Completeness and Definability in the Logic of Noncontingency

Abstract

Hilbert-style axiomatic systems are presented for versions of the modal logics K$ \Sigma$, where $ \Sigma$ $ \subseteq$ {D, 4, 5}, with noncontingency as the sole modal primitive. The classes of frames characterized by the axioms of these systems are shown to be first-order definable, though not equal to the classes of serial, transitive, or euclidean frames. The canonical frame of the noncontingency logic of any logic containing the seriality axiom is proved to be nonserial. It is also shown that any class of frames definable in the noncontingency language contains the class of functional frames, and dually, there exists a greatest consistent normal noncontingency logic.