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

222

MATHEMATICAL LOGIC

[PART I

The above is the law of tautology for the logical addition of classes.
*22-57.. F.av3=3v a [*4-31. *1011]
*22-58. F.aC.a/3./C a v/ [*P:13.*2'2]
*22-59. F:aCy./3Cry.=-.cUv/Cy
Dem.
F. *22-1 F:: Hp x E a x.~, y e: xE,8:)x. x E ry
[*10-22]:. ( x):.xe..a ey r: XE/.). D y:.
[*4-77.*10-271]:.(x):.x a.av.XE: ).Xy:.
[*22-34.*10-413] rr:.(x): x e a v3.). xEry::)I-. Prop
The analogue of *4'78, i.e.
a C /. v. a Cy: a a C/3v y.
is false. We have only
a C /. v. a Cry:.a C/ ry.
A similar remark applies to the analogue of *4-79. Cf. *22-64-65.
*22-6. 1:. xea v/.: a Cry.3Ciy. C.y x e
Dem.
F.*22x59.:)-:. CyCry./3C7.):xEav/3.).xey:.
[Comm] )F:.xea8v/.):C(aCy./3Cry.).xery:.
[*10Oll21] D F:. x e a vI8.:): a Cy. e Cy.:Y. x ey (1)
F.*1O1. )F:.aCry./Cry.)..xey::):aCctv/3./3Cav/3.).xeav/3:
[*22'58]:):xeav/ (2)
F. (1).(2). ) F. Prop
*22-61. F:aC/3.).aC/3vy [*22-44-58]
*22-62. F: a C/.8.a V/=/8
Dem.
F. *472. ) F:: XE a..XE /3::. XE a a v. xE/3: XE:.
[*22-34]:. XE Ua 1. - XE / (1)
F. (1). *10-271. D F a C._:. xE a w P3- =. xE8:.
[*20-43] E:.au/3= F.::)F.Prop
*22-621. F:aC/3. u.arn/3=a [*4-71]
The proof proceeds as in *22-62. The proposition *22-621 is one of' the
most useful propositions in the present number.
*22-63. F: a v (a /3) = a [*4-44]
The process of obtaining *22-63 from *4-44 is of the same kind as thc
process employed in the proofs that have been written out in this number.