Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

216

MATHEMATICAL LOGIC

[PART I

*21'632. If, for some R, there is a proposition fR, then there is a function
fR, and vice versa.
[Proof as in *20-632]
*21-633. "Whatever possible relation R may be, f(R, S) is true whatever
possible relation S may be " implies "whatever possible relation S may be,
f (R, S) is true whatever possible relation R may be."
[Proof as in *20-633]
*21-64. F:. (R).fR: (R). gR: ).fS. gS [Proof as in *20-64]
*21'7. F: (ag):fR. R.g!R [Proof as in *20-7]
*21'701. I: ({g):f(R, x). — RX. g (R, x) [Proof as in *20'701]
*21'702. I: (qg):f(x, R).-R,X g! (R, ) [Proof as in *20-702]
*21'703.: (23g):f(R, S). —R,S g! (R, S) [Proof as in *20-703]
*21'704. I: (g):f(R, a). -R,a. g! (R, a) [Proof as in *20'703]
*217705. I: ({g):f/(a, R). —,R. (a, R) [Proof as in *20'703]
*21'71. F:. R = S. -: g! R.. g! S [Proof as in *20'71]
From the above propositions it appears that relations, like classes, have
all the formal properties which they would have if they were symbols having
a meaning in isolation. Hence unless a symbol occurs in a way in which
only a relation can occur significantly, we do not need to decide whether it
stands for a relation or not. This result, like the corresponding result for
classes mentioned at the end of *20, is important as giving greater generality
to our propositions than they would otherwise possess. The results obtained
in *20 and *21 for classes and relations whose members or terms are neither
classes nor relations can be extended, by mere repetition of the proofs, to
classes of classes, classes of relations, relations of classes, relations of relations,
and so on.