Peirce’s 1870 “Logic Of Relatives” • Comment 11.12

Since functions are special cases of dyadic relations and since the space of dyadic relations is closed under relational composition — that is, the composition of two dyadic relations is again a dyadic relation — we know that the relational composition of two functions has to be a dyadic relation. If the relational composition of two functions is necessarily a function, too, then we would be justified in speaking of functional composition and also in saying that the space of functions is closed under this functional form of composition.

Just for novelty’s sake, let’s try to prove this for relations that are functional on correlates.

The task is this — We are given a pair of dyadic relations:

\begin{array}{lll}  P \subseteq X \times Y & \text{and} & Q \subseteq Y \times Z  \end{array}

The dyadic relations P and Q are assumed to be functional on correlates, a premiss that we express as follows:

\begin{array}{lll}  P : X \gets Y & \text{and} & Q : Y \gets Z  \end{array}

We are charged with deciding whether the relational composition P \circ Q \subseteq X \times Z is also functional on correlates, in symbols, whether P \circ Q : X \gets Z.

It always helps to begin by recalling the pertinent definitions:

For a dyadic relation L \subseteq X \times Y, we have:

\begin{array}{lll}  L ~\text{is a function}~ L : X \gets Y & \iff & L ~\text{is}~ 1\text{-regular at}~ Y.  \end{array}

As for the definition of relational composition, it is enough to consider the coefficient of the composite relation on an arbitrary ordered pair, i\!:\!j. For that, we have the following formula, where the summation indicated is logical disjunction:

(P \circ Q)_{ij} ~=~ \sum_k P_{ik} Q_{kj}

So let’s begin.

P : X \gets Y, or the fact that P ~\text{is}~ 1\text{-regular at}~ Y, means that there is exactly one ordered pair i\!:\!k \in P for each k \in Y.

Q : Y \gets Z, or the fact that Q ~\text{is}~ 1\text{-regular at}~ Z, means that there is exactly one ordered pair k\!:\!j \in Q for each j \in Z.

As a result, there is exactly one ordered pair i\!:\!j \in P \circ Q for each j \in Z, which means that P \circ Q ~\text{is}~ 1\text{-regular at}~ Z, and so we have the function P \circ Q : X \gets Z.

And we are done.

This entry was posted in Graph Theory, Logic, Logic of Relatives, Logical Graphs, Mathematics, Peirce, Relation Theory, Semiotics and tagged , , , , , , , . Bookmark the permalink.

One Response to Peirce’s 1870 “Logic Of Relatives” • Comment 11.12

  1. Pingback: Survey of Relation Theory • 3 | Inquiry Into Inquiry

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s