Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/169

 SECTION B] THEOR Y OF ONE APPARENT VARIABLE 123456

147 Dem. I-. *10'1 . * 1-6 . ::> I- :. p. v. (.1;) . cp:JJ : ::>. p v cpy :. [ *10'11] ::> I- :. (y) :. p. v. (x). cpx : ::>. P v cp y :. [*10'12J ::> I- :. p. v. (x). cpx: ::>. (y). p v cpy (1) 1-. *10'12. ::> 1-:. (y) .pv cpy.::>:p. v. (x). cpa; (2) I-. (1) . (2) . ::> I-. Prop. !! :x · : p · ::> · (:x) · cf>a: ["'10'2 ;p J This proposition is much more used than *10"2. Dern. I- . *10'1 . ::> I- : (x). cpa:. vr. ::> . cpy. yy. (1) [*3'26] ::> . cpy : [*10-11 J ::> I- :. (y) : (.1:) . cpa; . 'o/a; . ::> . cpy :. [*10'21] ::> I- :. (a:) . cpa: . t:JJ. ::> . (y) . cpy (2) I- . (1) . *3' 2 7 . ::> I- :. (x) . cpx . '0/ JJ . ::> . ,y z :. [*10-11] ::> I- :. (z) : (x) . cpa:. yx. ::> . yz :. [ *10' 21 ] ::> I- :. (x) . cpa; . t 3J . ::> . (z) . '0/ z (3 ) I- . (2). (3). Comp . ::> I- :. (x). cpa;. tx. ::> : (y) . cpy : (z). 'o/z (4) 1-. *10'14'11 . ::> 1-:. (y) :.(X).cp.l:: (.1:.'). 'o/IT:::>. cf>!J .yy:. [ *10' 21 ] ::> f- :. (x) . cpr : (:L) . '0/ x : ::> . (y) . cp y . yy ( 5) I- . (4) . (5) . ::> I- . Pro p The above proposition is true whenever it is significant; but, as was pointed out in connexion with *10'14, it is not always significant when "(x) . cpx : (x) . 'lrx" is significant. stituent X (x, u, v, ...), where X is an elementary function and y, z, _.. u, v, ... are either constants or apparent variables, then cf>f£ and yx take arguments of the sante type. This can be proved in each particular case, though not generally, provided that, in obtaining cp and" frOln x' X is only submitted to negations, disjunctions and generalizations. The process may be illustrated by an example. Suppose cpa; is (Y).X(x, y).::>.Ox, and yx isfx. ::>.(y). X (x, y). By the definitions of *9, cpx is (y).X (x, y) v Ox, and ""X is (y).fxvx(x, y). Hence since the primitive ideas (x). Fx and (x) .Fx only apply to functions, there are functions X (1£, y) v Ox, Jf£ v X (5;, y). Hence there is a proposi- tion x (a, b) v Oa. Hence, since "p v q" and "r-.Jp" are only significant 10-2
 * 10'2. I- :. (x) . p v cpX. = : p. v . (x) . cpX
 * 10'22. I- :. (x) . cpx. ,yx . - : (x) . cpx : (x) . yx
 * 10'221. If cpx contains a constituent X (x, y, z, ...) and +x contains a con-