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 / twitter.com/JonAlanSchmidt On Sat, Aug 23, 2025 at 5:04 AM Jack Cody <[email protected]> wrote: > Jon, Edwina, > > Jon, I appreciate your pushing. I did anticipate most, if not all, of your > objections. > Formal Table.PNG > > <https://1drv.ms/i/c/d04faa036421906b/EXKXTYjIJG1Nog1W1WdXgAoBR9g5fK25IgDQ0YmhtdNj-g> > 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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