This note was published in Dialogue, (1974), Vol. 13, no. 1, pp. 111-112.

ON  REDUCING  THE  NUMBER  OF  POSSIBLE  WORLDS

NORMAN SWARTZ
Simon Fraser University

In his paper, "The Two Main Problems of Philosophy"[Note 1], Professor N.L. Wilson offers an inductive argument for the thesis "that there is only one possible world … possibility, actuality and necessity collapse into each other."[p. 200].

He proceeds by posing several questions of the form,"What would the world be like if everything of such-and-such a sort were instead of so-and-so a sort?" In each case he shows that "the world would be exactly as it is."[p. 203]. For each question posed, there is an infinity-fold reduction in the number of possible worlds. Thereupon, Wilson concludes the section of his paper devoted to modalities by stating: "There must be more reductions. … At any rate these reductions seem to me to be sufficiently striking to justify us in at least conjecturing that there is only one possible world."[p. 212].

In spite of the fact that Wilson (successfully I think) effects several infinity-fold reductions of the number of possible worlds, he far from succeeds in reducing the number to one. Quite the contrary: an infinity remains. For it does little good to support Wilson's radical reduction thesis by finding whole families of reductions, if at the same time there exist means by which to generate vast hosts of different possible worlds. The trouble with Wilson's reduction-formula is that in each instantiation of it in his hands at least one universal quantifier occurs, e.g. "What would the world be like if every event occurred ten years before it actually occurred?"[p. 202, italics added]. But one need only substitute the strictly-particular quantifier, "some but not all (or every)", for the universal one to get very different results. For example we would get instead, "What would the world be like if some but not every event occurred ten years before it actually occurred?" Rather than the world being 'exactly as it is' we would find the world somewhat or even greatly different. Indeed for a world consisting of n original events, there are n! (i.e., n x [n-1] x [n-2] x … x 2 x 1) different ways of arranging them. Far from reducing the number, we can if we like in this way multiply the number of possible worlds astronomically.

Wilson tries to bolster his argument by also deriving his conclusion from considerations of 'Carnap-like' state-descriptions. In effect, he argues that all state-descriptions which can be obtained from one another by the permutation of individual constants describe one and the same world. (This is hardly news. Carnap himself called such state-descriptions "isomorphic" and assigned equi-probabilities to their disjunctions (the so-called structure-descriptions) treating them, as it were, as 'super'-state-descriptions). What is crucial for our purposes, however, is that the list of all the state-descriptions for any specified language will always contain some state-descriptions which are not isomorphic to some others. The proof is trivial. All state-descriptions in a language contain the same number of conjuncts. If there are n terms in a state-description, then there must be one state-description with no negated terms, some with one, some with two, and so on, all the way through to the one state-description which consists wholly of (n) negated terms. But since it is a necessary (but not sufficient) condition for the isomorphism of state-descriptions that they have the same number of negated terms, it follows that not all state-descriptions in a language can be isomorphic. Again, far from supporting Wilson's thesis, we can see that playing the state-description game undermines it. For even though some (but not all) state-descriptions in a given language will have isomorphs, we have proved that every state-description has at least one non-isomorph. In non-technical language this means that for any possible world described by a Carnapian state-description, it can be demonstrated that there exists at least one possible world which is not identical to it.

In sum, if the modalities, possibility, actuality and necessity, are to be collapsed into one another, then some far stronger arguments than Wilson's will be required.

Note
1.  Dialogue,  XII (1973) No. 2, pp. 199-216.[ Resume ]