The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed development of higher-order logic, including a comprehensive discussion of its semantics. Professor Shapiro demonstrates the prevalence of second-order notions in mathematics is practised, and also the extent to which mathematical concepts can
be formulated in second-order languages . He shows how first-order languages are insufficient to codify many concepts in contemporary mathematics, and thus that higher-order logic is needed to fully
reflect current mathematics. Throughout, the emphasis is on discussing the philosophical and historical issues associated with this subject, and the implications that they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic as might be gained from a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics
will find this a thought-provoking discussion of some of the central issues in this subject.
'In this excellent treatise Shapiro defends the use of second-order languages and logic as framework for mathematics. His coverage of the wide range of logical and philosophical topics required for understanding the controversy over second-order logic is thorough, clear, and persuasive. Shapiro recognizes that it is unlikely that he has had the last word on these controversial philosophical subjects. Nevertheless, his book is certainly an excellent place to
start work on them.'
Michael D. Resnik, University of North Carolina, History and Philosophy of Logic, 14 (1993)
`Clearly written and well produced, making it a pleasure to read . . . Shapiro's excellent book is for just about everyone. Logicians and philosophers of mathematics will certainly want to read it; but metaphysicians and philosophers of science and of language will greatly benefit, too.'
PART I: ORIENTATION; Terms and questions; Foundationalism and foundations of mathematics; PART II: LOGIC AND MATHEMATICS; Theory; Metatheory; Second-order logic and mathematics; Advanced metatheory; PART III: HISTORY AND PHILOSOPHY; The historical triumph of first-order languages; Second-order logic and rule-following; The competition; Index.