Recent investigations of the logic of supervenience have stipulated that the supervenience base, or set of "subvening" properties, be closed under various property-forming operations. For example, Kim has assumed that the base is dosed under Boolean operations (disjunction, negation, etc.)) and Bacon has assumed that the base is closed under an operation he calls "diagonalizing" or "resplicing. "2 1 shall argue here that it is generally reasonable to assume closure under disjunction and conjunction, but not always under negation or resplicing.
Kim distinguishes two principal varieties of supervenience, strong and weak. In the definitions below, you may think of the A-properties as the properties "above" (the supervening or determined properties) and the B-properties as the properties "below" (the subvening or determining properties).
Weak supervenience, definition 1: A-properties weakly supervene on B-properties = Df Necessarily, for any two individuals x and y, if x and y share all their B-properties, they share all their A-properties. (Or: if x and y are B-indiscernible, they are A-indiscernible.)
Strong supervenience: A-properties strongly supervene on B-properties --Df Necessarily for any individual x and A-property a, if x has a, there is a B-property b such that (i) x has b, and (ii) necessarily, whatever has b has a. (Or, every A-property has a necessitating ground among the B-properties.)