A lattice of the paracomplete calculi


Janusz Ciuciura


Paracomplete logic is intended to cope with the problem of vagueness, or uncertain and incomplete data. It deals with the situation when some propositions and their negations are allowed to be simultaneously false, which is obviously impossible in the classical and many non--classical propositional logics. In paracomplete logic, such classical laws as tertium non datur or consequentia mirabilis are not generally accepted. This implies that the logic is defined negatively.
In this paper, we introduce a family of the paracomplete calculi that will be defined in a Hilbert-style formalization. We propose the so-called bi--valuational semantics and prove the key metatheorems for the calculi. We also discuss a generalization of the paracomplete calculus $QD^{1}$ to the hierarchy of related calculi. 




Non-classical logics


Arruda, Alves, 1979 – Arruda, A.I., Alves, E.H. “Some remarks on the logic of vague- ness”, Bulletin of the Section of Logic, 1979, Vol. 8, No. 3, pp. 133–138.
Arruda, Alves, 1979 – Arruda A.I., Alves, E.H. “A semantical study of some systems of vagueness logic”, Bulletin of the Section of Logic, 1979, Vol. 8, No. 3, pp. 139– 144.
Batens et all, 1999 – Batens, D., De Clercq, K., Kurtonina, N. “Embedding and in- terpolation for some paralogics. The propositional case”, Reports on Mathematical Logic, Vol. 33, pp. 29–44.
Beall, 2017 – Beall, Jc. “There is no logical negation: true, false, both and neither”, Australasian Journal of Logic, 2017, Vol. 14, No. 1, pp. 1–29.
Bolotov et all, 2018 – Bolotov, A., Kozhemiachenko, D., Shangin, V. “Paracomplete Logic Kl – Natural Deduction, its Automation, Complexity and Applications”, Journal of Applied Logics, 2018, Vol. 5, No. 1, pp. 221–261.
Carnielli, Coniglio, 2016 – Carnielli, W., Coniglio, M.E., Paraconsistent Logic: Con- sistency, Contradiction and Negation. Logic, Epistemology, and the Unity of Sci- ence. Vol. 40. Springer International Publishing, 2016.
Carnielli, Marcos, 1999 – Carnielli, W., Marcos, J., “Limits for paraconsistent calculi”, Notre Dame Journal of Formal Logi, 1999, Vol. 40, No. 3, pp. 375–390.
Ciuciura, 2015 – Ciuciura, J., “A Weakly-Intuitionistic Logic I1”, Logical Investiga- tions, 2015, Vol. 21, No. 2, pp. 53—60.
Ciuciura, 2019 – Ciuciura, J. “Paraconsistency and Paracompleteness”, Logical Invest- igations, 2019, Vol. 25, No. 2, pp. 46–60.
da Costa, Marconi, 1986 – da Costa, N.C.A., Marconi, D., “A note on paracomplete logic”, in: Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti, Serie 8, 1986, Vol. 80, No. 7–12, pp. 504–509.
Ferna ́ndez, Coniglio, 2003 – Ferna ́ndez, V.L., Coniglio, M.E., “Combining Valuations with Society Semantics”, Journal of Applied Non-Classical Logics, 2003, Vol. 13, No. 1, pp. 21–46.
Karpenko, Tomova, 2017 – Karpenko, A., Tomova, N., “Bochvar’s three-valued logic and literal paralogics: Their lattice and functional equivalence”, Logic and Logical Philosophy, 2017, Vol. 26, No. 2, pp. 207–235.
Lopari ́c, da Costa, 1984 – Lopari ́c, A., da Costa, N.C.A., “Paraconsistency, Paracom- pleteness, and Valuations”, Logique et Analyse, 1984, Vol. 27, No. 106, pp. 119–131.
Nowak, 1998 – Nowak, M. “Kripke semantics for some paraconsistent logics”, Logica Trianguli, 1998, Vol. 2, pp. 87–101.
Petrukhin, 2018 – Petrukhin, Y. “Generalized Correspondence Analysis for Three- Valued Logics”, Logica Universalis, 2018, Vol. 12, No. 3–4, pp. 423–460.
Pogorzelski, Wojtylak, 2008 – Pogorzelski, W.A., Wojtylak, P., Completeness Theory for Propositional Logics, Studies in Universal Logic, Birkha ̈user Basel, 2008.
Popov, 2002 – Popov, V.M. “On a three-valued paracomplete logic”, Logical Investig- ations, 2002, Vol. 9, pp. 175–178. (In Russian)
Sette, Carnielli, 1995 – Sette, A.M., Carnielli, W.A. “Maximal weakly-intuitionistic logics”, Studia Logica, 1995, Vol. 55, No. 1, pp. 181–203.
Wo ́jcicki, 1988 – Wo ́jcicki, R. Theory of Logical Calculi. Basic Theory of Consequence Operations, Synthese Library, Vol. 199, Springer Netherlands, 1988.