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

INTRODUCTION.

THE mathematical logic which occupies Part I of the present work has
been constructed under the guidance of three different purposes. In the first
place, it aims at effecting the greatest possible analysis of the ideas with
which it deals and of the processes by which it conducts demonstrations,
and at diminishing to the utmost the number of the undefined ideas
and undemonstrated propositions (called respectively primitive ideas and
primitive propositions) from which it starts. In the second place, it is
framed with a view to the perfectly precise expression, in its symbols, of
mathematical propositions: to secure such expression, and to secure it in the
simplest and most convenient notation possible, is the chief motive in the
choice of topics. In the third place, the system is specially framed to solve
the paradoxes which, in recent years, have troubled students of symbolic
logic and the theory of aggregates; it is believed that the theory of types, as
set forth in what follows, leads both to the avoidance of contradictions, and to
the detection of the precise fallacy which has given rise to them.
Of the above three purposes, the first and third often compel us to adopt
methods, definitions, and notations which are more complicated or more
difficult than they would be if we had the second object alone in view. This
applies especially to the theory of descriptive expressions (*14 and *30) and
to the theory of classes and relations (*20 and *21). On these two points,
and to a lesser degree on others, it has been found necessary to make some
sacrifice of lucidity to correctness. The sacrifice is, however, in the main
only temporary: in each case, the notation ultimately adopted, though its
real meaning is very complicated, has an apparently simple meaning which,
except at certain crucial points, can without danger be substituted in
thought for the real meaning. It is therefore convenient, in a preliminary
explanation of the notation, to treat these apparently simple meanings as
primitive ideas, i.e. as ideas introduced without definition. When the
notation has grown more or less familiar, it is easier to follow the more
complicated explanations which we believe to be more correct. In the body
of the work, where it is necessary to adhere rigidly to the strict logical order
R. & W. 1