Scott Dixon, forthcoming in Analysis

I noticed the title of this just because millennia ago, as an MPhil student, I wrote an essay on the arguments against converse relations. Cringe prevents me rereading it now. But at the time “are there converse relations, or just nonsymmetric relations looked at from the Tibetan side?" struck me as pretty much the maximally abstract question. So it’s interesting to see that the literature remains active, despite various turns to more substantively impactful metaphysics.

Anyway. The converse of a relation is basically just that relation put into the passive voice (the converse of ‘x loves y’ is ‘y is loved by x’). Directionalism, the view that they are distinct relations, has an intuitive Occam’s-razor problem: why? The converse relation applies exactly where the proverse one does—as I vaguely remember putting it in 2005, they are always in the same photos—so why treat them as distinct?

Williamson's 1985 challenge to directionalism, which Dixon is targeting, makes a linguistic argument out of the impossibility of separating the two. If both relations exist, says Williamson, it’s indeterminate which of them any relational predicate refers to. Imagine two languages with the term ‘loves’, but in one it expresses the relation ‘loves’ and in the other ‘is loved by’. How can you tell which language uses the predicate which way? Well, you could look at the order of the argument places, but of course those can be permuted as well: so that the language where ‘loves’ means ‘is loved by’ also puts the lover in first position. (Abstract metaphysics me manque beaucoup.) Then it seems there is no way to tell: in particular, since the two are extensionally equivalent there’s no way of telling empirically. Such indeterminacy would be bad, and Williamson takes it that our language does not have it, and the only way to avoid it seems to be to treat the two relations as identical after all.

Dixon’s response appeals to the notion of plural reference, applied to the relations expressed by predicates: plural expression. (Additional autobiographical note: Alex Oliver, who is half of the Oliver and Smiley that Dixon cites re matters plural, supervised that MPhil essay on converse relations. I’m not sure whether that shows a deep connection between the issues or just that the community of people who enjoy this sort of stuff is small.) Dixon’s suggestion is that there’s no indeterminacy about which relation “loves” expresses: because it plurally expresses both! 

That rather neatly addresses Williamson’s argument, I think. But the price is reinforcing the Occam worry: if even our relational predicates don’t distinguish between relations and their converses, why should we? The only real reason for ever thinking them distinct, it seems to me, was that we express them linguistically in differing ways (‘loves’ vs ‘is loved by’). But now it turns out we don’t even do that: both predicates express both relations, they just flip the argument places for some reason.

So while Dixon’s suggestion looks to me like a technical success, it's a rather pyrrhic victory. It shows us how we can express both relations and their converses without indeterminacy, but leaves us even more puzzled about why we would want to.