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

SECTION A]

PRIMITIVE IDEAS AND PROPOSITIONS

101

*1*5. H: pv(qvr) D.qv(pvr) Pp.
This principle states: "If either p is true, or 'q or r' is true, then either
q is true, or 'p or r' is true." It is a form of the associative law for logical
addition, and will be called the " associative principle." It will be referred to
as "Assoc." The proposition
p v (q v r). D. (p v q) vr,
which would be the natural form for the associative law, has less deductive
power, and is therefore not taken as a primitive proposition.
*1'6. F:.q)r. ):pvq.D.pvr Pp.
This principle states: "If q implies r, then 'p or q' implies 'p or r."' In
other words, in an implication, an alternative may be added to both premiss
and conclusion without impairing the truth of the implication. The
principle will be called the " principle of summation," and will be referred to
as "Sum."
*1'7. If p is an elementary proposition, U-p is an elementary proposition. Pp.
*1'71. If p and q are elementary propositions, p v q is an elementary proposition. Pp.
*1'72. If fp and rp are elementary propositional functions which take
elementary propositions as arguments, pvfrp is an elementary propositional
function. Pp.
This axiom is to apply also to functions of two or more variables. It is
called the " axiom of identification of real variables." It will be observed
that if q and r are functions which take arguments of different types, there
is no such function as " x v rx," because 0 and s cannot significantly have
the same argument. A more general form of the above axiom will be given
in *9.
The use of the above axioms will generally be tacit. It is only through
them and the axioms of *9 that the theory of types explained in the Introduction becomes relevant, and any view of logic which justifies these axioms
justifies such subsequent reasoning as employs the theory of types.
This completes the list of primitive propositions required for the theory
of deduction as applied to elementary propositions.