A Generalisation of a Refutation-related Method in Paraconsistent Logics

Adam Trybus

doi:10.12775/LLP.2018.002

Authors

Adam Trybus The University of Zielona Góra

DOI:

https://doi.org/10.12775/LLP.2018.002

Keywords

paraconsistency, refutation, maximality

Abstract

This article describes a refutation method of proving maximality of three-valued paraconsistent logics. After outlining the philosophical background related to paraconsistent logics and the refutation approach to modern logic, we briefly describe how these two areas meet in the case of maximal paraconsistent logics. We focus on a method of proving maximality introduced in [34] and [37] that has the benefit of being simple and effective. We show how the method works on a number of examples, thus emphasising the fact that it provides a unifying approach to the search for maximal paraconsistent logics. Finally, we show how the method can be generalised to cover a wide range of paraconsistent logics. We also conduct a small experimental setting that confirms the theoretical results.

Author Biography

Adam Trybus, The University of Zielona Góra

Institute of Philosophy, Chair of Logic and Methodology of Science

References

Alves, E.H., “The first axiomatization of paraconsistent logic”, Bulletin of the Section of Logic 21, 1 (1992): 19–20.

Arieli, O., and A. Avron., “Three-valued paraconsistent propositional logics”, pages 91–129 in J.-Y. Beziau, M. Chakraborty, and S. Dutta (eds.), New Directions in Paraconsistent Logic, Springer, 2015. DOI: 10.1007/978-81-322-2719-9_4

Arieli, O., A. Avron, and A. Zamansky, “Maximally paraconsistent threevalued logics”, pages 210–218 in F. Lin, U. Sattler, and M. Truszczynski (eds.), Proceedings of the Twelfth International Conference on the Principles of Knowledge Representation and Reasoning, The AAAI Press, Menlo Park, California, 2010.

Arruda, A.I., “Aspects of the historical development of paraconsistent Logic”, pages 99–130 in G. Priest, R. Roultey, and X. Norman (eds.), Paraconsistent Logic: Essays on the Inconsistent, Philosophia Verlag, 1989.

Batens, D., “Paraconsistency and its relation to worldviews”, Foundations of Science 3 (1999): 259–283.

Batens, D., “A deneral characterization of adaptive logics”, Logique et Analyse 173–175 (2002): 45–68.

Batóg, T., Dwa paradygmaty matematyki, Wydawnictwo Naukowe UAM, Poznań, 2000.

Caicedo, X., “A formal system for the non-theorems of the propositional calculus”, Notre Dame Journal of Formal Logic 19, 1 (1978): 147–151. DOI: 10.1305/ndjfl/1093888218

Carnap, R., Formalization of Logic, Harvard University Press, 1943.

Ciuciura, J., “Paraconsistency and Sette’s calculus P1”, Logic and Logical Philosophy 24 (2015): 265–273. DOI: 10.12775/LLP.2015.003

Gabbay, D.M., and A.S. d’Avila Garcez, “Logical modes of attack in argumentation networks”, Studia Logica 93, 2–3 (2009): 199–230. DOI: 10.1007/s11225-009-9216-z

Ganeri, J., “Jaina logic and the philosophical basis for pluralism”, History and Philosophy of Logic 23, 4 (2002): 267–281. DOI: 10.1080/0144534021000051505

Goranko, V., “Proving unprovability in some normal modal logics”, Bulletin of the Section of Logic 20, 1 (1991): 23–29.

Goranko, V., “Refutation systems in modal logics”, Studia Logica 53, 2 (1994): 299–324. DOI: 10.1007/BF01054714

Hałkowska, K., “A note on matrices for systems of nonsense-logics”, Studia Logica 48, 4 (1989): 461–464. DOI: 10.1007/BF00370200

Hałkowska, K., and A. Zając, “O pewnym trójwartościowym systemie rachunku zdań”, Acta Universitatis Wratislaviensis. Prace Filozoficzne. Logika, 13 (1988): 40–51.

