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

SECTION A]

EQUIVALENCE AND FORMAL RULES

121

of these ideas alone. We shall give the name of a truth-function to a function
f(p) whose argument is a proposition, and whose truth-value depends only
upon the truth-value of its argument. All the functions of propositions with
which we shall be specially concerned will be truth-functions, i.e. we shall
have
p- q. D. f ()-f (q).
The reason of this is, that the functions of propositions with which we deal
are all built up by means of the primitive ideas of *1. But it is not a
universal characteristic of functions of propositions to be truth-functions.
For example, "A believes p" may be true for one true value of p and false
for another.
The principal propositions of this number are the following:
*4'1. ': pO.. -. q p~p
*4'11.:p- q.-. p- q
These are both forms of the "principle of transposition."
*4'13. F.p =~(~p)
This is the principle of double negation, i.e. a proposition is equivalent to
the falsehood of its negation.
*4-2.. p - p
*4'21. F:p- q.-. q-p
*4'22. p q. q - r.. p r
These propositions assert that equivalence is reflexive, symmetrical and
transitive.
*4'24. F:p.=.p.p
*4'25. F:p.-.pvp
I.e. p is equivalent to "p and p" and to "p or p," which are two forms of
the law of tautology, and are the source of the principal differences between
the algebra of symbolic logic and ordinary algebra.
*4*3.:. q.. q.p
This is the commutative law for the product of propositions.
*4'31. F:pvq.=.qvp
This is the commutative law for the sum of propositions.
The associative laws for multiplication and addition of propositions,
namely
*4'32. F:(p.q).r.-. p.(q.r)
E*433. F:(pvq)vr.-.pv(qvr)
The distributive law in the two forms