Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...

This book is collection of the the author’s previously published papers on the philosophy of modality and tense and it also includes three unpublished papers. The author provides an exposition and defence of certain positions for which he is well-known: the intelligibility of modality de re; the primitiveness of the modal; and the primacy of the actual over the possible. He also argues for some less familiar positions: the existence of distinctive forms of natural and normative necessity, not reducible to (...) any form of metaphysical necessity; the need to make a distinction between the worldly and unworldly, analogous to the distinction between the tensed and the tenseless; and the viability of a nonstandard form of realism about tense, which recognizes the tensed character of reality without conceding there is any privileged standpoint from which it is to be viewed. (shrink)

There has been very little discussion of the appropriate principles to govern a modal logic of plurals. What debate there has been has accepted a principle I call (Necinc); informally if this is one of those then, necessarily: this is one of those. On this basis Williamson has criticised the Boolosian plural interpretation of monadic second-order logic. I argue against (Necinc), noting that it isn't a theorem of any logic resulting from adding modal axioms to the plural logic PFO+, and (...) showing that the most obvious formal argument in its favour is question begging. I go on to discuss the behaviour of natural language plurals, motivating a case against (Necinc) by developing a case that natural language plural terms are not de jure rigid designators. The paper concludes by developing a model theory for modal PFO-f which does not validate (Necinc). An Appendix discusses (Necinc) in relation to counterpart theory. Of course, it would be a mistake to think that the rules for "multiple pointing" follow automatically from the rules for pointing proper. Max Black—The Elusiveness of Sets In some influential articles during the 1980s George Boolos proposed an interpretation of monadic second-order logic in terms of plural quantification [4, 5]. One objection to this proposal, pressed by Williamson [22, 456-7], focuses on the modal behaviour of plural variables, arguing that the proposed interpretation yields the wrong results in respect of the modal status of atomic predications. In the present paper I will present this objection and argue against it. In the course of developing the argument, I will have cause to consider the under-investigated question of how a logic for plurals should be extended to incorporate modal operators. (shrink)

The transcendental aesthetic -- Arithmetic and the categories -- Remarks on pure natural science -- Two studies in the reception of Kant's philosophy of arithmetic: postscript to part I -- Some remarks on Frege's conception of extension -- Postscript to essay 5 -- Frege's correspondence: postscript to essay 6 -- Brentano on judgment and truth -- Husserl and the linguistic turn.

What is predicativity? While the term suggests that there is a single idea involved, what the history will show is that there are a number of ideas of predicativity which may lead to different logical analyses, and I shall uncover these only gradually. A central question will then be what, if anything, unifies them. Though early discussions are often muddy on the concepts and their employment, in a number of important respects they set the stage for the further developments, and (...) so I shall give them special attention. NB. Ahistorically, modern logical and set-theoretical notation will be used throughout, as long as it does not conflict with original intentions. (shrink)

On reading the last sentence, did you interpret me as saying falsely that everything — everything in the entire universe — was packed into my carry-on baggage? Probably not. In ordinary language, ‘everything’ and other quantifiers (‘something’, ‘nothing’, ‘every dog’, ...) often carry a tacit restriction to a domain of contextually relevant objects, such as the things that I need to take with me on my journey. Thus a sentence of the form ‘Everything Fs’ is true as uttered in a (...) context C if and only if everything that is relevant in C satisfies the predicate ‘F’; ‘everything’ ranges just over the contextually relevant things. Such generality is restricted in a context-relative way. (shrink)

