## Relations & Their Relatives : 19

I would not want the dyadic case to detain us too long, as often happens when we frame a simple example for the purpose of illustration and then fail to rise beyond it.

I raised the example of biblical brothers simply as a way of illustrating the distinction between a relation proper, like that symbolized by the formula “x is y’s brother” and any of its elementary relations, like the ordered pair (Cain, Abel).

There are, however, a few more points that could be illustrated within the scope of this simple example.

Recall that we had a universe of discourse X consisting of biblical figures and a 2-place relation B forming a subset of the cartesian product X × X such that (xy) is in B if and only if x is a brother of y.

The “biblical brother relation” B would contain a large number of elementary dyadic relations or ordered pairs (x, y), for example:

(Abel, Cain), (Isaac, Ishmael), (Esau, Jacob), (Benjamin, Joseph), …
(Cain, Abel), (Ishmael, Isaac), (Jacob, Esau), (Joseph, Benjamin), …

Because B is a symmetric relation, each unordered pair {xy} makes its appearance as two ordered pairs, (xy) and (yx).

The extension of the elder brother relation E would have the pairs:

(Cain, Abel), (Ishmael, Isaac), (Esau, Jacob), (Joseph, Benjamin), …

Peirce regarded a set of tuples as an “aggregate” or “logical sum” and would have written the above subset of B in the following way:

B  =  Abel:Cain +, Isaac:Ishmael +, Esau:Jacob +, Benjamin:Joseph +, …
+, Cain:Abel +, Ishmael:Isaac +, Jacob:Esau +, Joseph:Benjamin +, …

So what does all this — the distinction between relations in general and elementary relations plus the analysis of relations in general as sets or sums of elementary relations — imply for the case of triadic relations in general and sign relations in particular?

It means that non-trivial examples of triadic relations are aggregates, logical sums, or sets of many elementary triadic relations or triples.

As a result, the classification of single triples and their components gets us only so far in the classification of triadic relations proper, and except in very special cases not very far at all.

## Relations & Their Relatives : 18

The immediate task is to get clear about the critical relationship between relations as sets and elementary relations as elements of those sets.  What’s at stake is understanding the extensional aspect of relations.  Beyond its theoretical importance, the extensional aspect of relations is the interface where relations make contact with empirical phenomena and ground logical theories in observational data.

The relationship between tokens and types, under one pair of terms or another, has been pervasive in science and knowledge-oriented philosophy from the time of Plato and Aristotle at least, arising from the observation that knowledge is of forms and generalities, not haecceities or individuals in themselves.

There is a communication problem that arises here, because the words “token” and “type” tend to be used differently outside Peirce studies, referring to objects that aren’t always signs.  So I have found it less confusing to use more neutral terms, like “instance of a type” or “element of a set”.

In that sense, we can say that the ordered pair (Cain, Abel) is an instance of the type B, where B is a particular subset of all ordered pairs of biblical figures.

## Relations & Their Relatives : 17

I think a few people are making this harder than it needs to be.

Let’s put aside potential subtleties about elementary vs. individual vs. infinitesimal relatives and simply use “elementary relative” to cover all cases at a first approximation.  One of the advantages of this approach is the analogy it highlights between elementary relations in the logic of relatives and elementary transformations in linear algebra, affording a bridge to practical applications of relation theory.

The time has come for a concrete example.  Suppose we have a universe of discourse X consisting of biblical figures.

Linguistic phrases like “brother of __” or “x is y’s brother” and many others may be used to indicate a dyadic relation B forming a subset of X × X such that (x, y) is in B if and only if x is a brother of y.

It is often convenient to use Peirce’s notation x:y for the ordered pair (x, y).  Among other things it’s easier to type on the phone.

In the universe X of biblical figures, Cain:Abel is an elementary relation in the brotherhood relation B.

But Cain:Abel also belongs to the relation E indicated by “elder brother of” and again to the relation S indicated by “slayer of”.  So the elementary relation by itself does not completely determine the general relation or general relative term under which it may be considered.

This means that classifying relations is a task at a categorically higher level than classifying elementary relations.

In the special case of triadic sign relations, almost all the literature so far has tackled only the case of elementary sign relations.

## Relations & Their Relatives : 16

Re: Peirce List Discussions • (1)(2)

First off, we need to be clear about the difference between objects and signs:

• Relations are formal objects of discussion and thought while relative terms are signs employed to denote relations.  (The shorthand term “relative” is short for “relative term”.)
• The default meaning for “relative term” is general relative term, that is, a term whose denotation extends over many objects.
• The default meaning for “relation” is general relation, that is, an aggregate, collection, or set of elementary relations.

Next, we need to be clear about the distinction between relatives (= general relatives) and elementary relatives:

• Note.  There is a distinction in Peirce’s usage between elementary relatives and individual relatives, but if we factor in what he says about the Doctrine of Individuals and recognize that we are dealing with abstract forms then it becomes a “distinction without a difference”.  So I will tend to use the terms interchangeably.

Here is one place where Peirce exhibits his appreciation of the critical difference between relatives in general and elementary or individual relatives.

### Chapter 3. The Logic of Relatives

#### §4. Classification of Relatives

225.   Individual relatives are of one or other of the two forms

$\begin{array}{lll} \mathrm{A : A} & \qquad & \mathrm{A : B}, \end{array}$

and simple relatives are negatives of one or other of these two forms.

226.   The forms of general relatives are of infinite variety, but the following may be particularly noticed.

Relatives may be divided into those all whose individual aggregants are of the form $\mathrm{A : A}$ and those which contain individuals of the form $\mathrm{A : B}.$  The former may be called concurrents, the latter opponents.

It needs to be appreciated that classifying relations is vastly more complex than classifying elementary or individual relations.

In particular, classifying sign relations is vastly more complex than classifying elementary or individual sign relations, which is just about all the entire literature on sign taxonomy has been able to touch from Peirce’s time to ours.

## Survey of Relation Theory • 2

In this Survey of previous blog and wiki posts on Relation Theory, relations are viewed from the perspective of combinatorics, in other words, as a topic in discrete mathematics, with special attention to finite structures and concrete set-theoretic constructions, many of which arise quite naturally in applications.  This approach to relation theory, or the theory of relations, is distinguished from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other.

## Peirce’s Categories • 3

Recent travels and other travails (dental work) have scattered my thoughts to the four winds, so let me just document a few bits from my current state of mind in case I can get back to it someday.

Here is the figure I drew to illustrate a recurring theme from Peirce.

Normative science rests largely on phenomenology and on mathematics;
metaphysics on phenomenology and on normative science.

— Charles Sanders Peirce, Collected Papers, CP 1.186 (1903)
Syllabus : Classification of Sciences (CP 1.180–202, G-1903-2b)

Here are a few additional resources that I find useful by way of establishing a foothold on these shores.

## Peirce’s Categories • 2

According to Peirce, it is logic that draws on both mathematics and phenomenology.

At any rate, Peirce takes the distinctive position that normative science, which includes logic, “rests largely on” phenomenology and mathematics.  Unless there is a case to be made for a practical difference between drawing on and resting on, as those phrases are intended in the present setting, I would have to say they mean the same thing.

I discussed the relationship among these sciences in a previous post and drew the following figure to illustrate it.

Normative science rests largely on phenomenology and on mathematics;
metaphysics on phenomenology and on normative science.

— Charles Sanders Peirce, Collected Papers, CP 1.186 (1903)
Syllabus : Classification of Sciences (CP 1.180–202, G-1903-2b)

The following post contains a longer excerpt from Peirce’s Classification of the Sciences.