Algebraic Polymodal Logic: A Survey

This is a review of those aspects of the theory of varieties of Boolean algebras with operators (BAO’s) that emphasise connections with modal logic and structural properties that are related to natural properties of logical systems. It begins with a survey of the duality that exists between BAO’s and relational structures, focusing on the notions of bounded morphisms, inner substructures, disjoint and bounded unions, and canonical extensions of structures that originate in the study of validity-preserving operations on Kripke frames. This duality is then applied to polymodal propositional logics having finitary intensional connectives that generalise the Box and Diamond connectives of unary modal logic. Issues discussed include validity in canonical structures, completeness and incompleteness under the relational semantics, and characterisations of logics by elementary classes of structures and by finite structures. It turns out that a logic is strongly complete for the relational semantics iff the variety of algebras it defines is complex , which means that every algebra in the variety is embeddable into a full powerset algebra that is also in the variety. A hitherto unpublished formulation and proof of this is given (Theorem 5.6.1) that applies to quasi-varieties. This is followed by an algebraic demonstration that the temporal logic of Dedekind complete linear orderings defines a complex variety, adapting Gabbay’s model-theoretic proof that this logic is strongly complete.

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