Could there be exactly two things?

Many philosophers think that, necessarily, any material objects have a fusion (let’s call that doctrine “Universalism”). In this paper I point out a couple of strange consequences of Universalism and related doctrines, and suggest that they are strange enough to constitute a powerful argument against those views. By “thing” I mean material thing. We all believe that there actually are many more than two things, but could there have been exactly two? More generally, what are the ns such that there could have been exactly n things? Pre-theoretically, the following conjecture is quite attractive: Primitive Cardinality: For any n, there could have been exactly n things. For instance, we naturally judge that there could have been exactly two things, and exactly three, . . . Attractive as it is, many philosophers are committed to denying Primitive Cardinality. For instance, many philosophers believe in Universalism: Necessarily, any material objects have a fusion.1 It can be shown that Universalism implies that there could not have been exactly two things, given the following minimal assumptions: Reflexivity: Necessarily, everything is a part of itself. Principle of Weak Supplementation: Necessarily, if x is a proper part of y then some part of y does not overlap with x (x is a proper part of y just in case x is part of y and x 6= y).2 Notice that these three principles (Universalism, Reflexivity and the Principle of Weak Supplementation) are substantially weaker that Classical Extensional Mereology (CEM). Many philosophers believe not only that any objects have a sum, but also that any objects have a unique sum. In other words, many philosophers believe not only in Universalism but also in: Uniqueness: Necessarily, if x and y are sums of the Fs, then x = y.3 Indeed, given only the following principle Transitivity: Necessarily, if x is part of y and y is part of z, then x is part of z; c © 2006 Kluwer Academic Publishers. Printed in the Netherlands. 2things-3.tex; 24/12/2006; 15:22; p.1