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

218

MATHEMATICAL LOGIC

[PART I

I a. a u b is in the class whenever a and b are in the class.
I b. a n b is in the class whenever a and b are in the class.
II a. There is an element A such that a v A = a for every element a.
II b. There is an element V such that a n V = a for every element a.
III a. av b = b v a whenever a, b, a u b and b u a are in the class.
III b. a n b = b n a whenever a,, a n b and b n a are in the class.
IV a. a u (b n c)= (a b)n (a u c) whenever a, b, c, a u b, a v c, b n c, a u (b n c),
and (a v b) n (a u c) are in the class.
IVb. a (bvc) = (a n b)v(a n c)whenevera, b,c, a b, a n c, b vc, a n(b uc),
and (a n b) u (a n c) are in the class.
V. If the elements A and V in postulates IIa and IIb exist and are
unique, then for every element a there is an element - a such that
a v - a = V and a - a = A.
VI. There are at least two elements, x and y, in the class, such that x $ y.
The form of the above postulates is such that they are mutually independent, i.e. any nine of them are satisfied by interpretations of the symbols
which do not satisfy the remaining one.
For our purposes, "K" must be replaced by "Cls." A and V will be the
null-class and the universal class, which are defined in *24. Then the above
ten postulates are proved below, as follows:
I a, in *22'37, namely "F. a v /3 e Cls"
I b, in *22'36, namely "F. a / e CCls"
I a, in *24 24, namely ". a u A = a"
II b, in *24'26, namely "F. a n V = a"
III a, in *22'57, namely " F. a v / = 8 v a"
III b, in *22-51, namely "F. a A n = /n a"
IV a, in *22'69, namely " F. (a v/3) n (a v y) = a u (, n y)
IV b, in *22'68, namely "F. (a,/) u (a n y) = a n (/ uv y)
V, in *24'21'22, namely "F.a n-a=A" and "F. a u-a=V"
VI, in *24-1, namely " F. A V "
Hence, assuming Huntington's analysis of the postulates for the formal
algebra of logic, the propositions proved in what follows suffice to establish
that this algebra holds for classes. The corresponding propositions of *23
and *25 prove that it holds for relations, substituting Rel, v, A, A,
for Cls, u, n, A, V.