Fibred BDI Logics: Completeness Preservation in the Presence of Interaction Axioms

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 ⇒□2 ϕ in terms of canonical models. To be more precise, let Λa , Λb , Λc , Λd be canonical normal modal logics and $\Lambda_{abcd} = \Lambda_{a} \circledcirc \Lambda_{b} \circledcirc \Lambda_{c} \circledcirc \Lambda_{d}$ be the logics obtained by fibring/dovetailing Λa , Λb , Λc , Λd . Then we show that $\Lambda_{abcd} \oplus \Diamond_{a} \Box_{b} \varphi \Rightarrow \Box_{c} \Diamond_{d} \varphi$ is characterised by the class of fibred models satisfying the condition ∀ ω ∈ W,∀ $\mathfrak{f} \in$ F, $\mathfrak{M}^{ac}$ (ω ) $\sqsubseteq_{N}$ $\mathfrak{M}^{bd}(\omega)$ .