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

SECTION B]

DESCRIPTIONS

189

*14-171. H:. (x) (ox)-=. -=:! b. 3. x! (0x) (Ox)
Dem.
F.,14-17. D F:. 0x) (Ox) =. =: b! b.! (?x) (ox) (1). *10'1. *121.) F:.! b. D),. r! (x) (zx): D): b =b.. (x) (x) = b:
[*13-15] D:(7x) (4x)=b (2)
F. (1). (2). ). Prop
*14'18. F:. E! (?x) (x). ): (x). x.. * (,x) (x)
Dem.
F.101'. F: (x)... b:
[Fact] ~:. ). - ~. = b: (x).kx: 3: ~;. =_.x = b:kb:
[Fc1035] DF::(3b): /x.=-b.x=:.(x).tx:.:(b.):x.x..=b:Ib:.
[o10..1 28]:. (ab): x.-. ~ = b: (x).,x: D: (b): Ox.-x.x = b: b:.
[*1,os] D F:: (ab): Ox. =-x*. = b:.(.). *:.: (ab): O. —. x* =:b:.
[*14-1-11] D F:. E! (ix) (x): (x). #x: D: (ix) (bx):. ) F. Prop
The above proposition shows that, provided (ix) (x) exists, it has
(speaking formally) all the logical properties of symbols which directly
represent objects. Hence when (ix) (x) exists, the fact that it is an incomplete symbol becomes irrelevant to the truth-values of logical propositions
in which it occurs.
*14'2.. (ix) (x = a) = a
Dem.
F. *14 101. 3 F:. (z) (x = a) = a.: (ab): x = a. =. x = b: b =a:
[*13'195]: x = a. =-x. x = a (1)
F.(1). Id. )D. Prop
*14-201.: E! (ix) (x).. (gx). x
Denm.
F. *14'11.):. Hp. D: (b): x.. x =b:
[*10o1]: (gb):.b.. b = b:
[*13'15] D: (tb). bb:. D F. Prop
*14'202. F:. Ox. x=. = b:=: (ix) (cx)= b::: x. =-. b = x:=: b=(ix) (cx)
Dem.
F. *14 1. ) F:. (? x-) (Q = b. -: (gc): fx. -x. x= c: c= b:
[*13-195] -: x.. x = b:. ). Prop
[The second half is proved in the same way as the first half.]
*14-203. k:. E! (1X) (x) (x). x: (rx. *y. 3,y. x = y
Dem.
k. *14.12.201. D k:. E! (x)(Qbx). D: (ax) *.: fx. dy. ).y. x= y (1)