Possible worlds are part of the landscape of contemporary philosophy, but there is much disagreement about precisely what they are. Some philosophers, wary of talk about anything nonactual, want to reduce them to actual but abstract objects. We shall call their position actualism. On the other hand, those who accept merely possible worlds, whom we shall call possibilists, have generally also felt comfortable with abstracta, particularly sets, and have employed both freely. This paper has two aims: (1) to show that combining possibilism with the classical conception of set leads to awkward theoretical difficulties, and (2) to show how alternatives to the classical conception of set avoid them.
My reasoning is therefore addressed primarily to the possibilist and is of general interest only to the extent that possibilism itself is. There are signs that possibilism is gaining respectability. David Lewis has made a rather radical version of it famous, 1 and at least one introductory logic text 2 is based on a possibilistic metaphysics. Still, many philosophers remain sceptical, in part, I think, because they mistakenly identify possibilism with Lewis' realism. Therefore, before proceeding to my main points I would like, first, to explain and defend possibilism in general and, second, to show that it does not entail Lewis' realism. In doing so, I hope to allay possible misgivings about possibilism and to enhance interest in what I have to say later.
It is sometimes charged that possibilism is simply unintelligible. But in the absence of further support or explanation, this charge hardly constitutes a rational objection. Indeed, it might tell more against its authors than against the theory at which it is aimed. One thinks, for example, of similar charges initially levelled against noneuclidean geometry, the theory of transffmite numbers, and most of modern physics. The only attempt I know of to show that possibilism is unintelligible is the claim that it entails the self-contradictory assertion, 'There are things which do not exist. '3 But this assertion is ambiguous; and there is, as I'll explain shortly, a sense in which it is not