Relations & Their Relatives : 15

Definitions and examples for relation composition and the two types of relation reduction that commonly arise can be found in the following articles:

Peirce’s idea of reducibility and irreducibility is the more fundamental concept, having to do with the question of whether relations can be derived from others by relational composition, and this type of operation is invoked in every variety of formal construction.  Consequently, projective reducibility does nothing to defeat Peirce’s thesis about the primal nature of triadic relations.

But people sometimes confuse the two ideas of reducibility, compositional and projective, so it’s good to clarify the differences between them.  Projective reducibility, when you can get it, is more of a “consolation prize” for dyadic reductionists, who tend to ignore the fact that you can’t do anything constructive without triadic relations being involved the mix.  Still, it’s a useful property and good to recognize it when it occurs.

Relations & Their Relatives : 14

I constructed the “Ann and Bob” examples of sign relations back when I was enrolled in a Systems Engineering program and had to explain how triadic sign relations would naturally come up in building intelligent systems possessed of a capacity for inquiry.  My adviser asked me for a simple, concrete, but not too trivial example of a sign relation and after cudgeling my wits for a while this is what fell out.  Up till then I had never much considered finite examples before as the cases that arise in logic almost always have formal languages with infinite numbers of elements as their syntactic domains if not also infinite numbers of elements in their object domains.

The illustration at hand involves two sign relations:

• $L_\text{A}$ is the sign relation that captures how Ann interprets the signs in the set $S = I = \{ {}^{\backprime\backprime}\text{Ann}{}^{\prime\prime}, {}^{\backprime\backprime}\text{Bob}{}^{\prime\prime}, {}^{\backprime\backprime}\text{I}{}^{\prime\prime}, {}^{\backprime\backprime}\text{you}{}^{\prime\prime} \}$ to denote the objects in $O = \{ \text{Ann}, \text{Bob} \}.$
• $L_\text{B}$ is the sign relation that captures how Bob interprets the signs in the set $S = I = \{ {}^{\backprime\backprime}\text{Ann}{}^{\prime\prime}, {}^{\backprime\backprime}\text{Bob}{}^{\prime\prime}, {}^{\backprime\backprime}\text{I}{}^{\prime\prime}, {}^{\backprime\backprime}\text{you}{}^{\prime\prime} \}$ to denote the objects in $O = \{ \text{Ann}, \text{Bob} \}.$

Each of the sign relations, $L_\text{A}$ and $L_\text{B},$ contains eight triples of the form $(o, s, i)$ where $o$ is an object in the object domain $O,$ $s$ is a sign in the sign domain $S,$ and $i$ is an interpretant sign in the interpretant domain $I.$  These triples are called elementary or individual sign relations, as distinguished from the general sign relations that generally contain many sign relational triples.

If this much is clear we can move on next time to discuss the two types of reducibility and irreducibility that arise in semiotics.

To be continued …

Inquiry, Signs, Relations • 1

Human spontaneous non-demonstrative inference is not, overall, a logical process.  Hypothesis formation involves the use of deductive rules, but is not totally governed by them;  hypothesis confirmation is a non-logical cognitive phenomenon:  it is a by-product of the way assumptions are processed, deductively or otherwise.  (Sperber and Wilson, p. 69).

From a Peircean standpoint this raises the question of abductive reasoning and its role in the cycle of inquiry.

As I read him, Peirce began with a quest to understand how science works, which required him to examine how symbolic mediations inform inquiry, which in turn required him to develop the logic of relatives beyond its bare beginnings in De Morgan.  There are therefore intimate links, which I am still trying to understand, among his respective theories of inquiry, signs, and relations.

There’s a bit on the relation between interpretation and inquiry and a bit more on the three types of inference — abduction, deduction, induction — in the following paper and project report.

Reference

• Sperber, Dan and Wilson, Deirdre (1995), Relevance : Communication and Cognition, Second Edition, Blackwell, Oxford, UK.

Relations & Their Relatives : 13

The facts about relational reducibility are relatively easy to understand and I included links to relevant discussions in my earlier survey of relation theory.

The following article discusses relational reducibility and irreducibility in general terms and gives concrete examples of reducible and irreducible triadic relations of the sort we find in mathematics and semiotics, illustrating the two types of reducibility that usually come up in these settings.

These examples were introduced in the following articles on triadic relations and sign relations and I believe one could learn a lot from their careful consideration.

Relations & Their Relatives : 12

In viewing the structures of relation spaces, even the smallest dyadic cases we’ve been exploring so far, no one need feel nonplussed at the lack of obviousness in this domain.  Anyone who spends much time doing mathematics will discover how far from being that advertised brand of purely à priori, non-empirical, non-observational, non-nitty-gritty practice it really is.  This is especially true of combinatorics, where a would-be theorist for the lack of a good theory about a species of combinatorial creatures will proceed like a seventeenth century naturalist, collecting specimens of combinatorial fauna and flora until their natures impress themselves on a thus-prepared mind.  Just as I’ve been doing here.

Relations & Their Relatives : 11

In discussing the “combinatorial explosion” of dyadic relations that takes off in passing from a universe of two elements to a universe of three elements, I made the following observation:

Looking back from the ascent we see that the two-point universe $\{ \mathrm{I}, \mathrm{J} \}$ manifests a type of formal degeneracy (loss of generality) compared with the three-point universe $\{ \mathrm{I}, \mathrm{J}, \mathrm{K} \}.$  This is due to the circumstance that the number of “diagonal” pairs, those of the form $\mathrm{A\!:\!A},$ equals the number of “off-diagonal” pairs, those of the form $\mathrm{A\!:\!B},$ so the two-point case exhibits symmetries that will be broken as soon as one adds another element to the universe.

There are two types of symmetry that we might be talking about in this setting and it behooves us to keep them distinctly in mind:

1. There is the symmetry exhibited by pairs of the form $\mathrm{A\!:\!A}$ versus the asymmetry exhibited by pairs of the form $\mathrm{A\!:\!B}.$
2. There is the number of pairs of the form $\mathrm{A\!:\!A}$ versus the number of pairs of the form $\mathrm{A\!:\!B}$ and whether those numbers are equal or not.

The type of symmetry (“sameness in measure”) motivating the above observation is the second type, where the number of pairs on the diagonal is equal to the number of pairs off the diagonal.  That is the symmetry that will be broken when we pass from the 2-point universe to the 3-point universe.

Relations & Their Relatives : 10

Here is the series of blog posts on Chapter 3 (The Logic of Relatives) from Peirce’s 1880 “Algebra of Logic” up to the point where I left off on May Day.

Up to this point we are still dealing mainly with dyadic relations and as interesting as those may be, especially to a graph theorist, the level of complexity it takes for the first signs of semiosis to get up and running does not come into play until we reach the playing field of triadic relations.