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

416

PROLEGOMENA TO CARDINAL ARITHMETIC

[PART II

*62-26. F.R=e R
Dem.
F. *32'18. -: xRy. -.xeR'y.
[*30-33.*32-12] =. (a). x e a. aRy.
[*34'1] -. x (e R) y: ) F. Prop
*62 3; F. e"c = s'K
Dem.
F. *371. F. "K X(. ea}). aK. Xe a
[(*40'02)] s'. D F. Prop
*62'31. F. 62 K = S'C
Note that, since e is not a homogeneous relation, i.e. not one in which
referent and relatum belong to the same type, 62 is strictly meaningless.
For if we have x Ea. a K, the two e's have different meanings, and do not
therefore properly give x e2 K. But it is convenient to allow 62, on the understanding that the ambiguity of e is to be differently determined for the two
factors in the product e ie, namely the second e must make both referent and
relatum belong to the next type above that to which they respectively belong
for the first e.
Dem.
F.*32'13. 13. e' - 62' ( (X E2 K)
[*34'5] = {(ga). xe aa e K
[(*40-02)] = S K
*62-32.. s = e = E [*30-41. *62-3-31. *37 1]
*62-33. F. =IrCls
Dem.
F. *622. *30-3. D F:8e a.-. 8 = a.
[*20-41] =-., = a. a e Cls.
[*50-1.*35-101] -. /(I Cls) a: D F. Prop
The use of *20'41 in the above proof depends upon the fact that a is
merely an abbreviation for an expression of the form (lrz).
*62-34. F. P, = sg'(P I e)
Dem.
-F. *37'101. (*37'01). D F:. aP/3. _: a = x {(gy). y e 3. xPy}
[*34-1] = 2 {x (P e) }3:
[*32-1-23] -: a {sg'(P 1 e)}:. F. Prop