This paper introduces a new format for reasoning with prioritized stan- dards of normality. It is applicable in a broad variety of contexts, e.g. dealing with (possibly conflicting) prioritized belief bases or combining different reasoning methods in a prioritized way. The format is a gener- alization of the standard format of adaptive logics (see [4]). Every logic that is formulated within it has a straightforward semantics in the style of Shoham’s selection semantics (see [22]) and a dynamic proof theory. Fur- thermore, it can count on a rich meta-theory that inherits the attractive features of the standard format, such as soundness and completeness, re- flexivity, idempotence, cautious monotonicity, and many other properties.