Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

TRANSFORMATION 165
regular instruction, but it is, I believe, destined by its
intrinsic importance to win such recognition.
Relations, as we are presently to see, are determined
by propositional functions of two or more variables, and
are accordingly described as dyadic, triadic (3-cornered),
tetradic (4-cornered),..., n-adic (n-cornered), and so
on. The most important ones are the dyadic relations.
What is meant by a dyadic relation? I will answer
as clearly and simply as I can and will do so by the help
of two familiar examples. Consider the two propositional
functions: (I) 2x+3y-I=0; (2) x is a parent of y.
Each of these is said to determine a dyadic relation.
What is the relation determined by (I)? We see that
(t) will be satisfied if, for example, we replace x by I and
y by -- and so we say that the ordered pair (I, -4) is a
couple of verifiers of (I); another such couple is (O, 1);
there are, you see, infinitely many such couples; the set,
or class, of all the couples of verifiers of (I) is said to be
the relation determined by (I). Each of the couples may
be called an element or constituent of the relation. What
is the relation determined by (2)? Suppose John Smith is
the father of Bill Smith, then the ordered pair (John
Smith, Bill Smith) is a couple of verifiers of (2); the class
of all such verifying couples is the relation determined by
(2). In the light of these simple illustrations you will
rightly understand that a dyadic relation is the class of
all the couples (x, y) that verify (satisfy) some propositional function F(x, y) containing two (and only two)
variables, say, x and y.
It is necessary to note carefully the following distinction in usage: in ordinary function-theory-say, in algebra
or in analytical geometry-it makes no essential difference
whether the x-terms in a propositional function come first