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.
*
Premise (External Principle): Subjects cannot be fully described by
words/general predicates and require indices (LOIs).
*
Formal Investigation: What happens when we build a formal system (Beta) that
uses indices (LOIs) and general predicates (Words)?
*
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.
*
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
________________________________
From: [email protected] <[email protected]> on behalf of
Jon Alan Schmidt <[email protected]>
Sent: Saturday, August 23, 2025 10:29 PM
To: Peirce-L <[email protected]>
Subject: Re: [PEIRCE-L] Peirce and Incompleteness -- Why the Parsimony of
"Credit"?
Jack, List:
I found the "Formal Table" image linked below more helpful than your post
itself and will focus on what I think is the key line in it.
JRKC: Target Proposition ϕ = ∃x ∀P ¬U(P,x) = "There exists a subject that no
predicate uniquely describes"
Again, Peirce's statement in CP 5.525 actually amounts to the stronger claim
that no subject of any proposition is describable using only words, so it must
be indicated or found instead. This is built into Beta/Gamma EG by denoting
every subject of every graph with a heavy line of identity as an indexical
sign. Its irremediable indeterminacy is iconically represented by the fact that
no matter how many words are attached to it, there is always room to attach
even more words, such that "every line of identity ought to be considered as
bristling with microscopic points of teridentity" (SS 199, 1906 Mar 9; see also
CP 4.583, LF 3/1:285-6, 1906).
Another issue that I notice this time is the inclusion of quantification over
predicates, thus requiring second-order logic, which frankly goes beyond my
competence and interest. Peirce provided a few hints toward implementing it in
Gamma EG, but as far as I know, no one has ever worked out the details; so I
still do not see how ϕ can be expressed in L, except trivially for this
non-self-referencing version by attaching a line of identity to the word
"indescribable" directly on the sheet, signifying "something is indescribable."
Likewise, Peirce's stronger claim can be scribed by attaching a line of
identity to the word "describable" within a shaded area, signifying "nothing is
describable."
Obviously, neither of these is a theorem of Beta/Gamma EG that can be derived
directly from the blank, so L neither proves nor disproves either one of them.
Is that all you have been trying to demonstrate? Again, Peirce's statement in
CP 5.525 is a logical principle that applies to all propositions, not something
derived within a formal system. It highlights the necessity of indices to
denote their subjects, and of collateral experience/observation for
interpreters to understand them. In its original context, his point is that the
thing-in-itself can neither be indicated nor found, and no one can ever have
any collateral experience/observation of it, so it is "meaningless surplusage."
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 Sat, Aug 23, 2025 at 5:04 AM Jack Cody
<[email protected]<mailto:[email protected]>> wrote:
Jon, Edwina,
Jon, I appreciate your pushing. I did anticipate most, if not all, of your
objections.
<https://1drv.ms/i/c/d04faa036421906b/EXKXTYjIJG1Nog1W1WdXgAoBR9g5fK25IgDQ0YmhtdNj-g>
[https://ukwest1-mediap.svc.ms/transform/thumbnail?provider=spo&farmid=188926&inputFormat=PNG&cs=OTE5OWJmMjAtYTEzZi00MTA3LTg1ZGMtMDIxMTQ3ODdlZjQ4fFNQTw&docid=https%3A%2F%2Fmy.microsoftpersonalcontent.com%2F_api%2Fv2.0%2Fdrives%2Fb!lwR8cgTI1E2aKZh9t-jAqqjMU1xldHhLj_BG1XNqW06K5qeznDLVQr2fcvlp5V5v%2Fitems%2F01CEFMJ5TSS5GYRSBENVG2EDKW2VTVPAAK%3Ftempauth%3Dv1e.eyJzaXRlaWQiOiI3MjdjMDQ5Ny1jODA0LTRkZDQtOWEyOS05ODdkYjdlOGMwYWEiLCJhcHBpZCI6IjkxOTliZjIwLWExM2YtNDEwNy04NWRjLTAyMTE0Nzg3ZWY0OCIsImF1ZCI6IjAwMDAwMDAzLTAwMDAtMGZmMS1jZTAwLTAwMDAwMDAwMDAwMC9teS5taWNyb3NvZnRwZXJzb25hbGNvbnRlbnQuY29tQDkxODgwNDBkLTZjNjctNGM1Yi1iMTEyLTM2YTMwNGI2NmRhZCIsImV4cCI6IjE3NTU5NjEyMDAifQ.kW8jbhMnIEBrkw6sSIR0DSauiyv8GdLpq1LHp2KOpyDFCvsUP5e43CEVkiSCTac0jGKKLaa6mvKkp6LDxEqujgUnod_QjqgSa96QMIUmD9S4wyg_W1MeOssTRB7nDr3hYUu_Lt7ve2XBuqcu7OQK0qSrqu8CMjIQDhku1zenL7IfqSoEDQD5liBx-CE8hT7QDiplf9WggZX9i2RMyISBx5Mky1lZdAqU4XwiRYXD04mr2g9EIfLy5EwVAXGZ4kcQOJNN7vNFG8aGXkNY71ISD_mp2pdHj7ns7WWzXH289ItHWDY1kMySE239s6sOSsCEtHqsfz0M4anTyPAQf0WIUq4luDJRO8NhJXn3M8FlJisf3JYkUmMnB4om2_FZcbXYGthGHqFRLiEpYuumjitU2Uz5nHRWTRc9X6M30_D61s3ZFDAlxP6bFgamXLKum9vG.jV9XyStn47rKyREjHwIL0UqnhQAVtlxP6XOG8_LpyRQ%26version%3DPublished&width=176&height=176&cb=63891539980]
[https://res-1.cdn.office.net/assets/mail/file-icon/png/cloud_blue_16x16.png]Formal
Table.PNG
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
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