At i, A is not F
I've thought a bit more about the comments Michael Fara left last week, and I don't find my points very convincing any more. The following is partly a correction, but mostly just thinking out loud about a more general semantic question.
The general question is how to interpret sentences of the form
1) At i, A is F
2) At i, A is not F
where 'i' denotes something like a time or a place or a world. There are a dozen proposals for interpretations of (1) in the temporal case, invoking temporal parts or relations to times or whatever. Most of these proposals can be applied to other indices as well. But let's put that aside. Suppose we understand how to interpret instances of (1) in easy cases. The hard cases I have in mind are cases where A doesn't exist exactly once at i. The precise definition of these cases depends on the question I've put aside, but I hope it is reasonably clear what I mean anyway. Not existing exactly once at i means either not existing at i at all, or multiply existing at i. Plausible examples of the first kind: I do not exist in 1758; I do not exist on Alpha Centauri; I do not exist at any world containing only empty space-time. Controversial examples of the second kind: if I get split into two persons tonight, I will doubly exist tomorrow; if river R has two branches where is crosses the border to country C, R doubly exists at the border to C; if at some world, two people resemble me to exactly the same degree in all extrinsic and intrinsic respects, I doubly exist at that world.
(There is an additional problem created by the fact that in ordinary discourse, i is usually not a point but a region in space, time or the modal realm. The additional problem is what to say if A is F at only some points in the region. This problem seems related to the one I'm interested in here, and hence should be considered in a solution, but since I'm not going to present a solution, I hope it's okay if I just ignore it.)
Most instances of (1) are false if A doesn't exist at i. For example, it's not true that I I'm 6 ft tall at Alpha Centauri, nor that I am married there or writing a blog entry. It's also not true that I'm doing any of these in 1758. However, there are exceptions: Even though Frege doesn't exist in 2004, and Sherlock Holmes doesn't exist at our world at all, both are nevertheless admired and thought about at this world in 2004. Properties like being admired and being thought about that can be instantiated in absentia are interesting -- do they have an official name? --, but I'd like to put them aside for now as well.
So we're left with properties F for which (1) is intuitively false if A doesn't exist at i. What if A multiply exists at i and only some of the A at i are F? These cases are rare, and my semantic intuitions aren't clear about them. I'm somewhat inclined to say (1) is false then as well. There is definitely something wrong with saying "at the border to C, R is blue" if only one of the two branches of R at C is blue. But the problem may be pragmatic rather than semantic. On the other hand, if both branches are blue, the statement looks more acceptable to me, even though there is still a pragmatic problem, since the statement somehow implies that there is only one branch of R at C. (If you know that there are two branches, you should say that both branches of R at the border to C are blue.) So I'm somewhat inclined to say that
A1) (1) is true iff A exists at least once at i and all A existing at i are F (at i).
But where my intuitions are so weak, I'm happy to be guided by a good theory. If linguists have a good theory on which, say,
A2) (1) is true iff A exists at least once at i and at least one A existing at i is F (at i),
or
A3) (1) is true iff A exists exactly once at i and all A existing at i are F (at i),
the latter treating "at i, A" like Russell treated "the A at i", I wouldn't object.
(As I said, I presuppose that we know what is meant by "A is F at i" in easy cases. Given such an interpretation, (A1)-(A3) could be stated more informatively, e.g.: "A1') (1) is true iff A has at least one counterpart at i and all counterparts of A at i are F".)
What about (2)? In easy cases, it behaves just like the negation of (1): Since it is not the case that I was 6 ft tall 10 minutes ago, it is true that that 10 minutes ago, I wasn't 6 ft tall.
But in the tricky cases, this is less clear. Here it seems to me that there is always something wrong with saying (2). I shouldn't say that R isn't blue at the border to C if I know that R doesn't even reach the border to C, nor if I know that it has two branches at the border. Saying (2) seems to imply that the present case is an easy case. But ignoring pragmatics, is "at the border to C, R isn't blue" true or false if R doesn't reach the border to C? I don't know. Similarly for temporal and modal examples, and similarly for the cases of multiple existence at an index.
Here I'm not even sure I would follow theory wherever it leads. Rather, what I'd expect of a good theory is to tell me that not only are there pragmatic problems with uttering (2) in the hard cases, but also that they give rise to semantic ambiguity. The ambiguity is familiar if we use (A3) as the interpretation of tricky (1) cases. Then it's the old scope ambiguity of negated definite descriptions: By saying that at the border to C, R isn't blue, you could either mean that there is exactly one R existing at the border to C and it isn't blue, or that it is not the case that there is exactly one R existing at the border to C which is blue. Quite similar scope ambiguities can arise from (A1) and (A2), however. For instance, on (A1), (2) could be analysed either as "A exists at least once at i and all A existing at i are not F" or as "A exists at least once at i and not all A existing at i are F" or, finally, as "it is not the case that A exists at least once at i and all A existing at i are F".
At any rate, it seems to me that the semantics of (2) in the hard cases is far from clear. This is where I now disagree with my previous posting. Fara and Williamson assume that "it is not the case that actually, A is F" is always equivalent to "actually, A is not F", even in the tricky cases. But if "actually" is an operator of the "at i" kind then this isn't obvious at all, at least not to me. Nor is it obvious that the assumption is wrong, though. If one wants to use a simple 'actually' operator in modal logic, it is of course fine to stipulate that it behaves like that. And if that yields a nice formal system, this is a point in favour of interpreting even our ordinary idiom in that way. But it's not more than that, I'd say. If another nice formal system comes up that contradicts the assumption, it isn't fair to say the other system rejects logical truths.