Disentangling FDE-Based Paraconsistent Modal Logics

摘要

AbstractThe relationships between various modal logics based on Belnap and Dunn’s paraconsistent four-valued logicFDEare investigated. It is shown that the paraconsistent modal logic$$mathsf{BK}^Box $$BK□, which lacks a primitive possibility operator$$Diamond $$?, is definitionally equivalent with the logic$$mathsf{BK}$$BK, which has both$$Box $$□and$$Diamond $$?as primitive modalities. Next, a tableau calculus for the paraconsistent modal logicKN4introduced by L. Goble is defined and used to show thatKN4is definitionally equivalent with$$mathsf{BK}^Box $$BK□without the absurdity constant. Moreover, a tableau calculus is defined for the modal bilattice logicMBLintroduced and investigated by A. Jung, U. Rivieccio, and R. Jansana.MBLis a generalization ofBKthat in its Kripke semantics makes use of a four-valued accessibility relation. It is shown thatMBLcan be faithfully embedded into the bimodal logic$$mathsf{BK}^Box imes mathsf{BK}^Box $$