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

viii

PREFACE

that we have shown) that it is possible to construct a mathematical logic
which does not lead to contradictions. It should be observed that the whole
effect of the doctrine of types is negative: it forbids certain inferences which
would otherwise be valid, but does not permit any which would otherwise be
invalid. Hence we may reasonably expect that the inferences which the
doctrine of types permits would remain valid even if the doctrine should
be found to be invalid.
Our logical system is wholly contained in the numbered propositions,
which are independent of the Introduction and the Summaries. The
Introduction and the Summaries are wholly explanatory, and form no part
of the chain of deductions. The explanation of the hierarchy of types in
the Introduction differs slightly from that given in *12 of the body of the
work. The later explanation is stricter and is that which is assumed
throughout the rest of the book.
The symbolic form of the work has been forced upon us by necessity:
without its help we should have been unable to perform the requisite
reasoning. It has been developed as the result of actual practice, and is not
an excrescence introduced for the mere purpose of exposition. The general
method which guides our handling of logical symbols is due to Peano. His
great merit consists not so much in his definite logical discoveries nor in the
details of his notations (excellent as both are), as in the fact that he first
showed how synlbolic logic was to be freed from its undue obsession with the
forms of ordinary algebra, and thereby made it a suitable instrument for
research. Guided by our study of his methods, we have used great freedom
in constructing, or reconstructing, a symbolism which shall be adequate to
deal with all parts of the subject. No symbol has been introduced except
on the ground of its practical utility for the immediate purposes of our
reasoning.
A certain number of forward references will be found in the notes and
explanations. Although we have taken every reasonable precaution to secure
the accuracy of these forward references, we cannot of course guarantee their
accuracy with the same confidence as is possible in the case of backward
references.
Detailed acknowledgments of obligations to previous writers have not
very often been possible, as we have had to transformn whatever we have
borrowed, in order to adapt it to our system and our notation. Our chief
obligations will be obvious to every reader who is familiar with the literature
of the subject. In the matter of notation, we have as far as possible followed
Peano, supplementing his notation, when necessary, by that of Frege or by
that of Schrdder. A great deal of the symbolism, however, has had to be
new, not so much through dissatisfaction with the symbolism of others, as
through the fact that we deal with ideas not previously symbolised. In all