Fibred BDI logics: Completeness preservation in the presence of interaction axioms

Abstract

In [6, 9] the authors have shown how to combine propositional BDI logics using Gabbay's fibring methodology and in [11, 10] they outlined a tableaux proof procedure for the fibred BDI logic. In this paper we provide a proof related to completeness preservation of the combined BDI logic in the presence of interaction axioms of the form □ 1 φ ⇒ □ 2φ in terms of canonical models. To be more precise, let Λ a, Λ b, Λ c, Λ d be canonical normal modal logics and Λ abcd = Λ a ⊙ Λ b ⊙ Λ c ⊙ Λ d be the logics obtained by fibring/dovetailing Λ a, Λ b, Λ c, Λ d. Then we show that Λ abcd ⊕ ◇ a□ bφ ⇒ □ c◇ dφ is characterised by the class of fibred models satisfying the condition ∀ ω ∈ W, ∀ F ∈ F, M ac (ω) ⊆ N M bd (ω). © 2011 Springer-Verlag.