Numbers and other mathematical objects are exceptional in having no locations in space or time or relations of cause and effect. This makes it difficult to account for the possibility of the knowledge of such objects, leading many philosophers to embrace nominalism, the doctrine that there are no such objects, and to embark on ambitious projects for interpreting mathematics so as to preserve the subject while eliminating its objects. This book cuts through a host of technicalities that have obscured previous (...) discussions of these projects, and presents clear, concise accounts of a dozen strategies for nominalistic interpretation of mathematics, thus equipping the reader to evaluate each and to compare different ones. The authors also offer critical discussion, rare in the literature, of the aims and claims of nominalistic interpretation, suggesting that it is significant in a very different way from that usually assumed. (shrink)

This book is concerned with `the problem of existence in mathematics'. It develops a mathematical system in which there are no existence assertions but only assertions of the constructibility of certain sorts of things. It explores the philosophical implications of such an approach through an examination of the writings of Field, Burgess, Maddy, Kitcher, and others.

This paper criticizes George Boolos's famous use of plural quantification to argue that monadic second-order logic is pure logic. I deny that plural quantification qualifies as pure logic and express serious misgivings about its alleged ontological innocence. My argument is based on an examination of what is involved in our understanding of the impredicative plural comprehension schema.

Michael Dummett and Dag Prawitz have argued that a constructivist theory of meaning depends on explicating the meaning of logical constants in terms of the theory of valid inference, imposing a constraint of harmony on acceptable connectives. They argue further that classical logic, in particular, classical negation, breaks these constraints, so that classical negation, if a cogent notion at all, has a meaning going beyond what can be exhibited in its inferential use. I argue that Dummett gives a mistaken elaboration (...) of the notion of harmony, an idea stemming from a remark of Gerhard Gentzen's. The introduction-rules are autonomous if they are taken fully to specify the meaning of the logical constants, and the rules are harmonious if the elimination-rule draws its conclusion from just the grounds stated in the introduction-rule. The key to harmony in classical logic then lies in strengthening the theory of the conditional so that the positive logic contains the full classical theory of the conditional. This is achieved by allowing parametric formulae in the natural deduction proofs, a form of multiple-conclusion logic. (shrink)

Reviewed Works:Gaisi Takeuti, Proof Theory.Georg Kreisel, Proof Theory: Some Personal Recollections.Wolfram Pohlers, Contributions of the Schutte School in Munich to Proof Theory.Stephen G. Simpson, Subsystems of $\mathbf{Z}_2$ and Reverse Mathematics.Solomon Feferman, Proof Theory: A Personal Report.

There is little doubt that a second-order axiomatization of Zermelo-Fraenkel set theory plus the axiom of choice (ZFC) is desirable. One advantage of such an axiomatization is that it permits us to express the principles underlying the first-order schemata of separation and replacement. Another is its almost-categoricity: M is a model of second-order ZFC if and only if it is isomorphic to a model of the form Vκ, ∈ ∩ (Vκ × Vκ) , for κ a strongly inaccessible ordinal.

The aim of this essay is to show that the subject-matter of ontology is richer than one might have thought. Our route will be indirect. We will argue that there are circumstances under which standard first-order regimentation is unacceptable, and that more appropriate varieties of regimentation lead to unexpected kinds of ontological commitment.

George Boolos's employment of plurals to give an ontologically innocent interpretation of monadic higher‐order quantification continues and extends a minority tradition in thinking about quantification and ontological commitment. An especially prominent member of that tradition is Stanislaw Leśniewski, and shall first draw attention to this work and its relation to that of Boolos. Secondly I shall stand up briefly for plurals as logically respectable expressions, while noting their limitations in offering ontologically deflationary accounts of higher‐order quantification. Thirdly I shall focus (...) on the key idea of ontological commitment and investigate its connection with the idea of truth‐making. Fourthly I shall consider how different interpretations of quantification may sideline Boolos's work, but finally I shall largely support his analysis of quantification involving nominal expressions, while arguing, in the spirit of Arthur Prior, that non‐nominal quantification is non‐committing. (shrink)

The thesis that, in a system of natural deduction, the meaning of a logical constant is given by some or all of its introduction and elimination rules has been developed recently in the work of Dummett, Prawitz, Tennant, and others, by the addition of harmony constraints. Introduction and elimination rules for a logical constant must be in harmony. By deploying harmony constraints, these authors have arrived at logics no stronger than intuitionist propositional logic. Classical logic, they maintain, cannot be justified (...) from this proof-theoretic perspective. This paper argues that, while classical logic can be formulated so as to satisfy a number of harmony constraints, the meanings of the standard logical constants cannot all be given by their introduction and/or elimination rules; negation, in particular, comes under close scrutiny. (shrink)

This paper investigates the claim that the second-order consequence relation is intractable because of the incompleteness result for SOL. The opponents’ claim is that SOL cannot be proper logic since it does not have a complete deductive system. I argue that the lack of a completeness theorem, despite being an interesting result, cannot be held against the status of SOL as a proper logic.

Suppose you hold the following opinions in the philosophy of logic. First-order predicate logic is expressively inadequate to regiment concepts of mathematic and natural language; logicism is plausible and attractive; set theory as an adjunct to logic is unnatural and ontologically extravagant; humanly usable languages are finite in lexicon and syntax; it is worth striving for a Tarskian semantics for mathematics; there are no Platonic abstract objects. Then you are probably already in cognitive distress. One way to decease your unhappiness, (...) short for embracing Platonism, is to accept higher-order logic and look, as did Arthur Prior, for a plausible way to neutralize the ontological commitment to abstract entities that this acceptance appears to entail. (shrink)