  This paper puts forth a class of algebraic structures, relativized Boolean
algebras (RBAs), that provide semantics for propositional logic in which
truth/validity is only defined relative to a local domain. In particular, the
join of an event and its complement need not be the top element. Nonetheless,
behavior is locally governed by the laws of propositional logic. By further
endowing these structures with operators -- akin to the theory of modal
Algebras -- RBAs serve as models of modal logics in which truth is relative. In
particular, modal RBAs provide semantics for various well known awareness
logics and an alternative view of possibility semantics.
