In this paper, I will describe a technique for generating a novel kind of semantics for a logic, and explore some of its consequences. It would be natural to call the semantics produced by the technique in question ‘many-valued'; but that name is, of course, already taken. I call them, instead, ‘plurivalent'. In standard logical semantics, formulas take exactly one of a bunch of semantic values. I call such semantics ‘univalent'. In a plurivalent semantics, by contrast, formulas may take one or more such values (maybe even less than one). The construction I shall describe can be applied to any univalent semantics to produce a corresponding plurivalent one. In the paper I will be concerned with the application of the technique to propositional many-valued (including two-valued) logics. Sometimes going plurivalent does not change the consequence relation; sometimes it does. I investigate the possibilities in detail with respect to small family of many-valued logics.