Generalized Kripke semantics for the Lambek–Grishin calculus

In this article, we present relational semantics for the Lambek–Grishin calculus and various extensions. Following the approach of generalized Kripke semantics described in Gehrke (2006, Studia Logica, 84, 241–275), we consider semantics based on the generalized Kripke frames naturally associated with the algebraic semantics of the logics in question via their representation theory. This approach is based on canonicity and correspondence as in the classical modal logic setting. Traditional Kripke semantics for the Lambek–Grishin calculus have the drawback that, as soon as additional axioms or additional connectives are present, one may have to start over to obtain semantics for such richer logics. The advantage of our approach via canonical extensions of LG-algebras is that each additional axiom that lifts to the canonical extension can be handled in a modular way, whereas additional connectives modularly slot in as additional relational components. All groups of axioms presented by Grishin in Grishin (1983, Symmetric categorial Grammar) are canonical, and we obtain Sahlqvist-style correspondence results for each of these. The modular set-up allows us to augment these results by the correspondence results for associativity, commutativity, weakening and contraction given in Dunn, Gehrke, and Palmigiano (2005, J. Symbol. Logic., 70, 713–740), as well by results for additional connectives such as lattice operations and linear logic-type negation. This allows a clear comparison of the various logics and a fully modular family of completeness results.