Opaque predicates, veiled sets and their logic

Motivated by considerations in the foundations of quantum mechanics and inspired by the literature on vague predicates, we introduce the concept of an opaque predicate. While in the case of vague predicates there is a kind of indeterminacy with respect to the predicate, in the sense that the vagueness concerns whether a well-determined object satisfies it or not, in the case of opaque predicates the indeterminacy is with regard to the objects which should satisfy them. In other words, their extensions are not well-defined, despite the fact that the conditions for an object to satisfy the predicates are well-known. We suggest that such opaque predicates (and more generally, what we call opaque relations) can be characterized by a logic which encompasses a semantics founded in quasi-set theory, and call their extensions veiled sets.