*21. GENERAL THEORY OF RELATIONS.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
The following propositions are the analogues of *20·6 ff., and have a similar purpose.
*21·6.
*21·61.
*21·62. When is true, whatever possible argument of the form may be, ( is true. [Proof as in *20·62]
*21·63.
*21·631. If "" is significant, then if is of the same type as , "" is significant, and vice versa.
[Proof as in *20·631]
*21·632. If, for some , there is a proposition , then there is a function , and vice versa.
[Proof as in *20·632]
*21·633. "Whatever possible relation may be, is true whatever possible relation may be" implies "whatever possible relation may be, is true whatever possible relation may be."
[Proof as in *20·633]
*21·64.
*21·7.
*21·701.
*21·702.
*21·703.
*21·704.
*21·705.
*21·71.
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.