Jack, List: What is your exact definition of "incompleteness" in this context? In the linked paper, Tarksi defines the decision problem as "whether there exists a mechanical means of deciding whether any given statement of a formal system is a theorem," and (more precisely) "whether the set of provable statements of a formal system is general recursive" (p. 24). He also states that "by completeness we mean simply that, given any formula [without free variables], either that formula or its negation is a theorem" (ibid.). He goes on to say, as quoted below, that the result "is negative ... for the general case of the predicate calculus" (p. 25), i.e., the predicate calculus is incomplete in the defined sense. As I understand it, Gödel's incompleteness theorem demonstrates that number theory is likewise incomplete in this sense.
I still do not see what these *decidability *results for sufficiently powerful *formal systems* have to do with the general logical principle stated by Peirce in CP 5.525--every proposition has a subject that must be indicated or found, because it cannot be described in words. This corresponds to the line of identity in the Beta part of Existential Graphs, which implements a version of the predicate calculus without free variables, as well as the indefinite pronoun "something" in ordinary English. Can you make the alleged connection explicit for me? I just came across the "proof" that you recently published on Zenodo ( https://zenodo.org/records/16681952), which purports to demonstrate that for every *symbolic *system, there is some truth-value or independently existing object that it cannot express. However, Peirce's whole point is that symbols *alone *are indeed insufficient for formulating propositions--indices are also required. Again, nobody is disputing that one individual person's *actual *representation of something is never identical to another individual person's *actual *representation of the same thing--a sign token always produces at least slightly different *dynamical *interpretants in different interpreters, because their minds have been determined by different previous signs, such that they have different *habits *of interpretation. The question is whether it *would *be possible, as an ideal limit in the infinite future, for an infinite community to have identical representations of *everything real* after infinite investigation and thus infinite experience--the *final *interpretant of every sign. Peirce, of course, says yes--not as a demonstrable fact, but as a methodological principle and regulative hope of inquiry. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt On Wed, Aug 20, 2025 at 3:58 AM Jack Cody <[email protected]> wrote: > On a more "fun" tangent, I have been experimenting with classical semiotic > ideas of experience and representation which no doubt can be read in > semeiotic qua object/interpretant/determination and so forth (in > classification). I've used the famous sequence in The Good, The Bad, and > The Ugly to make the point: > P(O)P Truth Table — Tuco (T), Blondie (B), Angel Eyes (AE) > > | Subject | Object | Prime (representation) | Non-identity to base > | Cross-prime inequality | > > |---------|--------|-------------------------------|----------------------|--------------------------------| > | T | B | B' (Blondie-as-to-Tuco) | B' != B > | B' != B'' | > | T | AE | AE' (AngelEyes-as-to-Tuco) | AE' != AE > | AE' != AE'' | > | B | T | T' (Tuco-as-to-Blondie) | T' != T > | T' != T'' | > | B | AE | AE'' (AngelEyes-as-to-Blondie)| AE'' != AE > | AE'' != AE' | > | AE | T | T'' (Tuco-as-to-AngelEyes) | T'' != T > | T'' != T' | > | AE | B | B'' (Blondie-as-to-AngelEyes)| B'' != B > | B'' != B' | > > Minimal consequences (also copy/pasteable): > > Incompleteness (prime != base): > T' != T, T'' != T, B' != B, B'' != B, AE' != AE, AE'' != AE > > Unique experience (cross-prime, same base): > T' != T'', B' != B'', AE' != AE'' > The table illustrates a core concept in social cognition and philosophy of > mind: an individual is not a single, fixed object but is constituted > differently in relation to others*.* Each person (the subject) has their > own unique representation (or model) of another person (the object). > *Subject:* The person who is doing the perceiving. > *Object:* The person who is being perceived. > *Prime (representation):* This is the Subject's internal representation > or mental model of the Object. The prime symbol ( ′ ) denotes that this is > a version *for* or *as seen by* the Subject. > > - B′ is "Blondie as seen by Tuco." > - T′′ is "Tuco as seen by Angel Eyes." > > *Non-identity to base:* This column states a fundamental rule: a person's > representation of another (*Prime*) is never identical to that other > person's base identity or their representation of themselves. B′ ≠ B > means "Tuco's version of Blondie is not the same as Blondie's version of > himself (or Blondie's 'true' self, if such a thing exists)." > *Cross-prime (same object, different subject):* This column states > another fundamental rule: two different subjects will have different > representations of the *same* object. Tuco's version of Angel Eyes (AE′) > is not the same as Blondie's version of Angel Eyes (AE′′). > *Summary of the Relations Shown:* > > 1. *Tuco's View:* > - He sees Blondie as B′. > - He sees Angel Eyes as AE′. > 2. *Blondie's View:* > - He sees Tuco as T′ (which is different from Tuco's view of himself > and Angel Eyes' view of Tuco). > - He