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Completeness and Definability in the Logic of Noncontingency

Hilbert-style axiomatic systems are presented for versions of the modal logics KΣ, where Σ ⊆ {D,4,5}, with non-contingency 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 non-contingency logic of any logic containing the seriality axiom is proved to be non-serial. It is also shown that any class of frames definable in the non-contingency language contains the class of functional frames, and dually, there exists a greatest consistent normal non-contingency logic.