 The reasoning here is basically the same as that given in  Section 3.2  above for why supervenience does not guarantee entailment. Both weakm and strongm supervenience say i) that it is necessary that everything that has an A-property has some B-property or other, and ii) that that B-property entails the A-property. But neither i) nor ii) follows from the possible world versions of either weak or strong supervenience—unless B is assumed to be closed under the Boolean operations of complementation, infinitary conjunction, infinitary disjunction, and property-forming operations involving quantification (McLaughlin 1995). Without that assumption, the possible worlds versions allow things with A-properties to lack B-properties altogether, and a fortiori to lack any B-property that entails their A-properties. Of course, possible worlds supervenience requires that each pair of B-less intraworld (weak) or interworld (strong) individuals must have the same A-properties, but the possible world versions allow B-less individuals to have A-properties. The modal operator versions do not.