First published in 1937. Routledge is an imprint of Taylor & Francis, an informa company.

Published in 1903, this book was the first comprehensive treatise on the logical foundations of mathematics written in English. It sets forth, as far as possible without mathematical and logical symbolism, the grounds in favour of the view that mathematics and logic are identical. It proposes simply that what is commonly called mathematics are merely later deductions from logical premises. It provided the thesis for which _Principia Mathematica_ provided the detailed proof, and introduced the work of Frege to a wider (...) audience. In addition to the new introduction by John Slater, this edition contains Russell's introduction to the 1937 edition in which he defends his position against his formalist and intuitionist critics. (shrink)

By a `denoting phrase' I mean a phrase such as any one of the following: a man, some man, any man, every man, all men, the present King of England, the present King of France, the center of mass of the solar system at the first instant of the twentieth century, the revolution of the earth round the sun, the revolution of the sun round the earth. Thus a phrase is denoting solely in virtue of its form. We may distinguish (...) three cases: (1) A phrase may be denoting, and yet not denote anything; e.g., `the present King of France'. (2) A phrase may denote one definite object; e.g., `the present King of England' denotes a certain man. (3) A phrase may denote ambiguously; e.g. `a man' denotes not many men, but an ambiguous man. The interpretation of such phrases is a matter of considerably difficulty; indeed, it is very hard to frame any theory not susceptible of formal refutation. All the difficulties with which I am acquainted are met, so far as I can discover, by the theory which I am about to explain. (shrink)

83. It should now be apparent that the familiar problems of reference and modality can be satisfactorily resolved without either abandoning substitutivity of identity in modal contexts or invoking an intensional ontology. It might be argued, however, that the suggested resolution avoids invoking an intensional ontology only because the modal predicates, e.g. Nec and Believes, are left unanalysed. This may well be true but it should be appreciated that the analysis of modal predicates is an issue which is independent of (...) both the problem of reference in modal contexts and the problem of the logical structure of propositions containing such contexts. These problems can be settled, as we have seen, without also settling how the modal predicates are to be analysed. Hence, if an intensional ontology is in the end required, it is required to analyse the modal predicates, not to solve the problems of reference and logical structure. This, we suggest, is an interesting and perhaps unexpected result. (shrink)

This paper consists principally of selections from a much longer work on the semantics of English. It discusses some problems concerning how to represent grammatical modifiers (e.g. slowly in x drives slowly) in a logically perspicuous notation. A proposal of Reichenbach's is given and criticized; then a new theory (apparently discovered independently by myself, Romain Clark, and Richard Montague and Hans Kamp) is given, in which grammatical modifiers are represented by operators added to a first-order predicate calculus. Finally some problems (...) concerning applications of adjectives to that-clauses and gerundive-clauses are discussed. (shrink)

Evidence is presented to show that the role of a generative grammar of a natural language is not merely to generate the grammatical sentences of that language, but also to relate them to their logical forms. The notion of logical form is to be made sense of in terms a natural logic, a logical for natural language, whose goals are to express all concepts capable of being expressed in natural language, to characterize all the valid inferences that can be made (...) in natural language, and to mesh with adequate linguistic descriptions of all natural languages. The latter requirement imposes empirical linguistic constraints on natural logic. A number of examples are discussed. (shrink)

I reject the contention that an important theoretical difference exists between formal and natural languages.