Lukasiewicz's 4-valued logic and normal modal logics

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 orderings 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) necessity 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.