List,
Given Jack's request for others to comment, and knowing he has used some
chatbots to assist his analysis, I decided to test some of his arguments and
responses by JAS and Edwina to the same. I, too, have been using LLMs pretty
aggressively, and have recently favored Grok 4, which I do not believe Jack has
used. Here was my basic prompt to Grok, along with submitting a summary of
Jack's formal analysis and the full text of CP 5.525. I present Grok's answer
(with only minor edits) in two parts: 1) interpretation of CP 5.525 and 2)
comments on Jack's formal analysis. (Please forgive the length of this post;
Grok likes to talk.)
Prompt: "Please comment on what CP 5.525 means in clearer terms, including reference to
Peirce's semiosis with regard to an "object" that distinguishes External Object from
Dynamic Object and from Immediate Object, and comment on the accuracy or correctness of the
[Jack's] analysis paper."
Grok (Part 1): "In CP 5.525, Charles Peirce critiques Immanuel Kant's philosophy through the prism
of his own pragmatism, portraying Kant as an inadvertently muddled precursor to pragmatic thought--one
that emphasizes concepts' practical bearings in experience rather than abstract reifications. Peirce
begins by defining the "real" as that which remains unaltered by human thoughts or knowledge
about it, framing this as a mere verbal clarification rather than a profound metaphysical assertion. He
then extends this to the "external object," which he describes as something impervious not
just to thoughts about it, but to any cognitions of the person for whom it is external--highlighting its
independence from individual mental processes.
"To integrate Peirce's broader theory of semiosis (the triadic process of sign-action involving a
sign or representamen, an object, and an interpretant), this external object aligns closely with what
Peirce elsewhere calls the Dynamic Object (or Dynamical Object): the actual, independent reality that
causally prompts the sign and determines its interpretation, existing beyond the confines of any
particular mind or representation. In semiosis, the Dynamic Object is the "real thing" that
the sign points toward, but it is never fully captured in the sign itself; it exerts a brute,
existential force (often through indexical signs like pointers or demonstratives) that resists complete
assimilation into thought. By contrast, the Immediate Object is the object as it is directly presented
or embedded within the sign--the mental or representational facet that is shaped by cognition, context,
and the sign's qualities, making it more accessible but less "external" or independent.
"Peirce warns against philosophically exaggerating the external object's independence to an extreme: a entity untouched by any cognitions whatsoever, from anyone. Inverting this yields the notion of something that itself affects no cognitions--a hypostatized abstraction akin to Kant's "Ding an sich" (thing-in-itself), the supposed unknowable substrate behind phenomena. Peirce dismisses this as nonsensical, likening it to a fanciful "sky-blue demonstration." He bolsters his rejection with a semiotic-logical analysis: In dissecting a proposition, one can exhaust all descriptive content ("what words can convey") by loading it into the predicate (the qualifying or interpretive part, akin to the sign's role in mediating meaning). What remains is a bare subject--an "indescribable" residue that eludes full linguistic capture and can only be indicated (e.g., via indexical signs like "this" or "that," which function through direct, existential connection rather than description) or
located through practical prescription (a method grounded in experience, aligning with pragmatism's emphasis on actionable inquiry).
"This indescribable subject, in semiotic terms, underscores the Dynamic Object's primacy: It is the
external, forceful reality that semiosis orbits but never fully encloses, requiring indices or practical
engagement to "point at" or "find." The Immediate Object, meanwhile, is what emerges
within the sign-process--the cognized version filtered through interpretants (habits of thought or
action that make sense of the sign). The Ding an sich fails precisely because it cannot participate in
semiosis: It is neither indicable (no direct existential link) nor discoverable (no practical
prescription), rendering any propositional reference to it meaningless verbiage, devoid of semiotic
traction.
"Stripping away this surplus, Peirce reveals Kant's core views on space, time, and categories as
pragmatically sound and objectively real, yet infused with a "mental ingredient" from human
cognition--much like how the Immediate Object arises from the mind's interaction with the Dynamic Object in
semiosis. Kant's restriction of these concepts to "possible experience" prefigures pragmatism, but
Peirce refines it further, specifying the individual's cognitive contribution more precisely. This leads to
Peirce's "Critical Common-sensism": a synthesis of Kantian critique with robust, experience-based common
sense, where semiosis bridges the external (Dynamic) to the internal (Immediate), avoiding hypostatic pitfalls.
