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.