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

184

MATHEMATICAL LOGIC

[PART I

*14'01. [(1x) (<$)]., (0x) (<x). =: (gb)::<*x. -.x = b:,b Df
14'02. E!(ix)(qx).=:(gb ): qx.-:.x=bb Df
*14'03. [(ix) (4x), (ix) (#x)].f/{(x)) (,x), (x) (x)l =:
[(x) (<x)] [(1x') (,$)]./{(x) ($), 0x) (~x)} Df
*14-04. [(7x) (,k)].f{(0) (<x), (0x) (~)} ~ =.
[(ix) (,x), (ix) (ox)]./ {(i) (ox), (ix) ( }x)J Df
*14'1. k:. [(?x) (ox)]. * ( x) (ox) (: (b):. -x. x = b: *b
[*4'2. (*14-01)]
In virtue of our conventions as to the scope intended when no scope is
explicitly indicated, the above proposition is the same as the following:
*14'101. F:. * (x) (bx). =-: (lb): bx. -. = b: fb [*14-1]
*14'11. F:. E! (x) (4bx). -: (ab): bx. —. x= b [*42. (*14-02)]
*14111. 1:. [(ix) (#x)].ft{(x) (x), (ix) (x)}. -:
(ab, c): Ox. -.x = b:.. -xx =c:f(b, c)
Demn.. *42. (*1404'03). ):: [(7x) (x)]. f (?x) ((x), (x) (x)}:.
[(ix) (#x)] [(ix) (Ox)].f{(ix) (4x), (ix) (x)}:.
[*14'1] -:. [(ix) (x)]:. (g)b): bx.. -. x= = b:ffb, (ix) (x)}:.
[*14'1] -:. ([c):.x.. x.x= c:. (gb): x. =x. x = b:f(b, c):.
[*11'55]-:. (jb, c): x...x=c: x -.. x = c:f(b, c):: 3 F. Prop
*14'112.:.f /{(ix) (+x), (7x) (#x)}. =:
(3b, c): bx. --.x *= b: *x.. -. =c:f(b, c)
[Proof as in *14-111]
In the above proposition, we assume the convention explained on p. 182,
after the statement of *14'03.
*14-113. I: [(ix) (kx)]./{(ix) (/x), (ix) (x)}. =.f{(ix) (qx), (ix) (#x)}
[*14'111-112]
This proposition shows that when two descriptions occur in the same
proposition, the truth-value of the proposition is unaffected by the question
which has the larger scope.
*14'12. k:. E! (x) (oQx). ): *x. fy. x,y. = y
Dem.
F. *14-11 2 F:. Hp. ): (gb): Ofx. -. x = b (1)
F-.*438. 10'1. *11-11-3.
F:. 4x.-. = b: 2: Ox y. y x = b.y = b.
[*13-172] )X,Y.x=y (2)
F. (2). *10'11'23. ):. (2b): Ox. -=x =.: b::. py. 3,,y. =y(3)
F. (1). (3). D F. Prop