On the System CB1 and a Lattice of the Paraconsistent Calculi

Janusz Ciuciura

doi:10.12775/LLP.2019.035

Autor

Janusz Ciuciura Uniwersytet Łódzki, Instytut Filozofii, Katedra Logiki i Metodologii Nauk https://orcid.org/0000-0001-9965-9822

DOI:

https://doi.org/10.12775/LLP.2019.035

Słowa kluczowe

paraconsistent logic, paraconsistency, hierarchy of the paraconsistent calculi

Abstrakt

In this paper, we present a calculus of paraconsistent logic. We propose an axiomatisation and a semantics for the calculus, and prove several important meta-theorems. The calculus, denoted as CB¹, is an extension of systems PI, C min and B¹, and a proper subsystem of Sette’s calculus P¹. We also investigate the generalization of CB¹ to the hierarchy of related calculi.

Bibliografia

Asenjo, F.G., and J. Tamburino, “Logic of antinomies”, Notre Dame Journal of Formal Logic 16, 1 (1975): 17–44. DOI: http://dx.doi.org/10.1305/ndjfl/1093891610

Avron, A., O. Arieli and A. Zamansky, Theory of Effective Propositional Paraconsistent Logics, Studies in Logic, Mathematical Logic and Foundations, vol. 75, College Publications, 2018.

Batens, D., “Paraconsistent extensional propositional logics”, Logique et Analys 23, 90–91 (1980): 195–234.

Carnielli, W., and M.E. Coniglio, Paraconsistent Logic: Consistency, Contradiction and Negation, Logic, Epistemology, and the Unity of Science, vol. 40, International Publishing, 2016. DOI: http://dx.doi.org/10.1007/978-3-319-33205-5

Carnielli, W., M.E. Coniglio and J. Marcos, “Logics of formal inconsistency”, Chapter 1, pages 1–93, in D.M. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, 2nd edition, vol. 14, Springer, 2007. DOI: http://dx.doi.org/10.1007/978-1-4020-6324-4_1

Carnielli, W., and J. Marcos, “Limits for paraconsistent calculi”, Notre Dame Journal of Formal Logi 40, 3 (1999): 375–390. DOI: http://dx.doi.org/10.1305/nd-jfl/1022615617

Ciuciura, J., “Paraconsistent heap. A Hierarchy of mbCn-systems”, Bulletin of the Section of Logic 43, 3/4 (2014): 173–182.

Ciuciura, J., “Paraconsistency and Sette’s calculus P1”, Logic and Logical Philosophy 24, 2 (2015): 265–273. DOI: http://dx.doi.org/10.12775/LLP.2015.003

Ciuciura, J., Hierarchies of the Paraconsistent Calculi (in Polish), Wydawnictwo Uniwersytetu Łódzkiego, Łódź, 2018.

da Costa, N.C.A., “On the theory of inconsistent formal systems”, Notre Dame Journal of Formal Logic 15, 4 (1974): 497–510. DOI: http://dx.doi.org/10.1305/nd-jfl/1093891487

da Costa. N.C.A., and J.-Y. Béziau, “Carnot’s logic”, Bulletin of the Section of Logic 22, 3 (1993): 99–105.

Jaśkowski, S., “Rachunek zdań dla systemów dedukcyjnych sprzecznych” (in Polish), Societatis Scientiarum Torunensis Sect. A, I, 5 (1948): 57–77. First English translation: “Propositional calculus for contradictory deductive systems”, Studia Logica 24 (1969): 143–157. Second one: “A propositional calculus for inconsistent deductive systems”, Logic and Logical Philosophy, 7 (1999): 35–56. DOI: http://dx.doi.org/10.12775/LLP.1999.003

Loparić, A., “A semantical study of some propositional calculi”, The Journal of Non-Classical Logic 3, 3 (1986): 73–95.

Pogorzelski, W.A., and P. Wojtylak, Completeness Theory for Propositional Logics, Studies in Universal Logic, Birkhäuser, Basel, 2008. DOI: http://dx.doi.org/10.1007/978-3-7643-8518-7

Qingyu, Z., “A weak paraconsistent conditional logic”, Journal of Non-Classical Logic 8, 1 (1991): 45–57.

Sette, A.M., “On the propositional calculus P 1 ”, Mathematica Japonicae 18, 3 (1973): 173–180.

Tuziak, R., “Paraconsistent extensions of positive logic”, Bulletin of the Section of Logic 25, 1 (1996): 15–20.

Urbas, I., “A note on ‘Carnot’s logic’”, Bulletin of the Section of Logic 23, 3 (1994): 118–125