Inquiry Into Inquiry

Re: Category Theory • Morgan Rogers

MR:

Okay, I may have mixed up the meanings of “object” and “interpretant” in my plain language translations above?  Re determination, I read “B is determined by A” as meaning the conjunction of

and

  ?

Whether this is right depends on the answers to my previous questions.

Dear Morgan,

Let’s look at the gloss I gave for Determination under the Definition of a Sign Relation.

Determination.  Peirce’s concept of determination is broader in several directions than the sense of the word referring to strictly deterministic causal-temporal processes.  First, and especially in this context, he is invoking a more general concept of determination, what is called formal or informational determination, as in saying “two points determine a line”, rather than the more special cases of causal and temporal determinisms.  Second, he characteristically allows for what is called determination in measure, that is, an order of determinism admitting a full spectrum of more and less determined relationships.

Other words for this general order of determination are structure, pattern, law, form, and the one arising especially in cybernetics and systems theory, constraint.  It’s what happens when not everything that might happen actually does.  (The stochastic mechanic or the quantum technician will probably quip at this point, At least, not with equal probability.)

Regards,

Jon

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Re: Peirce List • Robert Marty

RM:

Thank you for reminding me of the definition of a group that I have taught for 45 years … I think you work with the permutations of symmetrical groups that do not fit well with the interdependence of categories and which make us go out of the Peircian theory, which is not forbidden as long as we point it out.  I’ll look at the use you make of them when you’ve answered my previous questions with something other than a stream of links and the definition of a group!  (my Ph.D. Math is on Abelian Groups) … formulating my questions correctly takes me time, especially to grasp your thought … I would like a reciprocal … I always thought that you had the capacity to do it without giving up your certainties, but I must say that today I am disappointed …

Dear Robert,

Auld acquaintance is not forgot 🍻 I will convey your thanks to one who reminded me.

My reason for encoring mathematical groups as a class of triadic relations and elsewhere casting divisibility in the role of a dyadic relation was not so much for their own sakes as for the critical exercise my English Lit teachers used to style as “Compare and Contrast”.  For the sake of our immediate engagement, then, we tackle that exercise all the better to highlight the distinctive qualities of triadic relations and sign relations.

A critical point of the comparison is to grasp sign relations as collections of ordered triples — collections endowed with collective properties extending well beyond the properties of individual triples and their components.

Regards,

Jon

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Re: Category Theory • Morgan Rogers

MR:

Okay, this is hard to parse, but I’ve looked at it a few times now framed with discussion from a few different sources, and it seems that if we fix some sets of signs, of interpretants and of objects, and treating the sign relation as there are some reasonable restrictions/assumptions we could place on   For example:

1a.

“every sign means something to every interpretant”,

1b.

a weaker alternative,
“every sign means something to some interpretant”,

2a.

“every interpretant has a name for every object”,

2b.

a weaker alternative,
“every object has at least one name assigned to it by each interpretant,”

and so on.

However, none of these seem strictly necessary to me;  there could be meaningless symbols or nameless objects.  Does Peirce assume any of these things or similar?  If not, I suspect the answer to my question regarding mathematical distinguishing features of sign relations is that there aren’t any:  that any ternary relation can be understood as a sign relation if one squints hard enough.

As far as meaningless signs go, we do encounter them in theoretical analysis (“resolving conundra” and “steering around nonsense”) and empirical or computational applications (“missing data” and “error types”).  The defect of meaning can affect either denotative objects or connotative interpretants or both.  In those events we have to generalize sign relations to what are called sign relational complexes.

Signless objects are a different matter since cognitions and concepts count as signs in pragmatic semiotics and Peirce maintains we have no concept of inconceivable objects.

If you fancy indulging in a bit of cosmological speculation you could imagine the whole physical universe to be a sign of itself to itself, making but this far downstream from the Big Bang we mortals usually have more pressing business to worry about.

In short, what we need sign relations for is not for settling big questions about cosmology or metaphysics but for organizing our thinking about object domains and constructing models of what goes on and what might go better in practical affairs like communication, inquiry, learning, and reasoning.

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It helps me to compare sign relations with my other favorite class of triadic relations, namely, groups.  Applications of mathematical groups came up just recently in the Laws of Form discussion group, so it will save a little formatting time to adapt the definition used there.

Cf: Animated Logical Graphs • 60

Definition 1.  A group is a set together with a binary operation satisfying the following three conditions.

1. Associativity.  For any
   we have
2. Identity.  There is an identity element such that
   we have
3. Inverses.  Each element has an inverse, that is, for each
   there is some such that

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Re: Semiotics, Semiosis, Sign Relations • Comment 1

Definitions tend to call on other terms in need of their own definitions, and so on till the process terminates at the level of primitive terms.  The main two concepts requiring supplementation in Peirce’s definition of a sign relation are the ideas of correspondence and determination.  We can figure out fairly well what Peirce had in mind from things he wrote elsewhere, as I explained in the Sign Relation article I added to Wikipedia 15 years ago.  Not daring to look at what’s left of that, here’s the relevant section from the OEIS Wiki fork.

• Sign Relations • Definition

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I opened a topic on Sign Relations in the Logic stream of Category Theory Zulipchat to work on Peirce’s theory of triadic sign relations in a category-theoretic framework.

I had been reading Peirce for a decade or more before I found a math-strength definition of signs and sign relations.  A lot of the literature on semiotics takes almost any aperçu Peirce penned about signs as a “definition” but barely a handful of those descriptions are consequential enough to support significant theory.  For my part, the definition of a sign relation coming closest to the mark is one Peirce gave in the process of defining logic itself.  Two variants of that definition are linked and copied below.

C.S. Peirce • On the Definition of Logic

Selections from C.S. Peirce, “Carnegie Application” (1902)

Reference

• Charles S. Peirce (1902), “Parts of Carnegie Application” (L 75), published in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by Charles S. Peirce, vol. 4, 13–73.  Online.

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