Combining multi-valued and modal logics into a single system is a long-standing concern in mathematical logic and computer science, see for example [7] and the literature cited there. Recent work in this trend [15, 17, 14] develops modal expansions of many-valued systems that are also inconsistencytolerant, along the tradition initiated by Belnap with his “useful four-valued logic” [3]. Our contribution continues on this line, and the specific problem we address is that of defining and axiomatizing the least modal logic over the four-element Belnap lattice. The problem was inspired by [5], but our solution is quite different from (and in some respects more satisfactory than) that of [5] in that we make an extensive and profitable use of algebraic and topological techniques. In fact, our algebraic and topological analyses of the logic have, in our opinion, an independent interest and contribute to the appeal of our approach. Kripke frames provide a semantics for modal logics that is both flexible with regards to intended applications and interpretations, and highly intuitive. When the non-modal part is multi-valued, though, one may wonder whether the accessibility relation between worlds should remain two-valued or be allowed to assume the same range of truth values as the logic itself. Starting from the point of view of AI applications, [7] argues forcefully that multiple values are an appropriate and useful modeling device. This is the approach taken in [5] and here, too. Our aim is to study the least modal logic over the Belnap lattice, that is, the logic determined by the class of all Kripke frames where the accessibility relation as well as semantic valuations are four-valued.

Four-valued modal logic: Kripke semantics and duality.

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.

Disentangling FDE-Based Paraconsistent Modal Logics

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Four-valued modal logic: Kripke semantics and duality

We employ the well-developed and powerful techniques of algebraic semantics and Priestley duality to set up a Kripke semantics for a modal expansion of Arieli and Avron's bilattice logic, itself based on Belnap's four-valued logic. We obtain soundness and completeness of a Hilbert-style derivation system for this logic with respect to four-valued Kripke frames, the standard notion of model in this setting. The proof is via intermediary relational structures which are analysed through a topological reading of one of the axioms of the logic. Both local and global consequence on the models are covered.

/pdf/kripke-semantics-for-modal-bilattice-logic-5fwjmj0fo9.pdf

Kripke Semantics for Modal Bilattice Logic

http://www.appliedlogictudelft.nl/wp-content/uploads/2014/02/U-Rivieccio_Paraconsistent-Modal-Logics_Applied-Logic-TU-Delft.pdf

Paraconsistent Modal Logics

Earlier algebraic semantics for Belnapian modal logics were defined in terms of twist-structures over modal algebras. In this paper we introduce the class of BK-lattices, show that this class coincides with the abstract closure of the class of twist-structures, and it forms a variety. We prove that the lattice of subvarieties of the variety of BK-lattices is dually isomorphic to the lattice of extensions of Belnapian modal logic BK. Finally, we describe invariants determining a twist-structure over a modal algebra.

E. I. Latkin

Papers

BK-lattices. Algebraic Semantics for Belnapian Modal Logics