Inquiry Into Inquiry

Re: Peirce List Discussion • Helmut Raulien

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 • G. Fuhrman • J.B. Downard • H. Raulien

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.

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.

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.

Resources

• Survey of Relation Theory
• Peirce’s 1870 Logic Of Relatives