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]<mailto:[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.
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