We consider two formalisations of the notion of a compositionalsemantics for a language, and find some equivalent statements in termsof substitutions. We prove a theorem stating necessary and sufficientconditions for the existence of a canonical compositional semanticsextending a given partial semantics, after discussing what features onewould want such an extension to have. The theorem involves someassumptions about semantical categories in the spirit of Husserl andTarski.

This is a book review of Cantorian set theory and limitations of size by Michael Hallett.

Provability, Computability and Reflection.

This book is essential for those interested in Russell, or in the philosophy of logic and mathematics. Only two of the fifteen selections are readily available elsewhere: "On Denoting" and "Mr Strawson on Referring." Amazingly, three substantial selections have never been published previously: "On the Substitutional Theory of Classes and Relations," "The Regressive Method of Discovering the Premisses of Mathematics," and "Is Mathematics Purely Linguistic?". Three other papers have appeared previously only in French : one of these is a better (...) introduction to the theory of types than the classic "Mathematical Logic as based on the Theory of Types." Finally, of the five remaining papers, which are not easily available, three are commentaries on Meinong, one is a note on the axiom of infinity, and the most substantial piece is a discussion of three notions of sets—in effect, Zermelo-Frankel, New Foundations, and Russell’s own "no-classes" theory, which required a theory of types applied to propositional functions. What is striking throughout these essays—all but one from the first decade of the century-is Russell’s persistent sympathy with some sort of realism in foundation theory, and his perception that the theory of definite descriptions, the "no-classes" proposals, and the theory of types, were very closely interrelated. Also included are three bibliographies. The chronological bibliography includes several still unpublished mss in the McMaster University Russell Archives.—J.L. (shrink)

Russell tries to give an account of influences that have shaped his philosophy, though there is no mention of the development of his ethical or social views. The last chapter is devoted to the replies to criticisms. As might be expected, a most readable book.--M. G.

The first edition of this now classical work appeared in 1953, the second heavily revised edition in 1961; this most recent edition is a revision in detail only of the previous one. The book is divided into three parts, the first two dealing with finite and infinite sets, infinite cardinals and their arithmetic, and related remarks on non-standard mathematics and the equivalence of various definitions of finitude. The third part considers ordered sets and isomorphism types, the special case of linearly (...) ordered sets, well-ordered sets in general, the relations of ordinals and cardinals. There is a very large bibliography and also a further list of literature pertaining to the companion volume. Fraenkel intended the book for philosophers as well as for mathematicians, and has kept to studying the most general and interesting problems of set theory; metamathematical results are kept in the background and the author works in an informal system of Zermelo-Fraenkel set theory. There is relatively little symbolism and Fraenkel's arguments are therefore more intuitive than rigorous, but clearly everything can be formalized in a proper language. There are many exercises in the book; these help to elaborate points only lightly touched upon in the main text and provide practice necessary for the growth of one's set-theoretical intuition. There are many alternative proofs of important theorems and alternative definitions for essential concepts; this lack of conceptual rigidity should especially please those who reject the now fairly widely held notion that Z-F set theory is the "true" one, the others proposed being ad hoc. Anyone who works his way through this delightfully written text will come away thoroughly prepared to attack more advanced work in the subject.—P. J. M. (shrink)

This book is a translation of some of the more important parts of the Grundgesetze of Frege: the introduction, the first part of the first volume which gives an exposition of the construction, rules, axioms of Frege's formal system, and two appendices, one of which is from the second volume and gives Frege's analysis of the paradox found by Russell in his system. The editor has provided a long introduction "for those not familiar with Frege," although it will benefit those (...) who have something more than acquaintance with his name as well. Frege's original two-dimensional symbolism has been preserved, but the editor has provided enough discussion in his introduction so as to make the going easier. Next to a complete translation of the Grundgesetze, this work is the most useful introduction to the magnum opus of the nineteenth century's greatest logician.—P. J. M. (shrink)

Reviewed Works:John R. Steel, A. S. Kechris, D. A. Martin, Y. N. Moschovakis, Scales on $\Sigma^1_1$ Sets.Yiannis N. Moschovakis, Scales on Coinductive Sets.Donald A. Martin, John R. Steel, The Extent of Scales in $L$.John R. Steel, Scales in $L$.