I’m not entirely sure of the quality or maturity of the following thought, so I propose it tentatively. Comments, and criticisms, are most welcome.
Let us suppose that the Principle of the Identity of Indiscernibles, which states, according to the Stanford Encyclopedia of Philosophy, “if, for every property F, object x has F if and only if object y has F, then x is identical to y,” is true. Now let us consider the multiverse hypothesis to be true. Can one derive a contradiction from these two? I think, very tenuously, that there may be an antinomy, and we can expose it by considering the following:
Suppose Universe U1 exists for a (relatively) short period of time, and involves no intelligent life with any free will, having certain values X for its initial physical constants which cause it to implode in on itself shortly after coming into existence. We might account for every event in such a universe in strictly deterministic vocabulary, and all its features and events could be extrapolated simply by elaborating the implications of X. Now suppose that there is another Universe U2 which exists, having the exact same properties X. U2 would just be U1, according to the Principle of the Identity of Indiscernibles. U1 and U2 would both have all and only the properties of the other, and thus share all and only the same predicates. However, it seems that the Physicist who describes the Multiverse may want to say that in the Multiverse Universes come about and exist with their precise properties by happenstance (even if there is some multi-world producing mechanism behind it). Thus, whatever mechanism is proposed for the production of Universes, presumably a function of the Multiverse-ensemble, it seems that mechanism can only produce one of each kind of universe (where ‘kind’ is defined by precise initial values for the physical constants). However, as I understand the Multiverse hypothesis, there is no proposed mechanism which could only produce one of each kind of universe. That might entail an absurdity though, and thus furnish us with a reductio ad absurdum against the Multiverse from the Principle of the Identity of Indiscernibles.
Perhaps I am wrong though. Maybe the Multiverse theorist who accepts the Principle of the Identity of Indiscernibles, will just say that the Multiverse gives rise to an infinite number of universes, each one of which uniquely has its own values and properties. Thus, no two universes can both have X initial constants and quantities by definition. The Mutiverse advocate would just have to inform their scientific language by a careful view to semantic considerations given the Principle of the Identity of Indiscernibles.
The problem here, though, is that if the physicist also believes in categorical free will, she will find it impossible to maintain that there is another universe in the Multiverse-ensemble which is identical to ours in all respects except for some contingent fact of the categorically libertarian kind; for example, that “Susie went out with John instead of Jacob” (and any consequences that fact entailed in conjunction with all other facts about the world). Can the infinitesimal variation of initial physical values assigned to the physical constants really account for such possible differences, even for the determinist? I’m not sure it can, though this is reaching far beyond my expertise at this point into the field of speculative physics.
Perhaps the simplest way to escape any such argument would be to deny the Principle of the Identity of Indiscernibles. Another way would be to appeal to the Copenhagen interpretation of Quantum Mechanics, or some other relevantly similar model granting impetus for an indeterministic physics, and argue that events do not follow deterministically from initial constants and quantities. Somebody may also object saying that my argument would work equally well against David Lewis’ extreme modal realism (according to which all logically possible worlds are actual worlds). That objection would be wrong headed though, as the difference between extreme modal realism and the multiverse hypothesis is that the first is a proposal about modal logic, and the latter is supposed to be a hypothesis of the empirical sciences.
If the Multiverse ensemble produces universes, assigning values to constants by some indiscriminate process/function, then it seems as though there can be no reason why it may not produce two universes of exactly the same kind. If this is so, then the Multiverse hypothesis is incompatible with the Principle of the Identity of Indiscernibles, since then two distinct universes would exist sharing all and only the same predicates with each other. Perhaps one could counter that two universes of the same kind would not necessarily share all and only the same predicates, as one event may obtain in one without obtaining in the other (if things happened indeterministically, or if there exists categorical freedom of the Libertarian variety). However, even if both indeterminism were true at some level, and libertarian freedom existed, the Multiverse still makes possible an absurdity (or what is an absurdity given the presumption of the Principle of the Identity of Indiscernibles). A similar absurdity argument is used when logicians say that there is no logically possible worlds with both an unstoppable force and an immovable object… Though I have my doubts about that as well – isn’t there a logically possible world where both exist, and it simply never happens that the unstoppable force meets the immovable object? In that case we should say that there is no logically possible world where an unstoppable force meets an immovable object. By analogy, then, one could simply say that no two universes of the same kind will end up having all and only the same predicates, even though the multiverse seems to allow for this as a physical possibility.
My whole point, though, is to obviate the antimony between physical possibility and logical possibility for the advocate of the Multiverse hypothesis who also accepts the principle of the identity of indiscernibles. I think that for somebody to claim something is physically possible, while it is not logically possible, is confused. I think this is what is going on in the case presented. Therefore, the advocate of the Multiverse hypothesis who also accepts the principle of the identity of indiscernibles must give up one or the other, or both, of those commitments.