I consider different versions of a structuralist view of mathematical objects, according to which characteristic mathematical objects have no more of a 'nature' than is given by the basic relations of a structure in which they reside. My own version of such a view is non-eliminative in the sense that it does not lead to a programme for eliminating reference to mathematical objects. I reply to criticisms of non-eliminative structuralism recently advanced by Keränen and Hellman. In replying to the former, (...) I rely on a distinction between 'basic' and 'constructed' structures. A conclusion is that ideas from the metaphysical tradition can be misleading when applied to the objects of modern mathematics. (shrink)

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.

When a debate seems intractable, with little agreement as to how one might proceed towards a resolution, it is understandable that philosophers should consider whether something might be amiss with the debate itself. Famously in the last century, philosophers of various stripes explored in various ways the possibility that at least certain philosophical debates are in some manner deficient in sense. Such moves are no longer so much in vogue. For one thing, the particular ways they have been made have (...) themselves undergone much critical scrutiny, so that many philosophers now feel that there is, for example, a Quinean response to Carnap, a Gricean reply to Austin, and a diluting proliferation of Wittgenstein interpretations.2 Be that as it may,3 there do of.. (shrink)

This paper argues for the thesis that, roughly put, it is impossible to talk about absolutely everything. To put the thesis more precisely, there is a particular sense in which, as a matter of semantics, quantifiers always range over domains that are in principle extensible, and so cannot count as really being ‘absolutely everything’. The paper presents an argument for this thesis, and considers some important objections to the argument and to the formulation of the thesis. The paper also offers (...) an assessment of just how implausible the thesis really is. It argues that the intuitions against the thesis come down to a few special cases, which can be given special treatment. Finally, the paper considers some metaphysical ideas that might surround the thesis. Particularly, it might be maintained that an important variety of realism is incompatible with the thesis. The paper argues that this is not the case. (shrink)

This paper argues for the thesis that, roughly put, it is impossible to talk about absolutely everything. To put the thesis more precisely, there is a particular sense in which, as a matter of semantics, quantifiers always range over domains that are in principle extensible, and so cannot count as really being ‘absolutely everything’. The paper presents an argument for this thesis, and considers some important objections to the argument and to the formulation of the thesis. The paper also offers (...) an assessment of just how implausible the thesis really is. It argues that the intuitions against the thesis come down to a few special cases, which can be given special treatment. Finally, the paper considers some metaphysical ideas that might surround the thesis. Particularly, it might be maintained that an important variety of realism is incompatible with the thesis. The paper argues that this is not the case. (shrink)

The Geach‐Kaplan sentence is alleged to be an example of a non‐first‐orderizable sentence, and the proof of the alleged non‐first‐orderizability is credited to David Kaplan. However, there is also a widely shared intuition that the Geach‐Kaplan sentence is still first‐orderizable by invoking sets or other extra non‐logical resources. The plausibility of this intuition is particularly crucial for first‐orderism, namely, the thesis that all our scientific discourse and reasoning can be adequately formalized by first‐order logic. I first argue that the Geach‐Kaplan (...) sentence is, in fact, not first‐orderizable even by invoking extra non‐logical resources, in any sense that is acceptable for first‐orderism and adequately corresponds to the sense in which the Geach‐Kaplan sentence is deemed to be non‐first‐orderizable simpliciter via Kaplan's proof. To address this problem on behalf of first‐orderism, I then propose an alternative conception of first‐orderizability in the sense of which the Geach‐Kaplan sentence and any other second‐order sentences become first‐orderizable by invoking extra non‐logical resources; furthermore, in certain circumstances, they are first‐orderizable without incurring any extra ontological commitment. My analysis also turns out to yield (as a biproduct) a significant enhancement of the so‐called paradox of plurality. (shrink)

Mereology, the theory of parts and wholes, is sometimes used as a framework for categorisation because it is regarded as ontologically innocent in the sense that the mereological fusion of some entities is nothing over and above the entities. In this paper it is argued that an adequate answer to the question of whether the thesis of the ontological innocence of mereology holds relies crucially on the underlying theory of reference. It is then shown that upholding the thesis comes at (...) high costs, viz. at the cost of a quite strong logical background theory or at paradoxical ways of predicating and counting. (shrink)

The Bounds of Logic presents a new philosophical theory of the scope and nature of logic based on critical analysis of the principles underlying modern Tarskian logic and inspired by mathematical and linguistic development. Extracting central philosophical ideas from Tarski’s early work in semantics, Sher questions whether these are fully realized by the standard first-order system. The answer lays the foundation for a new, broader conception of logic. By generally characterizing logical terms, Sher establishes a fundamental result in semantics. Her (...) development of the notion of logicality for quantifiers and her work on branching are of great importance for linguistics. Sher outlines the boundaries of the new logic and points out some of the philosophical ramifications of the new view of logic for such issues as the logicist thesis, ontological commitment, the role of mathematics in logic, and the metaphysical underpinning of logic. She proposes a constructive definition of logical terms, reexamines and extends the notion of branching quantification, and discusses various linguistic issues and applications. (shrink)

The object of this paper is to study the analogy, drawn both positively and negatively, between mathematics and fiction. The analogy is more subtle and interesting than fictionalism, which was discussed in part I. Because analogy is not common coin among philosophers, this particular analogy has been discussed or mentioned for the most part just in terms of specific similarities that writers have noticed and thought worth mentioning without much attention's being paid to the larger picture. I intend with this (...) analogy (looking at others' comparisons) to shed a little light on what is going on in mathematics, how one can understand it a bit other than experientially. This intention is philosophical and the way that I am attempting to accomplish it is also philosophical. I shall conclude my attempt to explain how it is possible and even natural for mathematics and fiction to have the analogy they have, taking it for granted, as argued in part I, that they are not to be identified. To this end I shall discuss philosophers' comparisons, mainly those of Hodes, Resnik, Tharp, and Wagner, who are the writers that seem to me to have written most thoughtfully and sufficiently extensively about fiction in making the comparison and of Körner, whose comparison is different. Whether either of these comparisons or the more general analogy is of permanent philosophical interest will have to be decided by philosophers now that they have had a fuller examination. I shall mention some other writers' reference to fiction, but not all; indeed, I am sure that I have not even found all the comparisons that there are. (shrink)

Jackendoff defends a mentalist approach to semantics that investigates conceptual structures in the mind/brain and their interfaces with other structures, including specifically linguistic structures responsible for syntactic and phonological competence. He contrasts this approach with one that seeks to characterize the intentional relations between expressions and objects in the world. The latter, he argues, cannot be reconciled with mentalism. He objects in particular that intentionality cannot be naturalized and that the relevant notion of object is suspect. I critically discuss these (...) objections, arguing in part that Jackendoff’s position rests on questionable philosophical assumptions. (shrink)