage. JRKC: Peirce's Gamma EG, with its dotted lines and ovals for reference and negation, is exactly designed for this kind of meta-discourse. The proof provides a formal model (γ(M)) for what those dotted lines are doing: they are allowing us to talk about the entire set of Beta-predicates. Unless and until you provide an *actual* graph in Gamma EG that represents your target proposition, including its quantification over predicates, it is strictly a conjecture on your part that it is even *capable* of being expressed in that formal system. Also, your chatbot mistakenly seems to be saying that *dotted *ovals are for negation, when in fact they are for enclosing graphs of propositions that are serving as subjects in other propositions. Thinly drawn ovals or (more iconically) shaded areas are for negation, as derived from the implication of falsity. JRKC: The proof's core conclusion is that neither of these graphs is a theorem derivable from the blank sheet. This is not a failure of the proof; it is the proof's successful conclusion. You have stated it perfectly: "L neither proves nor disproves either one of them." This is the structural incompleteness. It is already long- and well-established that first-order predicate logic, and thus Beta EG as a version of it, is semantically complete (all tautologies are theorems) but syntactically incomplete (some true propositions are not theorems); and that second-order logic, and thus Gamma EG, is both semantically and syntactically incomplete. These and other deductive formal systems are not intended to distinguish truth from falsity across the board, they simply ensure that a false conclusion can never be obtained from true premisses. For example, in Beta EG, the graph of your existential claim can be derived from the graph of Peirce's universal claim. JRKC: The need for "collateral experience" translates, in the model-theoretic framework, to the need to choose a specific model (M or N) to determine the truth of φ. The interpreter's "collateral experience" is their knowledge of the state of the world (the model) they are reasoning about. Likewise, this is not news to anyone familiar with Peirce's thought in general and EG in particular. The blank sheet is "considered as representing the universe of discourse, and as asserting whatever is taken for granted between the graphist and the interpreter to be true of that universe" (CP 4.396, LF 2/2:351, 1903). It corresponds directly to the commens (or commind) as "all that is, and must be, well understood between utterer and interpreter, at the outset, in order that the sign in question should fulfill its function. No object can be denoted unless it be put into relation to the object of the *commens*." (EP 2:478, SS 197, 1906 Mar 9). Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt On Sat, Aug 23, 2025 at 5:38 PM Jack Cody <[email protected]> wrote: > Jon, List, > Jon I go directly to your post (treated as an objection — albeit to > certain aspects of a provisional proof but insofar as you render them and I > retain such): > > Your objection contains several key claims. The proof structure addresses > each one not by dismissing them, but by formally demonstrating their > logical consequences. > ------------------------------ > *1. On Peirce's Stronger, Universal Claim* > > - *Your Point:* Peirce's claim is universal: "no subject of any > proposition is describable using only words." > - *Proof's Response:* The proof fully acknowledges this. The universal > claim is the philosophical premise. The proof then asks: *What is the > logical consequence of this premise for the formal system Beta?* > - The universal claim ("for all subjects, they are indescribable") > logically implies the existential claim ("there exists at least one > subject > that is indescribable"). > - The proof uses this *existential instantiation* (φ ≡ ∃S...) not > to replace Peirce's universal claim, but as a minimal, sufficient > condition > to demonstrate a specific formal result: *undecidability*. > - Demonstrating that Beta cannot decide this existential claim is a > powerful, formal consequence of the broader Peircean principle. It shows > that the system is inherently incapable of settling a question that > arises > directly from its own design philosophy. > > *2. On Quantification Over Predicates and Second-Order Logic* > > - *Your Point:* Quantification over predicates feels like second-order > logic and its implementation in Gamma EG is unclear. > - *Proof's Response:* This is precisely why the proof makes a strict > distinction between the *object language* (Beta) and the *metalanguage* > (Gamma). > - *Beta is first-order.* It does not and cannot quantify over > predicates. This limitation is the entire source of the problem. > - *Gamma is a metalanguage about Beta.