Disentangling FDE-Based Paraconsistent Modal Logics

Abstract

The relationships between various modal logics based on Belnap and Dunn’s paraconsistent four-valued logic FDE are investigated. It is shown that the paraconsistent modal logic $$\mathsf{BK}^\Box $$ , which lacks a primitive possibility operator $$\Diamond $$ , is definitionally equivalent with the logic $$\mathsf{BK}$$ , which has both $$\Box $$ and $$\Diamond $$ as primitive modalities. Next, a tableau calculus for the paraconsistent modal logic KN4 introduced by L. Goble is defined and used to show that KN4 is definitionally equivalent with $$\mathsf{BK}^\Box $$ without the absurdity constant. Moreover, a tableau calculus is defined for the modal bilattice logic MBL introduced and investigated by A. Jung, U. Rivieccio, and R. Jansana. MBL is a generalization of BK that in its Kripke semantics makes use of a four-valued accessibility relation. It is shown that MBL can be faithfully embedded into the bimodal logic $$\mathsf{BK}^\Box \times \mathsf{BK}^\Box $$ over the non-modal vocabulary of MBL. On the way from $$\mathsf{BK}^\Box $$ to MBL, the Fischer Servi-style modal logic $$\mathsf{BK}^\mathsf{FS}$$ is defined as the set of all modal formulas valid under a modified standard translation into first-order FDE, and $$\mathsf{BK}^\mathsf{FS}$$ is shown to be characterized by the class of all models for $$\mathsf{BK}^{\Box }\times \mathsf{BK}^{\Box }$$ . Moreover, $$\mathsf{BK}^\mathsf{FS}$$ is axiomatized and this axiom system is proved to be strongly sound and complete with respect to the class of models for $$\mathsf{BK}^{\Box }\times \mathsf{BK}^{\Box }$$ . Moreover, the notion of definitional equivalence is suitably weakened, so as to show that $$\mathsf{BK}^\mathsf{FS}$$ and $$\mathsf{BK}^{\Box }\times \mathsf{BK}^{\Box }$$ are weakly definitionally equivalent.