The Modalized Many-Valued Logic

The intermediate logic is a three-valued logic proposed by Zhu. A modalized many-valued logic will be proposed in this paper which the unary connective @_i is taken as a modality and a Gentzen-typed deduction system will be given so that the the system is sound and complete with the linearly many-valued semantics of the many-valued logic.

I. Introduction

There are many kinds of researches on many-valued logics. For example, the many values form a linear order ([6],[14]), a pseudo-Boolean algebra ([3],[15]), a lattice ([5],[12]), a bilattice ([2]). The corresponding deduction systems use the propositional logical language (including , etc.), extended logical language to include the value information, and hypersequents to represent many-valued information ([1],[3]).