* Its purpose is to make > statements *about* Beta's capabilities. The quantification over > predicates (∀p in φ) happens *here*, not inside Beta. This is not > "second-order logic" in the object language; it is "meta-logic." > - Peirce's Gamma EG, with its dotted lines and ovals for reference > and negation, is exactly designed for this kind of meta-discourse. The > proof provides a formal model (γ(M)) for what those dotted lines > are *doing*: they are allowing us to talk about the entire set of > Beta-predicates. > > *3. On How ϕ is Expressed in EG & The Role of the LOI* > > - *Your Point:* You can scribe a version of ϕ by attaching an LOI to > "indescribable" or by negating "describable." > - *Proof's Response:* You are absolutely correct, and the proof > structure agrees with you. This is exactly how the Gamma-level sentence > φ would be scribed. > - The graph for "something is indescribable" (an LOI attached to > "indescribable") is a direct scribing of ∃S ∀p ¬U(p, S). > - The graph for "nothing is describable" (an LOI attached to > "describable" within a shaded oval) is a scribing of the stronger, > universal claim ∀S ∃p U(p, S) (its negation). > - The proof's core conclusion is that *neither of these graphs is a > theorem derivable from the blank sheet.* This is not a failure of > the proof; it *is* the proof's successful conclusion. You have > stated it perfectly: *"L neither proves nor disproves either one of > them."* *This is the structural incompleteness.* > > *4. On the LOI's Role as an Indexical Sign* > > - *Your Point:* The LOI is an indexical sign representing irremediable > indeterminacy; adding more words never fully describes it. > - *Proof's Response:* The proof formally validates this. *Lemma 4.1 > (LOIs ≠ Predicative Description)* is a direct formalization of this > idea. > - The lemma proves that an LOI can assert existence (+S, the > "indication") *even in a model where that subject is describable* (e.g., > the singleton model N). > - Therefore, the LOI's function—existential indication—is *logically > independent* from the subject's describability. It does not solve > the problem of description; it merely marks the existence of a subject > that > the system's predicative resources may or may not be able to handle. > - The "irremediable indeterminacy" appears in models like M, where > the LOI indicates a subject that the predicates cannot uniquely capture > ( > -S). > > *5. On CP 5.525 as an External Logical Principle* > > - *Your Point:* Peirce's statement is an external logical principle, > not an internal theorem. It's about the necessity of indices and collateral > experience. > - *Proof's Response:* The proof does not try to derive Peirce's > principle *within* Beta. Instead, it *uses the principle as a premise* to > analyze Beta's structure. > 1. *Premise (External Principle):* Subjects cannot be fully described > by words/general predicates and require indices (LOIs). > 2. *Formal Investigation:* What happens when we build a formal > system (Beta) that uses indices (LOIs) and general predicates (Words)? > 3. *Result (Internal Theorem about the System):* This system is > necessarily structurally incomplete. It cannot decide, based on its own > rules, whether its indicatory and predicative resources are sufficient > to > describe a subject. > 4. *Conclusion:* The external principle explains the internal > limitation. The need for "collateral experience" translates, in the > model-theoretic framework, to the need to choose a specific *model* > (M or N) to determine the truth of φ. The interpreter's > "collateral experience" is their knowledge of the state of the world > (the > model) they are reasoning about. > > *Synthesis: What the Proof Demonstrates* > Your objection and the proof structure are in agreement. The proof > formalizes the following chain of reasoning: > > 1. *Peirce's Universal Principle:* All subjects are indescribable by > words alone and require indices. > 2. *Implementation in EG:* Beta implements this with Words > (predicates) and LOIs (indices). > 3. *Formal Consequence:* This design leads to a fundamental > *undecidability*. > - Beta can indicate subjects (LOIs) and describe properties (Words). > - But it cannot use its own resources to determine whether the > connection between indication and description is always possible. The > statement φ that questions this very connection is formally > undecidable. > 4. *Interpretation:* This undecidability is not a flaw but a > necessary structural feature of any system that separates indication from > description. It is the formal signature of the gap Peirce identified. > > Therefore, the proof does not contradict your objection; it *validates it > by showing its formal consequences*. It takes Peirce's philosophical > insight and demonstrates that it logically implies a specific, verifiable > limitation in the formal system he designed to embody that very insight. > (Note, I've had to program four different LLMs to deal with my logical > inputs which is going directly to/through Peircean grraph theory and other > [lots] of formal language). The primary proof structure is not even > included in that above rendering. I am merely dealing with your current > objections in a summation format — actual full-proof structure, which is > now very robust by anyone's standard, to follow as soon as I settle all > logical i's and t's. > > Best, > Jack >
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