Variant quantifiers in L₃-valued first-order logic

Abstract

Traditional first-order logic has four definitions for quantifiers, which are defined by universal and existential quantifiers. In L₃-valued (three-valued) first-order logic, there are eight kinds of definitions for quantifiers; and corresponding Gentzen deduction systems will be given and their soundness and completeness theorems will be proved.

Keywords

L ₃-valued (three-valued) first-order logic / sequent / quantifier / soundness theorem / completeness theorem

References

[1] Li W. Mathematical Logic: Foundations for Information Science. 2nd ed. Birkhäuser, 2010

[2] Zach R. Proof theory of finite-valued logics, Technical Report TUWE185.2-Z.1-93. Institut Für Computersprachen, Technische Universität Wien, 1993

[3] Avron A. The method of hypersequents in the proof theory of propositional non-classical logics. In: Hodges W, Hyland M, Steinhorn C, Truss J, eds. Logic: From Foundations to Applications European Logic Colloquium. Clarendon Press, 1994, 1–32

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

[5] Li W, Sui Y F. Multisequent gentzen deduction systems for B22-valued first-order logic. Journal of Artificial Intelligence Research, 2018, 7(1): 53

[6] Wansing H. The power of Belnap: sequent systems for SIXTEEN3. Journal of Philosophical Logic, 2010, 39(4): 369–393

[7] Urquhart A. Basic many-valued logic. In: Gabbay D M, Guenthner F, eds. Handbook of Philosophical Logic, vol.2. Springer Netherlands, 2001, 249–295

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

[9] Baaz M, Fermüller C G, Ovrutcki A, Zach R. MULTLOG: a system for axiomatizing many-valued logics. In: Voronkov A, ed. Logic Programming and Automated Reasoning. Springer Berlin Heidelberg, 1993, 345–347

[10] Bochvar D A, Bergmann M. On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus. History and Philosophy of Logic, 1981, 2: 87–112

[11] Fitting MC. Many-valued modal logics. Fundamenta Informaticae, 1999, 15: 235–254

[12] Gottwald S. A Treatise on Many-Valued Logics(Studies in Logic and Computation). Baldock: Research Studies Press, 2001

[13] Hähnle R. Advanced many-valued logics. In: Gabbay D, Guenthner F, eds. Handbook of Philosophical Logic, vol.2. Dordrecht: Kluwer, 2001, 297–395

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

[15] Novák V. A formal theory of intermediate quantifiers. Fuzzy Sets and Systems, 2008, 159: 1229–1246

[16] Straccia U. Reasoning within fuzzy description logics. Journal of Artificial Intelligence Research, 2001, 14: 137–166

[17] Li W, Sui Y F. The B₄-valued propositional logic with unary logical connectives ∼1/∼2/⊲. Frontiers of Computer Science, 2007, 11(5): 887–894

[18] Łukasiewicz J. Selected Works. Amsterdam: North-Holland, 1970

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Higher Education Press