TY - JOUR

T1 - Disentangling FDE-based paraconsistent modal logics

AU - Odintsov, Sergei P.

AU - Wansing, Heinrich

PY - 2017/9/23

Y1 - 2017/9/23

N2 - 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 BK□, which lacks a primitive possibility operator ◊, is definitionally equivalent with the logic BK, which has both and ◊ 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 BK□ 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 BK□ × BK□ over the non-modal vocabulary of MBL. On the way from BK□ to MBL, the Fischer Servi-style modal logic BKFS is defined as the set of all modal formulas valid under a modified standard translation into first-order FDE, and BKFS is shown to be characterized by the class of all models for BK□ × BK□. Moreover, BKFS is axiomatized and this axiom system is proved to be strongly sound and complete with respect to the class of models for BK□ x BK□. Moreover, the notion of definitional equivalence is suitably weakened, so as to show that BKFS and BK□ × BK□ are weakly definitionally equivalent.

AB - 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 BK□, which lacks a primitive possibility operator ◊, is definitionally equivalent with the logic BK, which has both and ◊ 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 BK□ 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 BK□ × BK□ over the non-modal vocabulary of MBL. On the way from BK□ to MBL, the Fischer Servi-style modal logic BKFS is defined as the set of all modal formulas valid under a modified standard translation into first-order FDE, and BKFS is shown to be characterized by the class of all models for BK□ × BK□. Moreover, BKFS is axiomatized and this axiom system is proved to be strongly sound and complete with respect to the class of models for BK□ x BK□. Moreover, the notion of definitional equivalence is suitably weakened, so as to show that BKFS and BK□ × BK□ are weakly definitionally equivalent.

KW - Belnap-Dunn modal logic

KW - Definitional equivalence

KW - First-degree entailment logic

KW - Modal bilattice logic

KW - Paraconsistent logic

KW - Standard translation

KW - Strong negation

KW - Tableau calculi

KW - SEMANTICS

UR - http://www.scopus.com/inward/record.url?scp=85029773059&partnerID=8YFLogxK

U2 - 10.1007/s11225-017-9753-9

DO - 10.1007/s11225-017-9753-9

M3 - Article

AN - SCOPUS:85029773059

VL - 105

SP - 1221

EP - 1254

JO - Studia Logica

JF - Studia Logica

SN - 0039-3215

IS - 6

ER -