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

SECTION C]

GENERAL THEORY OF CLASSES

207

*20'62. When fi9 is true, whatever possible argument of the form 2 ( )! z)
/3 may be, then (a).fa is true.
This is the analogue of *1011.
Dem.
F.10'11.. when f z( (! z)} is true, whatever possible argument 4
may be, then (4).f f (b! z)} is true, i.e. (by *20-07), (a).fa is true.
*20'63. F:. (a).p vfa. D:p. v (a).fa
This is the analogue of *1012.
Dem.
F. *4-2. (*20707). )
F:.(a).pvfa. -:().pv f {2(!z)}:
[*10-12] -. v. ().f f { (! z)}:
[(*20-07)] E: p. v. (a).fa:. D F. Prop
*20'631. If "fa" is significant, then if 3 is of the same type as a, "f/a" is
significant, and vice versa.
This is the analogue of *10'121.
Dem.
By *20-151, a is of the form (4)!z), and therefore, by *20-01, fa is a
function of )! z. Similarly / is of the form z (r! z), and f/ is a function of
Z! ^. Hence by applying *10121 to 4)! and 4! the result follows.
*20'632. If, for some a, there is a proposition fa, then there is a function fa,
and vice versa.
Dem.
By the definition in *20'01, f{2 (! z)} is a function of Z! 2. Hence the
proposition follows from *10'122.
*20'633. "Whatever possible class a may be, f(a, /3) is true whatever possible
class 3 may be" implies the corresponding statement with a and /3 interchanged except in "f(a, 3)." (The corresponding exception is to be understood in *11-07.)
This is the analogue of *11*07, and follows at once from *11'07 because
f(a, /) is a function of the defining functions of a and /3.
*20-64. F:. (a).fa: (a) ga: ).f/3. g
Dem.. *4'2. (*20'07). )
F. (a).fa: (a). ga:: (). f [{ (! z)}: () ). g {S (! z)}:
[*10'14] f:/{Z (! z)}. g {z (! z)}:. D h. Prop