The Metaphysical Significance of the Ugly Duckling Theorem

According to Satosi Watanabe’s “theorem of the ugly duckling”, the number of (possible) predicates satisfied by any two different particulars is a constant, which does not depend on the choice of the two particulars. If the number of (possible) predicates satisfied by two particulars is their number of properties in common, and the degree of resemblance between two particulars is a function of their number of properties in common, then it follows that the degree of resemblance between any two different particulars is a constant, which does not depend on the choice of the two particulars either. Avoiding this absurd conclusion requires questioning assumptions involving infinity in the proof or interpretation of the theorem, adopting a sparse conception of properties according to which not every (possible) predicate corresponds to a property, or denying that degree of resemblance is a function of number of properties in common. After arguing against the first two options, this paper argues for a version of the third which analyses degrees of resemblance in terms of degrees of naturalness of common properties. In the course of doing so, it presents a novel account of natural properties.