Wesley H. Holliday presents possibility semantics as a general approach to logical semantics that replaces possible worlds with possibly partial "possibilities." In classical possibility semantics, propositions are interpreted as regular open sets of a poset or as compact regular open sets of an upper Vietoris space. This approach allows for partial possibilities that may not settle the truth of a proposition or determine which disjunct is true. Possibilities are related by a refinement relation, where one possibility refines another if it contains all the information of the latter. This framework enables the interpretation of classical logic, modal logics, and intuitionistic logics, offering richer structures and avoiding nonconstructivity inherent in traditional semantics. The chapter surveys the representation of Boolean algebras, first-order logic, modal logics, and intuitionistic logic using possibility semantics, highlighting its benefits in overcoming incompleteness results and providing more flexible interpretations. It also discusses connections to other logics like inquisitive logic and explores the use of posets and topological spaces in semantics. The paper emphasizes the role of refinement, regular sets, and the duality between Boolean algebras and topological spaces in possibility semantics.Wesley H. Holliday presents possibility semantics as a general approach to logical semantics that replaces possible worlds with possibly partial "possibilities." In classical possibility semantics, propositions are interpreted as regular open sets of a poset or as compact regular open sets of an upper Vietoris space. This approach allows for partial possibilities that may not settle the truth of a proposition or determine which disjunct is true. Possibilities are related by a refinement relation, where one possibility refines another if it contains all the information of the latter. This framework enables the interpretation of classical logic, modal logics, and intuitionistic logics, offering richer structures and avoiding nonconstructivity inherent in traditional semantics. The chapter surveys the representation of Boolean algebras, first-order logic, modal logics, and intuitionistic logic using possibility semantics, highlighting its benefits in overcoming incompleteness results and providing more flexible interpretations. It also discusses connections to other logics like inquisitive logic and explores the use of posets and topological spaces in semantics. The paper emphasizes the role of refinement, regular sets, and the duality between Boolean algebras and topological spaces in possibility semantics.