The nonmonotonic Gentzen deduction systems for the propositional logic

Abstract

The traditional propositional logic is monotonic. With the same logical language as, the same valuation as and the validity of a sequent different from the traditional propositional logic, a propositional logic could be nonmonotonic. In this paper, the four Gentzen deduction systems G¹,G²,G³,G⁴ and their dualities G₁,G₂,G₃,G₄ will be given which are proved to be sound and complete with respect to the four definitions and their dualities of the validity of sequents, among which one is traditional and others are variations of the traditional one. Moreover, G¹, G³ are monotonic in both Г and Δ; and G², G⁴ are monotonic in Г and nonmonotonic in Δ. Dually, G₁, G₃ are nonmonotonic in both Г and Δ; and G₂, G₄ are nonmonotonic in Г and monotonic in Δ.