Supervenience and Infinitary Logic

The discussion of supervenience is replete with the use of infinitary logical operations. For instance, one may often find a supervenient property that corresponds to an infinite collection of supervenience-base properties, and then ask about the infinite disjunction of all those base properties. This is crucial to a well-known argument of Kim (1984) that supervenience comes nearer to reduction than many non-reductive physicalists suppose. It also appears in recent discussions such as Jackson (1998). Some philosophers have been troubled simply by the infinity of such a disjunction. Logicians tend to react somewhat differently. Infinitary logical operations have been studies in depth by the highly developed field of infinitary logic, and many of their properties are well-understood. On the other hand, as anyone who has ever worked with infinitary logic knows, it has proved difficult, messy, and filled with surprising pitfalls. Moreover, there are lots of different infinitary logics, displaying different characteristics, some better, some worse. Logicians are not likely to object to infinitary operations per se, but they are likely to ask whether their application to a metaphysical issue like supervenience work as smoothly as it is sometimes assumed. In this paper, I shall investigate the interaction between supervenience and infinitary logic. Supervenience has long been a point of contact between logic and metaphysics. In examining it, some philosophers have already questioned the metaphysical status of certain finitary logical operations. I shall show here that the step to infinitary logical operations raises significant metaphysical issues of its own. This step, I shall argue, is not an all-or-nothing deal. If we accept the use of infinitary logical operations, we still face hard choices about the strength of such operations to allow. This in turn forces us to confront difficult metaphysical