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
________________________________
From: [email protected] <[email protected]> on behalf of
Jon Alan Schmidt <[email protected]>
Sent: Wednesday, August 20, 2025 10:56 PM
To: Peirce-L <[email protected]>
Subject: Re: [PEIRCE-L] Peirce and Incompleteness -- Why the Parsimony of
"Credit"?
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<http://www.LinkedIn.com/in/JonAlanSchmidt> /
twitter.com/JonAlanSchmidt<http://twitter.com/JonAlanSchmidt>
On Wed, Aug 20, 2025 at 3:58 AM Jack Cody
<[email protected]<mailto:[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]<mailto:[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
_ _ _ _ _ _ _ _ _ _
► 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.
