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. 1. Vagueness and Opacity " Vagueness is a feature of scientific as of other discourse. " Max Black (1966) Peirce famously characterised vagueness in the following terms: " A proposition is vague when there are possible states of things concerning which it is intrinsically uncertain whether, had they been contemplated by the speaker, he would have regarded them as excluded or allowed by the proposition. By intrinsically uncertain we mean not uncertain in consequence of any ignorance of the interpreter, but because the speaker's habits of language were indeterminate " [29]. In this context, two issues immediately arise: (i) whether vagueness can be dismissed as merely a feature of 'natural' language which will effectively evaporate with the introduction of some formal system, and (ii) given a negative answer to (i), whether vagueness requires the use of some form of non-classical logic. These are the issues with which standard discussions of vagueness have been concerned. Thus Wright, for example, emphasises that vague predicates lack 'sharp boundaries' and argues forcefully that " ... the utility and point of the classifications expressed The authors would like very much to thank the two anonymous referees for their suggestions.