Maximally close possible worlds?

I was just reading Vann McGee’s “A Counterexample to Modus Ponens,” and my thoughts trailed off as I read the following passage:

“The most prominent logical theory of the subjunctive conditional is Robert Stalnaker’s account, according to which we test whether Φ⇒ψ is true in a possible world w by seeing whether ψ is true in the possible world most similar to w in which Φ is true”[1]

Something about Stalnaker’s account is attractive, but it raises a puzzling question: is it even coherent to suggest that there is one logically possible world which is ‘most like’ the actual world? Presumably such a world w* would be one in which all propositions share the same truth values as in the actual world, with the exception of one. However, since it could be any one, there are bound to be indefinitely many non-actual but possible worlds which are just as close to the actual world as each other (though none as close to each other as to the actual world). The problem gets worse, however, when one considers that any change in any one proposition’s truth-value will inevitably have a network-effect of turning over the truth values of other propositions. I can’t even think of a single proposition which, if its truth value were changed, wouldn’t entail that some other proposition’s truth-value would also change. So, is there a lower-bound to the set of propositions which any set of two different possible worlds fail to have in common? I’m not sure there is. Wouldn’t any difference at all in the propositional makeup of a world imply indefinitely many other differences?

We can demonstrate this by constructing a parody of Patrick Grim’s famous argument against the possibility of omniscience. Edward Wierenga explains the argument succinctly:

The argument (by reductio) that there is no set T of all truths goes by way of Cantor’s Theorem. Suppose there were such a set. Then consider its power set, ℘(T), that is, the set of all subsets of T. Now take some… truth t1. For each member of ℘(T), either t1 is a member of that set or it is not. There will thus correspond to each member of ℘(T) a further truth, specifying whether t1 is or is not a member of that set. Accordingly, there are at least as many truths as there are members of ℘(T). But Cantor’s Theorem tells us that there must be more members of ℘(T) than there are of T. So T is not the set of all truths, after all.[2]

Take world w to be the actual world, and world w* to be maximally close to w without being identical to ww* and w must have at least one difference in common – let’s say that w* differs from w with respect to the proposition p, such that while true in w*, p is not true in w. Now, take the set of any number of propositions true at w* including p, and take the power set of that set (and after adding its products to a new set, take the power set of the new set, and so on à la Patrick Grim), and one will generate new propositions which also differentiate w* and w. Thus, there are not only very many, and not only infinitely many, but indefinitely many differences (cashed out in terms of propositions with different truth-value assignments) between w and w*. With this kind of unfortunate situation, doesn’t it appear impossible to talk about ‘nearer’ or ‘farther’ possible worlds? We certainly can’t start talking about larger and smaller sets of indefinitely large sizes!

I expect that I am not alone in wanting desperately to talk about worlds in this way – as being closer to, or farther from, each other than others, and as being closer to, or father away from, the actual world than others. How can such talk be justified? I can only think of one way, which would be to remove what Pruss has called the ‘logical redundancies’ from the ‘big conjunctive contingent fact’ (which is the conjunction of all true propositions). The way he proposes to do this is to make the ‘big conjunctive contingent fact’  (or ‘BCCF’ for short) consist of only “the conjunction of all contingent first-order propositions.”[3] A ‘first-order’ proposition is one whose truth is not derived from any other(s). Thus, we should differentiate ‘primitive’ or ‘ground-level’ propositions from the propositions which ‘supervene’ upon them. This procedure may supply us with an implicit argument for some form of substance realism, since a first-order proposition is plausibly just a property borne by some thing, or a relation which obtains between things. However, I’m not being very systematic at the moment, so I’ll leave that thought hanging, as it were.

The take home thought of this article is that if we don’t eliminate the logical redundancies in the conjunction of all propositions at a world from the equation, then we can’t license talk of worlds being closer to, or farther from, the actual world than their peers. Moreover, to return to the first puzzle I raised, we might reduce the number of differences between w and w* by insisting that p (a first-order proposition), and p alone (i.e., none of the propositions derived from p or supervening upon p), is the whole difference between w and w*. This may not solve every puzzle, but it’s a good start. In particular it doesn’t really allow us to talk about one unique possible world which is maximally close to the actual world, but at least it does justify talking about classes of worlds which are nearer or farther away, and perhaps a unique class of maximally close worlds. This account holds promise for salvaging an amended account of subjunctive counterfactual conditionals, according to which Φψ is true in/at w just in case Φψ is true in most of the closest possible worlds to w.

[1] Vann McGee, “A counterexample to modus ponens,” in The Journal of Philosophy (1985): 466.

[2] Edward Wierenga, “Omniscience”, in The Stanford Encyclopedia of Philosophy (Winter 2012 Edition), Edward Zalta (ed.), accessed April 24, 2014: http://plato.stanford.edu/archives/win2012/entries/omniscience/

[3] Alexander R. Pruss, The principle of sufficient reason: A reassessment. (Cambridge University Press, 2006): 238.

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