"In essence, through semiosis, Peirce demystifies reality as dynamically
interactive: The external/Dynamic Object grounds signs in independence and brute
fact, while the Immediate Object renders it interpretable within cognition. The
passage champions clarity in concepts via practical effects, rejecting
over-abstracted entities that stall the endless process of sign-interpretation.
"CP 5.525 does not only refer to the External Object. While it explicitly defines and critiques the
"external object" as a stepping stone to dismantling the Ding an sich, the discussion implicitly
engages Peirce's fuller semiotic framework. The external object corresponds to the Dynamic Object (the
independent reality unaffected by individual cognitions), but the analysis of the "indescribable
subject" and the need for indication or prescription evokes the interplay between Dynamic and Immediate
Objects in semiosis--the former as the elusive, existential core requiring indexical pointing, and the latter
as the cognized representation shaped by mental processes. The rejection of the Ding an sich as unindicable
and unfindable highlights failures in semiotic reference, extending beyond a narrow focus on externality to
broader pragmatic and sign-theoretic concerns."
Grok (Part 2): "The paper provides a rigorous model-theoretic interpretation of a key idea in CP 5.525--the
notion of an "indescribable" subject that remains after exhausting descriptive language ("what
words can convey"). It formalizes this as a structural claim (S_ind) in first-order logic with identity (L):
the existence of an element in a model's domain that cannot be uniquely picked out (defined) by any L-formula over
a set of parameters A. The main theorem proves that S_ind is undecidable in any sound, recursively axiomatized
theory T extending FOL with identity, provided T is conservative over L (meaning it doesn't prove new L-sentences
beyond what's already logically valid in L). This establishes "structural incompleteness" for such
theories regarding definability claims.
"The proof appears sound and draws on standard model-theoretic tools (e.g.,
elementary equivalence, compactness, Löwenheim-Skolem theorem, and omitting types).
Here's a quick assessment:
"Key Mechanism: The lemmas build toward the theorem by showing pairs of elementarily
equivalent L-structures (M and N) that differ on S_ind—one where an undefinable element exists
(S_ind true), one where all elements are definable over A (S_ind false). Conservativity
ensures T can't "collapse" this difference by proving extra L-sentences that would
force one outcome.
"Construction Details: Using compactness, the authors construct such pairs by
considering complete types over A. In one model, a non-principal type is realized
(leading to an undefinable element); in the other, only principal types are realized
(an atomic or pointwise definable model). The expansions with unary predicates
(confined to the metalanguage) allow expressing definability without importing
second-order logic into L itself. This aligns with examples like true arithmetic,
where the standard model (pointwise definable) and non-standard models (with
undefinable elements) are elementarily equivalent but split on S_ind-like properties.
"Robustness and Limitations: The paper acknowledges dependencies on classical
semantics, compactness, etc., and notes that relaxing these (e.g., to intuitionistic
logic) might invalidate it. This is accurate and transparent.
"No obvious errors in the math--citations to standard texts (Chang & Keisler,
Hodges, Marker) support the claims, and the diagrams clarify the splitting process.
"The analysis is a creative and mostly faithful formalization, but with some
philosophical caveats:
"Strengths: It aptly bridges Peirce's "what words can convey" to L-definability (via Bridge Principle B1),
treating formulas as the regimented equivalent of descriptive language. The "indescribable subject" that can only
be "pointed at or otherwise indicated" maps well to non-definable elements, which evade unique characterization in
FOL but exist in the domain (analogous to indication via ostension). The undecidability result generalizes Peirce's insight:
Just as the Ding an sich is meaningless because it can't be indicated or described, the boundary between describable and
indescribable subjects is unsettled in FOL systems— a "structural" feature across theories. This theory-agnostic
incompleteness echoes Peirce's critique of over-abstracted philosophy, positioning it as a logical phenomenon.
