Another Useful Four-Valued Logic

• Zuoquan Lin
• Zhaocong Jia
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11062)

Abstract

We propose a four-valued logic with intuitive semantics by the connectives that is useful for understanding the contradictions in knowledge representation. The intuitive semantics reflects that any assertion has dual character by whose information for or against the judgment. The four-valued logic is weakly paraconsistent and has the weak consistency to capture whether or not the contradictions are reconcilable with information. The four-valued logic is a normal extension of classical logic in a sense that it contains the schemata of classical axioms. We then propose an axiomatization of the four-valued logic. The soundness and completeness of the axiomatization with respect to the semantics are proved. The usefulness of the four-valued logic is discussed.

Keywords

Four-valued logic Bilattice Paraconsistency Weak consistency Conflation

References

1. 1.
Anderson, A., Belnap, N.: Entailment: The Logic of Relevance and Necessity, vol. 1. Princeton University Press, Princeton (1975)
2. 2.
Arieli, O., Avron, A.: The value of the four values. Artif. Intell. 102(1), 97–141 (1998)
3. 3.
Belnap Jr., N.D.: A useful four-valued logic. In: Dunn, J.M., Epstein, G. (eds.) Modern Uses of Multiple-Valued Logic, pp. 5–37. Springer, Dordrecht (1977).
4. 4.
van Benthem, J., Pacuit, E.: Dynamic logics of evidence-based beliefs. Stud. Logica 99(1–3), 61–92 (2011)
5. 5.
Bergstra, J.A., Bethke, I., Rodenburg, P.: A propositional logic with 4 values: true, false, divergent and meaningless. J. Appl. Non-Class. Log. 5(2), 199–217 (1995)
6. 6.
De, M., Omori, H.: Classical negation and expansions of belnap-dunn logic. Stud. Logica 103(4), 825–851 (2015)
7. 7.
Fitting, M.: Bilattices and the semantics of logic programming. J. Log. Prog. 11(2), 91–116 (1991)
8. 8.
Font, J.M., Moussavi, M.: Note on a six-valued extension of three-valued logic. J. Appl. Non-Class. Log. 3(2), 173–187 (1993)
9. 9.
Font, J.M.: Belnap’s four-valued logic and De Morgan lattices. Log. J. IGPL 5(3), 1–29 (1997)
10. 10.
Ginsberg, M.L.: Multivalued logics: a uniform approach to reasoning in artificial intelligence. Comput. Intell. 4(3), 265–316 (1988)
11. 11.
Rosser, J.B., Turquette, A.R.: Multiple-Valued Logics. North Holland, Amsterdam (1952)
12. 12.
Kaluzhny, Y., Muravitsky, A.Y.: A knowledge representation based on the Belnap’s four-valued logic. J. Appl. Non-Class. Log. 3(2), 189–203 (1993)
13. 13.
Maruyama, Y.: Algebraic study of lattice-valued logic and lattice-valued modal logic. In: Ramanujam, R., Sarukkai, S. (eds.) ICLA 2009. LNCS (LNAI), vol. 5378, pp. 170–184. Springer, Heidelberg (2008).
14. 14.
Omori, H., Sano, K.: Generalizing functional completeness in Belnap-Dunn logic. Stud. Logica 103(5), 883–917 (2015)
15. 15.
Pietz, A., Rivieccio, U.: Nothing but the truth. J. Philos. Log. 42(1), 125–135 (2013)
16. 16.
Pynko, A.P.: Functional completeness and axiomatizability within Belnap’s four-valued logic and its expansions. J. Appl. Non-Class. Log. 9(1), 61–105 (1999)
17. 17.
Ruet, P., Fages, F.: Combining explicit negation and negation by failure via Belnap’s logic. Theor. Comput. Sci. 171(1), 61–75 (1997)
18. 18.
Tsoukias, A.: A first order, four-valued, weakly paraconsistent logic and its relation with rough sets semantics. Found. Comput. Decis. Sci. 27(2), 77–96 (2002)
19. 19.
Visser, A.: Four valued semantics and the liar. J. Philos. Log. 13(2), 181–212 (1984)
20. 20.
Wintein, S., Muskens, R.: A calculus for Belnap’s logic in which each proof consists of two trees. Log. Anal. 220, 643–656 (2012)
21. 21.
Wintein, S., Muskens, R.: A gentzen calculus for nothing but the truth. J. Philos. Log. 45(4), 451–465 (2016)