Harold R. (Hal)
Foster’s Prince Valiant
© Respective
copyright/trademark holders.
In an exactly analogous manner we introduce dual or
dyadic relations, i.e. relations
between two terms. Such relations will be called simply "relations";
relations between more than two terms will be distinguished as
multiple relations, or (when the number of their terms is specified) as triple,
quadruple,... relations, or as triadic, tetrad-al,... relations. Such relations
will not concern us until we come to Geometry. For the present, the only
relations we are concerned with are dual
relations.
Relations, like classes, are to be taken in extension, i.e. if R and S are relations which hold between the
same pairs of terms, R and S are to be identical. We may regard a
relation, in the sense in which it is required for our purposes, as a class of
couples; i.e. the couple (x, y) is to be one of the class of
couples constituting the relation R
if x has the relation R to y*. This view of relations as classes of
couples will not, however, be introduced into our symbolic treatment, and is
only mentioned in order to show that it is possible, so to understand the
meaning of the word relation; that a relation shall be determined by its extension.
Any function φ(x,
y) determines a relation R
between x and y. If we regard a relation as a class of couples, the relation
determined by φ(x, y) is the class of
couples (x, y) for which φ(x, y) is true. The relation determined
by the function φ(x, y) will be
denoted by
x^y^φ (x,
y).
We shall use a capital letter for a relation when it is
not necessary to specify the determining function. Thus whenever a capital
letter occurs, it is to be understood that it stands for a relation. The
propositional function "x has
the relation R to y" will be expressed by the
notation
xRy.
This notation is designed to keep as near as possible
to common language, which, when it has to express a relation, generally mentions
it between its terms, as in "x
loves y," "x equals y," "x is
greater than y," and so on. For
"relation" we shall write "Rel";
thus "R є Rel" means "R
is a relation.
"Owing to our taking relations in extension, we
shall have
├ :. x^y^φ (x, y) = x^y^y (x, y). φ(x,y) . ºx,y y(x,y),
i.e.
two functions of two variables determine the same relation when, and only when,
the two functions are formally equivalent.
We have ├ .
z {x^y^φ (x, y)} w . º .
φ (z, w),
i.e. "z
has to w the relation determined by
the function φ (x, y)" is
equivalent to φ (z,w);
├ :. R = x^y^φ (x, y) º : xRy . ºx,y .
φ(x,y),
├ :. R = S. =: xRy. ºx,y .
xSy,
├. x^y^(xRy) = R,
├ . {x^y^φ (x, y)} є Rel.
These
propositions are analogous to those previously given for classes. It results
from them that any function of two variables is formally equivalent to some
function of the form xRy; hence, in
extensional functions of two variables, variation of relations can replace
variation of functions of two variables.
Pencils and Inks: Jay Scott Pike © Respective copyright/trademark holders.
With relations between classes we reach
a new complexity that is so often the cause of plans running of the rails.
Whether it is of the wind, the sky, the buffalo herd and an enemy’s intentions,
a calculus of variables in the human mind may discern the playing out of events
with uncanny accuracy born of an easy understanding and appreciation of a life’s
totality in which all critical factors are known.
In today’s dominant culture this all-encompassing
perception is certainly elusive, but by means of this logical metaphysical system
we are exploring, one may cover all the bases in constructing propositional
equations in which we may test our hypotheses in such a manner as to clearly
see if they are in error, or indeed nonsense, and perceive the difference.
Harold R. (Hal)
Foster’s Prince Valiant
© Respective
copyright/trademark holders.
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