The neo-Fregean project of basing mathematics on abstraction principles faces “the bad company problem,” namely that a great variety of unacceptable abstraction principles are mixed in among the acceptable ones. In this paper I propose a new solution to the problem, based on the idea that individuation must take the form of a well-founded process. A surprising aspect of this solution is that every form of abstraction on concepts is permissible and that paradox is instead avoided by restricting what concepts (...) there are. (shrink)

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)

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)

This important book by a major American philosopher brings together eleven essays treating problems in logic and the philosophy of mathematics.

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.