Counterparts of sequences and multiple counterpart relations
Allen Hazen (1979, pp.328-330) pointed out a problem for Lewis's counterpart-theoretic interpretation of modal discourse: the fact that x is essentially R-related to y should be compatible with the fact that both x and y have multiple counterparts at some world, without all counterparts of x being R-related to all counterparts of y. But the latter is what Lewis's semantics requires for the truth of `necessarily xRy'.
Lewis's response, on pp.44f. of his 1983 postscript to 'Counterpart theory and quantified modal logic', was that a modal statement that is multiply de re, such as 'necessarily xRy', is better understood as a monadic de re statements about a sequence: 'necessarily, (x,y) is such that its first member is R-related to the second'. One can then say that although individually, x and y have many counterparts that are not R-related, the pair (x,y) only has R-related counterparts. It is not clear whether Lewis really means to claim that the syntax of quantified modal logic should only allow for monadic de re statements. More charitably, we could take him to suggest that when a formula is de re about several individuals, we need to invoke a counterpart relation between the corresponding pairs or triples etc., which may not be determined by the counterpart relation between single individuals. This proposal is also mentioned in Hazen's paper (on p.335) and endorsed in several more recent accounts, such as Kupffer's and Ghilardi's.
Interestingly, this idea turns out to be very helpful to solve a completely different problem when using counterpart semantics as a model theory for various modal logics. In the construction of canonical models, we would like to read off the counterpart relation between the individuals at two world (Henkin sets) from what the worlds "say" about the relevant individuals: if w' contains A(x') whenever w contains \Box A(x), then x' is a counterpart of x. But what if w contains \Box A(x,y)? Now we need to look at pairs (x',y') for which w' contains A(x',y').
However, when I tried to spell this out, I ran into some odd possibilities that cause trouble. For example:
- x has z as unique counterpart (at some given world), and the pair (x,y) has (u,v) as unique counterpart, where z is different from u.
- x has z as unique counterpart, but the pair (x,y) has both (z,u) and (v,u) as counterpart.
- The pair (x,x) has (v,u) as counterpart, but not (u,v).
Not only do these possibilities complicate the model theory, they also make no intuitive sense. Take the last example. We are looking at a pair of an object x and again x. Which counterparts can we find for "them" at the world under discussion? The supposed answer is that the first element of (x,x) has u as counterpart and the second v, and not the other way around. But what is that supposed to mean? The two elements are the same!
The proposal to extend counterparthood to sequences allows for possibilities that should be ruled out. What we actually want is a constraint on the counterparts that can be assigned to single individials. The individuals don't come in a fixed order, so we shouldn't think of counterparthood as relating ordered tuples. The better solution is to use sets of counterpart relations between individuals. If w is a world of eternal recurrence, and x and y have counterparts in each epoch, then one counterpart relation links x and y with their respective counterparts in the first epoch, another with their counterparts in the second, and so on. In the interpretation for the box, we then quantify over all counterpart relations.
This is precisely what is used in Kracht and Kutz 2002, which had always struck me as artificial. I was wrong. Looking back at Hazen's paper, it is also very close to his own proposal (on p.334f.), except that he has a set of counterpart functions, because he wants to validate the necessity of identity and get a traditional logic of 'actually'.