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

SUMMARY OF PART I.

IN this Part, we shall deal with such topics as belong traditionally to
symbolic logic, or deserve to belong to it in virtue of their generality. We
shall, that is to say, establish such properties of propositions, propositional
functions, classes and relations as are likely to be required in any mathematical
reasoning, and not merely in this or that branch of mathematics.
The subjects treated in Part I may be viewed in two aspects: (1) as a
deductive chain depending on the primitive propositions, (2) as a formal
calculus. Taking the first view first: We begin, in *1 (_nd in 0-'i), with
certain axioms as to deduction of one proposition or asserted propositional
function from another. From these primitive propositions, in Section A,
we deduce various propositions which are all concerned with four ways of
obtaining new propositions from given propositions, namely negation, disjunction, joint assertion and implication, of which the last two can be defined
in terms of the first two. Throughout this first section, although, as will
be shown at the beginning of Section B, our propositions, symbolically
unchanged, will apply to any propositions as values of our variables, yet it
will be supposed that our variable propositions are all what we shall call
elementary propositions, i.e. such as contain no reference, explicit or implicit,
to any totality. This restriction is imposed on account of the distinction
between different types of propositions, explained in Chapter II of the Introduction. Its importance and purpose, however, are purely philosophical, and
so long as only mathematical purposes are considered, it is unnecessary to
remember this preliminary restriction to elementary propositions, which is
symbolically removed at the beginning of the next section.
Section B deals, to begin with, with the relations of propositions containing apparent variables (i.e. involving the notions of "all" or "some")
to each other and to propositions not containing apparent variables. We
show that, where propositions containing apparent variables are concerned,
we can define negation, disjunction, joint assertion and implication in such a
way that their properties shall be exactly analogous to the properties of the
corresponding ideas as applied to elementary propositions. We show also
that formal implication, i.e. " (x). +x D fax " considered as a relation of 42 to
*X, has many properties analogous to those of material implication, i.e. "p ) q"