 For each semantics, specific conditions can be imposed on frames in order to secure the validity of particular axioms or argument schemata. Their effects on the logic of conditionals have been studied in detail. For illustration we give the examples of two specific axioms and their correspondence properties in the three frameworks. Identity requires that every \(A\)-closest world be indeed an \(A\)-world. Conditional excluded middle corresponds to a uniqueness condition: there is at most one closest \(A\)-world.