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

References

[1] Avron A . Natural 3-valued logics-characterization and proof theory. The Journal of Symbolic Logic, 1991, 56( 1): 276– 294

[2] Avron A . Gentzen-type systems, resolution and tableaux. Journal of Automated Reasoning, 1993, 10 : 265– 281

[3] Li W , Sui Y . Multisequent Gentzen deduction systems for B22-valued first-order logic.. Articial Intelligence Research, 2018, 7( 1): 53– 62

[4] Baaz M , Fermüller C G , Salzer G , Zach R . Labeled calculi and finite-valued logics. Studia Logica, 1998, 61 : 7– 33

[5] Fitting M C. Many-valued modal logics. Fundamenta Informaticae, 1991, 15(3−4): 235−254

[6] Zach R. Proof theory of finite-valued logics. Technical Report TUW-E185.2-Z.1-93, Institut Für Computersprachen, Technische Universität Wien, 1993

[7] Malinowski G. Many-valued logic and its philosophy. In: Gabbay D M, Woods D J, eds. Handbook of the History of Logic, Vol.8. The Many Valued and Nonmonotonic Turn in Logic. Elsevier, 2009

[8] Post E L . Determination of all closed systems of truth tables. Bulletin American Mathematical Society, 1920, 26 : 437–

[9] Li W. Mathematical Logic, Foundations for Information Science. Progress in Computer Science and Applied Logic, vol.25. Birkhäuser, 2010

RIGHTS & PERMISSIONS

Higher Education Press