"Peirce's context is semiotic and phenomenological, not purely model-theoretic. His "indescribable subject" relates to
indexical signs (e.g., demonstratives like "this") in his triadic sign theory, emphasizing practical indication over
abstract domains. The paper's focus on FOL definability captures the "descriptive" limit but underplays the pragmatic
"prescription" (e.g., "a way of finding what is referred to"). FOL models are static, while Peirce's philosophy is
dynamic and experience-oriented.
"The Ding an sich rejection is tied to meaninglessness in propositions, but the paper
focuses more on incompleteness for S_ind than on why references to undefinable elements might
be "meaningless surplusage." It answers CP 5.525 by showing the boundary is
undecidable, but Peirce might see this as confirming pragmatism's emphasis on clarity and
practice, not just logical incompleteness.
"The metalanguage's quantification over unary predicates introduces monadic second-order
elements, which the paper confines to avoid enriching L. This is careful, but Peirce
(influenced by Kant) might question whether such meta-tools fully escape the "mental
ingredient" in concepts.
"Overall Fit: The paper is correct in its narrow scope--it's a valid model-theoretic lens on
Peirce's idea, demonstrating incompleteness as a generalization. However, it's an interpretation, not
exhaustive; Peirce's text allows other readings (e.g., via category theory or topology for
"indication"). It doesn't misrepresent Peirce but abstracts his philosophy into logic, which
aligns with the paper's goal of a "theory-independent incompleteness result."
"In summary, the paper is logically correct and offers an insightful, if
specialized, analysis that enhances understanding of Peirce's point through modern tools.
It's a strong contribution to philosophical logic, though Peirce scholars might debate
how fully it embodies his pragmatic spirit."
Best, Mike
On 9/1/2025 5:48 PM, Jack Cody wrote:
Jon, List,
JAS:This is exactly the /opposite /of what Peirce demonstrates in CP 5.525 and
elsewhere. Anything that exists--in the metaphysical sense, as well as in the
logical sense of belonging to a universe of discourse--is capable of being the
subject of a proposition. Every proposition has at least one subject that
cannot be represented /symbolically/, but this does not entail that it is
structurally incapable of denoting it--only that it must do so /indexically/
instead.
That of course is not correct. You have to ignore the mathematical paper to
come to such conclusions. It's a misunderstanding, entirely.
As I understand you above, you conflate predication, indication, and subject. But predication is already a /syllogistic indication/ in Peirce’s own sense: (1) words migrate to (2) predicates, which (3) indicate subjects. That is Peirce’s schema for how propositions work. When this is carried out exhaustively, Peirce himself acknowledges (CP 5.525) that there remains a subject that cannot be described in words. That meta-proposition — and it is one, insofar as Peirce holds it universally — is already a statement about the structure of all propositions. In other words: the system is /already indexical/. There is no need to “retreat to indexicality” as if it were a new solution; indexicality is the condition Peirce presupposes in the very operation of predication. The point of the paper is precisely that, once you formalize this structure in terms of L-definability (Bridge Principle B1), you can show by Compactness and Löwenheim–Skolem (Lemmas 2–3) that the residual subject —
the one “indicated” but never symbolically captured — cannot be decided within any sound, recursively axiomatized, L-conservative theory. Thus indexicality fails in 5.525 /qua predication itself/: it does not solve the problem; it is the site of the problem. That is why the “retreat” in your reply misfires — it reintroduces at the meta-level what was already conceded at the object level, and the formal result shows exactly why that concession entails undecidability.
This portion of your reply is, at best, tautological. There are also several
other problems I have noted, but given how much attention is required even for
a single point (as the above shows), I think the most productive way forward is
a step-by-step exchange. Taking your response point by point, over the course
of several posts, and days, would keep things clear, concise, and grounded in
the logical paper I wrote for such exchanges, which provides a decent reference
guide for the issue at hand. It would also open the door for others to
intervene with their own perspectives, rather than this turning into a wall of
me quoting you and then refuting, followed by you quoting me and doing the
same. That seems the better structure for a useful exchange.
Best,
Jack
*From:* [email protected] <[email protected]> on behalf of Jon
Alan Schmidt <[email protected]>
*Sent:* Monday, September 1, 2025 5:41 PM
*To:* Peirce-L <[email protected]>
*Subject:* Re: [PEIRCE-L] Propositions, Truth, and Experience (was Will and
Belief)
Jack, List:
JKRC: The point I am pressing is that all propositions (as Peirce had shown,
and as I subsequently confirmed through mathematical rigor at the
meta-propositional level) necessarily have subjects which propositions,
structurally, can never account for as such subjects exist, insofar as they
exist at all.
