Strengthening Brady’s Paraconsistent 4-Valued Logic BN4 with Truth-Functional Modal Operators

Abstract

?ukasiewicz presented two different analyses of modal notions by means of many-valued logics: (1) the linearly ordered systems ?3,..., [InlineEquation not available: see fulltext.] ,..., $$\hbox {L}_{\omega }$$L?; (2) the 4-valued logic ? he defined in the last years of his career. Unfortunately, all these systems contain "?ukasiewicz type (modal) paradoxes". On the other hand, Brady's 4-valued logic BN4 is the basic 4-valued bilattice logic. The aim of this paper is to show that BN4 can be strengthened with modal operators following ?ukasiewicz's strategy for defining truth-functional modal logics. The systems we define lack "?ukasiewicz type paradoxes". Following Brady, we endow them with Belnap---Dunn type bivalent semantics.