Inquiry Into Inquiry

Re: Peirce List Discussion • HR

We have been considering special properties that a dyadic relation may have, in particular, the following two symmetry properties.

• A dyadic relation is symmetric if being in implies that is in
• A dyadic relation is asymmetric if being in implies that is not in

The first thing to understand about any symmetry of any relation is that it is a property of the whole relation, the whole set of tuples, not a property of individual tuples.

Many properties of dyadic relations can be made visually evident by arranging their ordered pairs in 2-dimensional arrays.  Let’s do this for our initial sample of biblical brothers, using the first three letters of their names as row and column labels.

The relation indicated by “brother of” is a symmetric relation.  The ordered pairs of are given below.

The relation indicated by “elder brother of” is an asymmetric relation.  The ordered pairs of are given below.

Re: Peirce List Discussion • HR

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 (x, y) 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 {x, y} makes its appearance as two ordered pairs, (x, y) and (y, x).

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.

Re: Peirce List Discussion • HR

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.

Re: Peirce List Discussion • GF • JBD • HR

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.