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

2.30

MATHEMATICAL LOGIC

[PART I

A and V, V being (in that case) the class whose only member is the one
individual. Our primitive propositions do not require the existence of more
than one individual.
*24'102'103 show that any function which is always true determines the
universal class, and any function which is always false determines the nullclass.
*24'21'22 give forms of the laws of contradiction and excluded middle,
namely "nothing is both a and not-a" (an-a=A) and "everything is
either a or not-a " (a v - a = V).
*24'23'24'26'27 give the properties of A and V with respect to addition and
multiplication, namely: multiplication by V and addition of A make no
change in a class (*24-26'24); addition of V gives V, and multiplication by
A gives A (*24'27'23). It will be observed that the properties of A and V
result from each other by interchanging addition and multiplication.
*24-3. F: a C /..c- -=A
I.e. " a is contained in 3" is equivalent to "nothing is a but not 3."
*24-311. F:aC-/3.=.an8/=A
I.e. "no a is a / " is equivalent to " nothing is both a and /."
*24-411. F:/3Ca. D.a=/3 v(a-/3)
*24'43. F:a-/3CY.-.aC/3vy
As a rule, propositions concerning V are much less used than the correlative propositions concerning A.
The properties of the existence of classes result from those of A, owing to
the fact that X! a is the contradictory of a = A, as is proved in *24'54. Thus
we have, in virtue of *24'3,
*24-55. F:~(aC C). -.!a-/I
I.e. "not all a's are /'s" is equivalent to "there are a's which are not /3's."
This is the familiar proposition of formal logic, that the contradictory of the
universal affirmative is the particular negative.
We have
*24-56. -:.!(av u ). -:!a.v.g!B
*24'561. F:!(anO). D.!a.!/
I.e; if a sum exists, then one of the summands exists, and vice versa; and
if a product exists, both the factors exist (but not vice versa).
The proofs of propositions in the present number offer no difficulty.