Lukasiewicz’s 4-valued Logic and Normal Modal Logics

Abstract. In this paper, we investigate the Lukasiewicz’s 4-valued modal logic based on the Aristotele’s modal syllogistic. We present a new interpretation of the set of algebraic truth values by introducing the truth and knowledge order-ings similar to those in Belnap’s 4-valued bilattice but by replacing the original Belnap’s negation with the lattice pseudo-complement instead. Based on it, we develop a formal modal Boolean algebra for Lukasiewicz’s system. We show that this modal algebra corresponds to the standard normal modal logic and develop an autoreferential Kripke-style semantics for it, where Lukasiewicz/Aristotele’s ”necessity ” operator is an existential additive instead of an universal (standard) multiplicative modal operator. Moreover, we show that the standard (modern) ne-cessity modal operator based on S4 accessibility relation (the truth partial order in Lukasiewicz/Belnap’s bilattice) is an identity and consequently is not useful in Lukasiewicz’s system.