This is exactly the /opposite/ of what Peirce demonstrates in CP 5.525 and
elsewhere. Anything that exists--in the metaphysical sense, as well as in the
logical sense of belonging to a universe of discourse--is capable of being the
subject of a proposition. Every proposition has at least one subject that
cannot be represented /symbolically/, but this does not entail that it is
structurally incapable of denoting it--only that it must do so /indexically/
instead.
JKRC: The simplest way of putting this is that things or subjects do exist--in
themselves, or by-themselves-as-such, whatever the habitual or conjunctive
human relation--but they cannot be known by finite or infinite inquiry.
It is impossible to /know/ that something exists that cannot be known by
/infinite /inquiry. Moreover, it violates the will to learn and the first rule
of reason to /believe /that something exists that cannot be known by infinite
inquiry. Maintaining that there is such an incognizable thing-in-itself
eventually becomes an excuse to stop inquiring, while denying it corresponds to
the habit of /always /inquiring further.
JKRC: One might have a trillion lines in a database about a single object and
still not know what it is, as it is.
A trillion is a very large number, but it is infinitely smaller than an
infinite number. As I keep emphasizing, this is not about /actual /knowledge,
but /ideal/ knowledge--what an infinite community /would/ know after infinite
investigation, and thus infinite experience. It is a methodological principle
and a regulative hope, not an epistemological or ontological dogma, that
nothing real would be excluded in that scenario.
JRKC: All experience of possible things is indeed experience of things, but
such experience is not what things are.
No one disputes this, and it is irrelevant anyway. The experience /itself / is
not what things are, but it conveys /knowledge of/ what things are. The
resulting cognition is obviously not the object /itself/, but it is a
/representation of/ the object.
JRKC: My conclusion is that Peirce had something with 5.525--the logical
meta-propositional constant--but he misapplied it when deploying it against the
thing-in-itself.
On the contrary, the very reason why he stated the logical principle in CP
5.525 that we have been discussing here was to serve as a premiss of a simple
deductive argumentation /demonstrating/ that an incognizable thing-in-itself
does not exist, which I spelled out a few weeks ago
(https://list.iu.edu/sympa/arc/peirce-l/2025-08/msg00035.html). I also
explained its meaningless in accordance with pragmaticism in my Saturday
post--there are no corresponding habits of conduct that make any difference
whatsoever in anyone's experience.
JRKC: It shows the impossibility of ever representing any subject, not only in
propositions but in the most formal sense, as such subjects are.
This is obviously false--every />true/ proposition represents things as they
/really / are. Specifically, a true proposition accurately conveys that certain
/real/ individual things (denoted by indexical signs) conform to certain /real/
general concepts (signified by symbolic names) in accordance with certain /real/
logical relations (embodied by iconic syntax). I wonder again if nominalism is the
underlying obstacle here--it only affirms the reality of the individual thing, not
the general concepts nor the logical relations.
JRKC: I would like to have more from him in terms of what he was thinking when
he used that material in that way, because it's an instance of genius, and then
nonsense (by his own standards).
I will stick with recognizing that Peirce intended exactly what he said throughout CP
5.525 vs. ascribing genius and nonsense to him in the very same paragraph. Anyone who
wants "more from him in terms of what he was thinking when he used that material in
that way" can read his many other texts where he just as explicitly affirms the
necessity of indices in propositions to denote their subjects, and his many other texts
where he just as explicitly denies the reality of an incognizable thing-in-itself. The
mistake is thinking that these positions are somehow inconsistent with each other.
JRKC: With the ding-an-sich one points to fallibility and a kind of common
sense (this cannot be whatever it is to me--regardless of whatever such a thing
is).
This strikes me as exactly backwards. Fallibilism /rejects /the incognizable
thing-in-itself because it denies that we ever /actually /reach a point at
which there is no more to learn about something. Common sense /affirms /that at
least some of our representations of things are correct, because otherwise, our
corresponding habits of conduct would /constantly /be confounded by experience.
