## ⚠ It’s A Trap ⚠

The most common mathematical trap I run across has to do with Triadic Relation Irreducibility, as noted and treated by the polymath C.S. Peirce.

This trap lies in the mistaken belief that every 3-place (triadic or ternary) relation can be analyzed purely in terms of 2-place (dyadic or binary) relations — “purely” here meaning without resorting to any 3-place relations in the process.

A notable thinker who not only fell but led many others into this trap is none other than René Descartes, whose problematic maxim I noted in the following post.

As mathematical traps go, this one is hydra-headed.

I don’t know if it’s possible to put a prior restraint on the varieties of relational reduction that might be considered, but usually we are talking about either one of two types of reducibility.

Compositional Reducibility. All triadic relations are irreducible under relational composition, since the composition of two dyadic relations is a dyadic relation, by the definition of relational composition.

Projective Reducibility. Consider the projections of a triadic relation $L \subseteq X \times Y \times Z$ on the coordinate planes $X \times Y, ~ X \times Z, ~ Y \times Z$ and ask whether these dyadic relations uniquely determine $L.$ If so, we say $L$ is projectively reducible, otherwise it is projectively irreducible.

### 4 Responses to ⚠ It’s A Trap ⚠

1. Poor Richard says:

Hearing no objection …

One argument for triadic irreducibility from empirical observation is that we can’t seem to do semantic computation (as far as I know) with combinations of dyadic associations. We have to use triples.

• Jon Awbrey says:

That is one of the founding observations of Peirce’s semiotics, or theory of signs. See the links on Sign Relations and the list of readings I collected at the end of the previous post. In ways I’m still trying to understand, Peirce’s theory of triadic sign relations is of a piece with his theory of 3-facet or 3-phase inquiry.

• Poor Richard says:

Jon, if I had the chops I’d work my way through the list of readings, but from here it goes over my pretty little head. I’m cheering on the sideline, though.

2. Jon Awbrey says:

I think we all find someday that something we really want to understand can’t be understood without slogging through some pretty rough trenches, and then there is nothing for it but to slog away.

But try reading the “Interpretation as Action : Risk of Inquiry” paper if nothing else. Anything Sue co-authored will be a far finer read than anything I ever wrote on my own.