How many possible worlds are there?

Abstract

(1) [It is possible that p] iff [there is a possible world w such that, at w, p]. I shall argue against (1). The idea is that, if we accept (1), we are committed to the existence of paradoxically many possible worlds. For the purposes of the argument, it does not matter whether by 'possible world' we mean 'way things might have been' (as in Stalnaker) or 'maximal possible state of affairs' (as in Plantinga) or 'spatio-temporal totality' (as in Lewis 1986), and correspondingly it does not matter what exactly we mean by the operator 'at w'. It is important, however, that the phrase 'there is a possible world w such that at w ...' is here meant to express existential quantification over some beings; it is not meant as a graphic reformulation of a phrase ('it is possible') which does not involve quantification. I shall presuppose the necessity of non-identity: there are no distinct things that could have been identical. There is a strong argument for the necessity of non-identity which parallels the well-known argument for the necessity of identity. Begin with VxVyL[x = y -> [Fx -> Fy]], which is a variation on the principle of the indiscernibility of identicals. By substitution we get VxVyD[x = y -> [@(x = x) -> @(x = y)]] where '@' means 'actually', as this operator has been studied in two-dimensional modal logic. Since Vx(x = x), we have Vx@(x = x) and so Vx~ @(x = x). Therefore VxVyO[x = y -> @(x = y)] and thus VxVy[O(x = y) -> O@(x = y)]. But VxVy[O@(x = y) -> @(x = y)] and VxVy[@(x = y) -> x = y]. Hence VxVy[O(x = y) -> x = y].l Problems about cardinality based on Cantor's proof that a set has fewer members than its power set have a distinguished history in discussions of possible worlds. There is the paradox presented in Davies 1981: 262 and Kaplan 1995. It concerns propositions, and in its best-known version it relies on the premiss that, given a thinker and a time, each proposition could have been the only proposition that was the content of a thought entertained by the thinker at that time.2 There is also Grim's argument that