Jaśkowski, S., “A propositional calculus for inconsistent deductive systems”, Logic and Logical Philosophy 7 (1999): 35–56. DOI: 10.12775/LLP.1999.003

Łukasiewicz, J., O zasadzie sprzeczności u Arystotelesa, Państwowe Wydawnictwo Naukowe, Warszawa, 1956.

Łukasiewicz, J., Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic, Clarendon Press, Oxford, 1957.

Łukasiewicz, J., “Two-valued logic” (1921), pages 89–109 in L. Borkowski (ed.),Jan Łukasiewicz. Selected Works, North Holland Publishing Company, Polish Scientific Publishers, 1970.

Pietryga, A., Status zasady sprzeczności w świetle logiki współczesnej, Aureus, 2004.

Poczobut, R., Spór o zasadę niesprzeczności, Towarzystwo Naukowe KUL, 2000.

Priest, G., In Contradiction, Kluwer Academic Publishers, 1987. DOI: 10.1007/978-94-009-3687-4

Priest, G., “Motivations for paraconsistent logic: The slippery slope from classical logic to dialetheism”, pages 223–232 in G. Priest and D. Batens (eds.), Frontiers of Paraconsisten Logic, Research Studies Press, Baldock, 2000.

Priest, G., “Vasil’iev and imaginarylogic”, History and Philosophy of Logic 21, 2 (2000): 135–146. DOI: 10.1080/01445340050064031

Pynko, P.A., “Extensions of Hałkowska–Zając three-valued paraconsistent logic”, Archive of Mathematical Logic 41, 3 (2002): 299–307. DOI: 10.1007/s001530100115

Restall, G., “Relevant and substructural logics”, pages 289–398 in D.M. Gabbay and J. Woods (eds.), Handbook of the History and Philosophy of Logic, volume 7, Elsevier, 2006. DOI: 10.1016/S1874-5857(06)80030-0

Sette, A.M., and E.H. Alves, “On the equivalence between two systems of paraconsistent logic”, Bulletin of the Section of Logic 24, 3 (1995): 155-157.

Sette, A.M., and E.H. Alves, “On the equivalence between some systems of non-classical logic”, Bulletin of the Section of Logic 25, 2 (1996): 68–72.

Skura, T., “A complete syntactical characterisation of the intuitionistic logic”, Reports on Mathematical Logic 23 (1989): 75–80.

Skura, T., “A new criterion of decidability for intermediate logics”, Bulletin of the Section of Logic 19, 1 (1990): 10–14.

Skura, T., “Syntactic refutations against finite models in modal logic”, Notre Dame Journal of Formal Logic, 35, 4 (1994): 595–605. DOI: 10.1305/ndjfl/1040408615

Skura, T., Aspects of Refutation Procedures in the Intuitionistic Logics and Related Modal Systems, Wydawnictwo Uniwersytetu Wrocławskiego, Wrocław, 1999.

Skura, T., “Maximality and refutability”, Notre Dame Journal of Formal Logic 45, 2 (2004): 65–72. DOI: 10.1305/ndjfl/1095386644

Skura, T., “Refutation systems in propositional logic”, pages 115–157 in D.M. Gabbay and F. Guenthner (eds.),Handbook of Philosophical Logic, volume 16, Springer, 2011. DOI: 10.1007/978-94-007-0479-4_2

Skura, T., Refutation Methods in Modal Propositional Logic, Semper, 2013.

Skura, T., and R. Tuziak, “Three-valued maximal paraconsistent logics”, Acta Universitatis Wratislaviensis. Logika 23 (2005): 129–134.

Suchoń, W., “Vasil’iev. What did he exactly do?”, Logic and Logical Philosophy 7 (1999): 131–141. DOI: 10.12775/LLP.1999.011

Tanaka, K., “Three schools of paraconsistency”, Australasian Journal of Logic 1 (2003): 28–42.

Wajsberg, M., “Untersuchungen über den Aussagenkalkül von A. Heyting”, Wiadomości Matematyczne 46 (1938): 429–435.

Wybraniec-Skardowska, U., “On the notion of function of the rejection of propositions”, Acta Universitatis Wratislaviensis. Logika 23 (2005): 179–202.