John William Waldrop has an interesting paper (PDF) in which he argues that Christopher Tomaszewski's argument that modal collapse arguments (such as those of R. T. Mullins) are generally invalid. Unlike Mullins, he rightly recognizes the importance of rigidity, but I think he still commits similar mistakes. For instance, he proposes this as a modification of Mullins's argument that is valid (with its formalization):
(1) Necessarily, God exists.
(2*) God is identical to the actual divine creative act.
(3*) Necessarily, the actual divine creative act exists.
Which can be formalized as follows:
(1) ◻∃x (x = God)
(2*) God = ℩x@Cx
(3*) ◻∃x (x = ℩y@Cy)
As I've noted before, classical divine simplicity is not formulated in terms of logical identity but in terms of noncomposition, but setting that aside, a problem with this is that, unless I am missing something important, Waldrop's formalization of (2*) doesn't say "God is identical to the actual divine creative act"; it says "God is identical to that which at the actual world is the divine creative act". Likewise, (3*) doesn't say "Necessarily, the actual divine creative act exists"; it says "Necessarily there is something such that it is identical to that which at the actual world is the divine creative act." There is no modal collapse here unless you assume that what at the actual world is the divine creative act is the divine creative act at every possible world -- that is, unless you already assume modal collapse.
Trying to splice an actual world operator into standard possible world semantics as used in modal metaphysics (in which possible worlds usually take the interpretation 'ways the actual world can be') is tricky business in any case. The way it's usually done, @ doesn't really mean 'actual world'; it merely locks a possible world in as your reference point. That is, there's nothing about the formalism that ties it to the actual world; it just treats one possible world as a reference point, and this then can be interpreted (if you want) as 'the way the world could be that (for whatever reason) we are taking to be the way the world is'. Thus if you took the above argument to imply modal collapse, any necessary existent would lead to modal collapse, because you could use @ to pick out any possible world as your reference point, and every necessary being would have to be identical to itself under some description that obtains at that possible world, so you can always have analogues of (1) and (2*).
Indeed, it doesn't just affect necessary beings. There are many Box operators, so for any subset of possible worlds you can define a Box operator restricted to those possible worlds (i.e., that would mean 'in every one of those possible worlds'). Every contingent being is 'necessary' if you restrict yourself entirely to those possible worlds whose description includes the proposition that it exists. But every contingent being can be identical to that being which has some descriptive property only in some of those possible worlds. For instance, I am identical to that being who writes this post, but as there are many other choices I could have made, I am not identical to a being with the predicate 'writes this post' for all the possible worlds in whose descriptions I am found. I am only so in this one (or perhaps this one and some very close neighbors). But then you can substitute 'Brandon' for 'God' and 'writes this post' for 'C' in the above premises, and for that restricted Box operator the argument will be valid. Modal collapse! I necessarily write this post for all the possible worlds to which I can be attributed! No, not really; the modal collapse was assumed by thinking that if 'Brandon' is intersubstitutable with what at the actual world is 'the person who writes this post', then 'Brandon' must be intersubstitutable with 'the person who writes this post at the actual world'.
Now, Waldrop to his credit considers something along these lines, although he muddies the water by trying to characterize it in terms of essentiality. He suggests that modal collapse arguments assume the following (and that this is what is really in dispute):
(E) Necessarily, something is a divine creative act only if it is essentially the unique divine creative act.
But this would make all modal collapse arguments question-begging; 'essentially' here has to mean ' in every possible world' and 'unique' has to mean 'one and the same', so this implies that there is one and only one possible world whose description includes a divine creative act. Waldrop tries to argue that (E) is plausible because if it is false, "there could have been something that would have been a divine creative act but would only accidentally be such". Since 'accidentally' here can only mean 'in some possible worlds and not others', Waldrop's argument for the plausibility of (E) is just that he thinks divine simplicity implies modal collapse. The way he tries to fill this out is "I am mystified at the suggestion that a divine creative act, an act that in fact makes the difference between a world in which God alone exists and a world in which God coexists with Richard Nixon and the rest of creation, could be other than a divine creative act." But Waldrop has made the same slip again: the claim, even in terms of identity, would not be that a divine creative act could be other than divine creative act, but that something that is identified with a divine creative act with respect to one possible world would not be identified with a divine creative act with respect to a different possible world. This is not baffling; it just means that 'divine creative act' is a relative term anchored by a contingent relatum, which we already know that it is ('creative' is relative to 'created'). If you assume that it is not, then of course you get modal collapse; you've just assumed that this term applies necessarily to anything to which it possibly applies.
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