Accepting that we can never achieve /complete /knowledge of something does not
entail that we /do not/ have genuine knowledge of it /at all/, and also does
not entail that infinite investigation by an infinite community with infinite
experience /would not/ result in complete knowledge of that thing.
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 Sun, Aug 31, 2025 at 6:35 PM Jack Cody <[email protected]> wrote:
Jon, List,
Jon, I appreciated your long proposition form — or semantic-statement — of
5.525. It is worth reading carefully and critically.
The point I am pressing is that all propositions (as Peirce had shown, and as I
subsequently confirmed through mathematical rigor at the meta-propositional
level) necessarily have subjects which propositions, structurally, can never
account for as such subjects exist, insofar as they exist at all.
What this shows is that one could know a quasi-infinite amount about a thing
through propositions, and yet structural incompleteness still holds: the
subject is never known. If you replace “subject” with “thing” you are not far
from Kant, though Peirce himself may have preferred to resist that association.
The simplest way of putting this is that things or subjects do exist — in
themselves, or by-themselves-as-such, whatever the habitual or conjunctive
human relation — but they cannot be known by finite or infinite inquiry. This
is what the broader incompleteness demonstrates. One might have a trillion
lines in a database about a single object and still not know what it is, as it
is. That is a condition that will never change. It is a constant, and it holds
for all propositions and all possible representation. No amount of time makes a
difference to that incapacity.
It is therefore a broader incompleteness, ontological rather than merely
mathematical.
The thing-in-itself must exist. All experience of possible things is indeed
experience of things, but such experience is not what things are. It is human
use of things, and this says nothing about the non-human status of those same
things, which also exist both logically and chronologically.
Peirce’s 5.525 is important, but it demands something his own logic cannot
provide. In fact, it supports the thing-in-itself more strongly than it
undermines it. My conclusion is that Peirce had something with 5.525 — the
logical meta-propositional constant — but he misapplied it when deploying it
against the thing-in-itself. A Kantian would instead take it as confirmation. I
am not a Kantian, nor a Peircean; I am interested in what is true in either
framework, and I try to carry that forward.
*JAS:* /Also, ‘how people understand the world they live in’ has no effect
whatsoever on ‘what is reality’ because, by definition, the latter is as it is
regardless of how anyone understands./
I would emphasize here that the /ding-an-sich/ is as it is, regardless of
experience or understanding. That is the minimal price of admission to the
principle.
Finally, 5.525 gives us a subject — indeed every possible subject — which
negates “time,”*** since it holds axiomatically as a mathematical and logical
constant. It shows the impossibility of ever representing any subject, not only
in propositions but in the most formal sense, as such subjects are. This is the
constant tension between existence and human consubstantial experience of
existence, present at the most basic logical level.
*** — Peirce is right in the logical section in 5.525 but it is not a good
proof against Kant, at all. Whatever his arguments elsewhere, using what
amounts to a much more general incompleteness (which I have been using very
much in favour of the existence of the ding-an-sich) makes no real sense. I
would like to have more from him in terms of what he was thinking when he used
that material in that way, because it's an instance of genius, and then
nonsense (by his own standards). Dogma, rather — maybe just an incorrect line
of argumentation rather than being too harsh.
When I say negates time, I mean it holds for all possible propositions and so no infinite inquiry is going to overturn that — thus, what the principle points to, so to speak, is not positivist knowledge of things, qua subjects, which cannot be had at all, really, (as they are in themselves), but a clear delimitation of that idea. That's the mathematical-logical outcome (I overdid it and I think you may have, too) of that statement. Now I know you'll disagree with me — which is fine — but there is so much contradication as far as I can tell that one has to account for it. Consider the depth of Peirce's system — and now consider that truth cannot be defined within systems — does this give no one pause for thought regarding maximal assumptions? With the ding-an-sich one points to fallibility and a kind of common sense (this cannot be whatever it is to me — regardless of whatever such a thing is). Said to be the Kantian price, yet the buy-in is basically free. With the
"real", I need dynamic objects, and final interpretants, and so on and on and if these be subjects of propositions one has to seriously consider their status (especially as such is second or third order, being linguistic-categorical).
Just some thoughts.