Necessitism is the view that necessarily everything is necessarily something; contingentism is the negation of necessitism. The dispute between them is reminiscent of, but clearer than, the more familiar one between possibilism and actualism. A mapping often used to ‘translate’ actualist discourse into possibilist discourse is adapted to map every sentence of a first-order modal language to a sentence the contingentist (but not the necessitist) may regard as equivalent to it but which is neutral in the dispute. This mapping enables (...) the necessitist to extract a ‘cash value’ from what the contingentist says. Similarly, a mapping often used to ‘translate’ possibilist discourse into actualist discourse is adapted to map every sentence of the language to a sentence the necessitist (but not the contingentist) may regard as equivalent to it but which is neutral in the dispute. This mapping enables the contingentist to extract a ‘cash value’ from what the necessitist says. Neither mapping is a translation in the usual sense, since necessitists and contingentists use the same language with the same meanings. The former mapping is extended to a second-order modal language under a plural interpretation of the second-order variables. It is proved that the latter mapping cannot be. Thus although the necessitist can extract a ‘cash value’ from what the contingentist says in the second-order language, the contingentist cannot extract a ‘cash value’ from some of what the necessitist says, even when it raises significant questions. This poses contingentism a serious challenge. (shrink)

Say that some things form a set just in case there is a set whose members are precisely the things in question. For instance, all the inhabitants of New York form a set. So do all the stars in the universe. And so do all the natural numbers. Under what conditions do some things form a set?

Some reasons to regard the cumulative hierarchy of sets as potential rather than actual are discussed. Motivated by this, a modal set theory is developed which encapsulates this potentialist conception. The resulting theory is equi-interpretable with Zermelo Fraenkel set theory but sheds new light on the set-theoretic paradoxes and the foundations of set theory.

Do mereological fusions have their parts necessarily? None of the axioms of non-modal formulations of classical mereology appear to speak directly to this question. And yet a great many philosophers who take the part-whole relation to be governed by classical mereology seem to assume that they do. In addition to this, many philosophers who make allowance for the part-whole relation to obtain merely contingently between a part and a mereological fusion tend to depart from non-modal formulations of classical mereology at (...) least when it comes to the axiom of Unique Fusion, which states that no two different mereological fusions ever fuse exactly the same objects. This is no coincidence. There are reasons of principle why one’s adherence to classical mereology should exert some pull towards the view that mereological fusions have their parts necessarily. There is, however, no direct route from the combination of classical mereology and propositional modal logic to the hypothesis that the part-whole relation obtains necessarily between a part and a mereological fusion. In order to bridge between a modal formulation of classical mereology and the hypothesis that fusions have their parts necessarily, one needs to strengthen the axiom of Unrestricted Fusion in a way that is agreeable to many philosophers on both sides of the debate. (shrink)

Identity is a modally inflexible relation: two objects are necessarily identical or necessarily distinct. However, identity is not alone in this respect. We will look at the relation that one object bears to some objects if and only if it is one of them. In particular, we will consider the credentials of the thesis that no matter what some objects are, an object is necessarily one of them or necessarily not one of them.

This chapter aims to resolve some of Nathan Salmon's puzzlement by clarifying the relationship between theses and questions about reference and theses and questions about necessity and possibility. It argues that while Saul Kripke defends metaphysical theses about the descriptive semantics of names, the way the reference relation is determined, and the capacities and dispositions of human beings and physical objects, his most important philosophical accomplishment is in the way he posed and clarified the questions, and not in the particular (...) answers that he gave to them. In response to the question about the capacities and potentialities of the things that peoples commonly refer to with names, Kripke defends the thesis that it makes sense to talk about the logical potential of an individual thing independently of how it is referred to, and that this potential is greater in certain ways, and less great in others, than some philosophers have supposed. (shrink)

Atomism denies that complexes exist. Common-sense metaphysics may posit masses, composite individuals and sets, but atomism says there are only simples. In a singularist logic, it is difficult to make a plausible case for atomism. But we should accept plural logic, and then atomism can paraphrase away apparent reference to complexes. The paraphrases require unfamiliar plural universals, but these are of independent interest; for example, we can identify numbers and sets with plural universals. The atomist paraphrases would fail if plurals (...) presuppose complexes: but an Appendix shows that reference to complexes is not required in the semantics of plurals. (shrink)

This chapter is a detailed study of predicativity in mathematics. It presents a number of historical versions predicativity requirements, looking for unifying ideas. The further development of the notions and requirements up to the present is traced, articulating connections among the different ideas. One underlying theme of the chapter is the motivations for the various requirements for rejecting impredicativity and the various ways of stating the requirement.