A Fully Internalized Sequent Calculus for Hybrid Categorial Logics

In this paper, a sequent calculus for a hybrid categorial type logic (HCTL) is obtained following Seligman’s internalization strategy (Seligman 2001). With this strategy, a sequent calculus for a hybrid language can be developed starting from a first order sequent calculus. Seligman exemplifies his strategy developing a calculus for hybrid modal logics, but rises the question of whether the strategy works in general. We investigate this issue in the particular case of categorial type logics. As categorial type logics lack Boolean structure, a successful hybridization would indicate that the strategy is indeed rather general and does not depend on the availability of Boolean connectives. In this paper, we will see that this is the case, and moreover, since we can easily arrive to an intuitionistic version of the calculus, we will see that a classical base is not needed either.