sees Angel Eyes as AE′′ (which is different from Angel Eyes' > view of himself and Tuco's view of him). > 3. *Angel Eyes' View:* > - He sees Tuco as T′′. > - He sees Blondie as B′′. > > Note, three base (person(s)) generate six necessary primes and no person's > primes (two unique for each one) can be the same as anyone else's without > contradicting identity principles. > > A fun way to explore semeiotics whilst illustrating certain points which > can be understood variously. > > The most important part to me, here, is that there are necessarily six > prime "people" (as far as I can tell) from three base person(s). As each > person "sees/experiences" two others, distinct, the mathematics is not > difficult. The larger question is what that means in more general terms. It > goes directly to relativity in prime representation as far as I can tell > but the base does not seem relative to me at all. Need more explication but > more a fun way of asking quesitons than a strict thesis. > > Best > Jack > > ------------------------------ > *From:* Jack Cody > *Sent:* Tuesday, August 19, 2025 8:09 AM > *To:* Peirce-L <[email protected]> > *Subject:* Peirce and Incompleteness -- Why the Parsimony of "Credit"? > > Dear List, > > I have been studiously preparing an on-list reply to a post made by JAS a > week or more ago. I would like to say, in advance, that I find it > incredibly interesting that Peirce is basically writing about incompletness > (Godel/Tarski) 50 years (1881) before Godel (1931?) renders his famous > theorems. Now Peirce's findings are proto-incompleteness but maximal within > that discovery period. > > There are a few things to note: one, Peirce establishes the truth-table > system and also the logic of number (article) which massively influences > not merely Tarski/Godel (more Tarski first hand and Godel second), but also > Peano et al in their work regarding the very system Godel will later use. > Tarski, at an address to Stanford in 1947 (where Godel and many other > famous logicians are present) cites Peirce's work directly (he wasn't sure > if it was Peirce or Frege — each had done something of note here but in > this instance it was indeed Peirce whom he meant. Peirce is aware, too, of > all those in the area of truth-tables or what would now be called "PA" and > you can find citations to all the canonical figures within Peirce's > writings (from the 19th through very early 20th centuries). > > Why is this interesting? Park the ding-an-sich for a minute. We do not all > agree. That's the subject of my larger thesis. However, before I even > arrive at that I have two more minor theses. One, a far more rigourously > formatted understanding of the above which gives Peirce his credit which I > believe has been seriously neglected over the years. I mean, I search > Google Scholar and so forth and there are some articles which are > interesting but nowhere is 5.525 ("It has been shown [3.417ff] that in the > formal analysis of a proposition, after all that words can convey has been > thrown into the predicate, there remains a subject that is > indescribable...") cited as a necessary example of proto-incompleteness. > That statement, logically, foreshadows so much in the semantic of Godel > and Tarski (and this before even citing Peirces schematic work on > truth-tables and also a kind of proto Peano Arithematic) . > I find it odd, basically, that of all the scholarship done on Peirce no > one, it seems, has made the obvious connection. If you take that section of > 5.525 and read Peirce's mathematical work as cited by Tarski > > "Now let us examine the decision problem in some elementary forms of > logic. First, the sentential calculus: for this there is the positive > result based on the two-valued truth-table method. I do not know who > actually is the author of this procedure - whether it was Frege or Peirce - > but what is important is that we do have this now classical result.20 For > the monadic functional calculus it is well known that the result is > positive.21 It is negative, however, for the general case of the predicate > calculus." > https://www.jstor.org/stable/421074 > it is very difficult not to understand the work Peirce was doing as > essential to incompleteness (what it would come to be called in the next > century). (I park the ding-an-sich to one side because there is an area, > here, where surely there is List agreement — there are two kinds of > incompleteness, to me at least, who has studied almost nothing but for > years now: maximal and minimal). Peirce is maximal in his logic ("after all > that words can convey have been thrown into [predicates of subjects] there > remains [subjects which are indescribable and thus we have > "incompleteness"]. 5.525. > > I have changed that wording, clearly, but it's not problematic to the > overall logical analysis. If one runs with Tarski and Godel here, one sees > immediately that you cannot derive, easily, anything other than > incompleteness (protean) from that which Peirce is speaking about. > > Anyway, this is overly long already but I wanted to throw it open for > others to consider as many diverse backgrounds exist here in > interdisciplinary fields. I'm just genuinely amazed, ding-an-sich or no (a > different day...), that such little work has been done here in terms of > threading the needly to/through Peirce and those exponents of > incompleteness theorems. > > Best, > Jack >
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