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

200

MATHEMATICAL LOGIC

[PART I

The following definitions merely extend to symbols representing classes
the definitions which have already been given for other symbols, with the
smallest possible modifications.
*20-07. (a).fa.-. ( *).f { (#! z)} Df
*20 071. (ga).fa.. (= ).( f {^z (O! z)} Df
*20072. [(a) (/a)].f(?a) ((a). = (a: (3): a.. a- y Df
*20-08. f /{a (a)}. =: (a): a. — a.! a:/(! a) Df
*20-081. a e*!$.=.r.!a Df
The propositions which follow give the most general properties of classes.
*20-1. 1:./{f (4z)}. -: (gb): cb! x. =. - 4x:f{c!!} [*42. (*20'01)]
*20-11. -:. x. -. xx: D:f{z (*z)}. -.f{z (X)
Dem.
F. *4'86. F:: Hp.:. O! x. -$. x::.! x. -. x. x:.
[*4-36]:.!.. x x:f{^! ^.}: -:!-..XX:f/q!zj:.
[*10-281]:. (0)! x. -x. x:/{f! z}:: (6):! X. - x%:f/{! }:.
[*20-1] ):.f{Z (*z)}.. f{ (Xz)}:: ) F. Prop
This proves that every proposition about a class expresses an extensional
property of the determining function of the class, and therefore does not
depend for its truth or falsehood upon the particular finction selected for
determining the class, but only upon the extension of the determining
function.
*20-111. F:.f(! ).. g (! z): (! z)}.. g (! z)}
Dem.
F. Fact. F:: Hp.:.! x. -=.!x:f(#!z):=:!!x.=.'!'!x:g(!^Z)::
[*10-11'21] D F:: Hp. D:. p!x. -=,.l!x:f(*! )): —,:!-.=-,.! x;: g(*l2):.
[*10-281] 2:.(gt):!-x._=.!x:f(q!z):-:(gf):~!x._.4!x:g(,!2):
[*20-1]./ (! ). (! )} (1)
F. (1). *10-1121. ) F. Prop
*20'112. F:. ([g):.f{z (! z)}.-. g! { ( (! z)}
Dem.
F. *12-1.::. (3[g):/(k! 2l).-=. g!(k! Z) (1). (1).*20111. D F. Prop
Thus the axiom of reducibility still holds for classes as arguments.