Abstract
A sequent is a pair (Γ, Δ), which is true under an assignment if either some formula in Γ is false, or some formula in Δ is true. In L3-valued propositional logic, a multisequent is a triple Δ∣Θ∣Γ, which is true under an assignment if either some formula in Δ has truth-value t, or some formula in Θ has truth-value m, or some formula in Γ has truth-value f. There is a sound, complete and monotonic Gentzen deduction system G for sequents. Dually, there is a sound, complete and nonmonotonic Gentzen deduction system G′ for co-sequents Δ: Θ: Γ. By taking different quantifiers some or every, there are 8 kinds of definitions of validity of multisequent Δ∣Θ∣Γ and 8 kinds of definitions of validity of co-multisequent Δ: Θ: Γ, 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.
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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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Wei Li is a PhD, Professor, PhD supervisor, and Academician of Chinese Academy of Sciences, China. He is a professor in the School of Computer Science and Engineering, Beihang University, China. His main research interests include computer science theory and software foundation, theory and practice of unstructured data, and analysis and processing of big data.
Yuefei Sui is a PhD, Professor, PhD supervisor, and a member of China Computer Federation. He is a professor in the Institute of Computing Technology, Chinese Academy of Sciences, and is a professor in School of Computer Science and Technology, University of Chinese Academy of Sciences, China. His main research interests include foundation of large-scale knowledge process and mathematical logic.
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Li, W., Sui, Y. Monotonicity and nonmonotonicity in L3-valued propositional logic. Front. Comput. Sci. 16, 164315 (2022). https://doi.org/10.1007/s11704-021-0382-0
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DOI: https://doi.org/10.1007/s11704-021-0382-0