Monotonicity and nonmonotonicity in L₃-valued propositional logic

Abstract

A sequent is a pair (\mathrm{\Gamma },\mathrm{\Delta }), which is true under an assignment if either some formula in \mathrm{\Gamma } is false, or some formula in \mathrm{\Delta } is true. In {\mathbf{L}}_3-valued propositional logic, a multisequent is a triple \mathrm{\Delta }\mathrm{\Theta }\mathrm{\Gamma }, which is true under an assignment if either some formula in \mathrm{\Delta } has truth-value \mathtt{\text{t}}, or some formula in \mathrm{\Theta } has truth-value \mathtt{\text{m}}, or some formula in \mathrm{\Gamma } has truth-value \mathtt{\text{f}}. There is a sound, complete and monotonic Gentzen deduction system \mathbf{G} for sequents. Dually, there is a sound, complete and nonmonotonic Gentzen deduction system {\mathbf{G}}^{\prime } for co-sequents \mathrm{\Delta }:\mathrm{\Theta }:\mathrm{\Gamma }. By taking different quantifiers some or every, there are 8 kinds of definitions of validity of multisequent \mathrm{\Delta }\mathrm{\Theta }\mathrm{\Gamma } and 8 kinds of definitions of validity of co-multisequent \mathrm{\Delta }:\mathrm{\Theta }:\mathrm{\Gamma }, and correspondingly there are 8 sound and complete Gentzen deduction systems for sequents and 8 sound and complete Gentzen deduction systems for co-sequents. Correspondingly their monotonicity is discussed.

Graphical abstract

Keywords

sequent / multisequent / gentzen deduction system / monotonicity / nonmonotonicity

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Acknowledgements

This work was supported by the Open Fund of the State Key Laboratory of Software Development Environment (SKLSDE-2010KF-06), Beijing University of Aeronautics and Astronautics, and by the National Basic Research Program of China (973 Program) ( 2005CB321901)

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