# Paraconsistency and Paracompleteness

## Abstract

A logic $\langle \mathcal{L},\vdash_{p}\rangle$ is said to be paraconsistent if, and only if $\{\alpha, \neg \alpha\} \nvdash_{p} \beta$, for some formulas $\alpha, \beta$. In other words, the necessary and sufficient (the latter is problematic) condition for a logic to be paraconsistent is that its consequence relation is not $\textit{explosive}$. The definition is very simple but also very broad, and this may create a risk that some logics, which have not too much in common with the $\textit{paraconsistency}$, are considered to be so. Nevertheless, the definition may still serve as a reasonable starting point for more thorough research.

Paracomplete logic can be defined in many different ways among which the following one may be of some interest: A logic $\langle \mathcal{L},\vdash_{q}\rangle$ is said to be paracomplete if, and only if $\{\beta \rightarrow \alpha, \neg \beta \rightarrow \alpha\} \nvdash_{q} \alpha$, for some formulas $\alpha, \beta$. But again, just as in the case of paraconsistent logic, the definition is very general and may be seen to overlap with the logics that have nothing in common with the \textit{paracompleteness}.

In the paper, we define some calculi of paraconsistent and paracomplete logics arranged in the form of hierarchies, determined by several criteria. We put central emphasis on logical axioms admitting only the rule of detachment as the sole rule of inference and on the so-called bi-valuation semantics. The hierarchies (no matter which one) are expected to shed some light on the aforementioned issue.

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How to Cite
Ciuciura J. Paraconsistency and Paracompleteness // Logicheskie Issledovaniya / Logical Investigations. 2019. VOL. 25. № 2. C. 46-60.
Issue
Section
Non-classical logics

## References

Arruda, Alves, 1979a – Arruda, A.I., Alves, E.H. “Some remarks on the logic of vagueness”, Bulletin of the Section of Logic, 1979, Vol. 8, No. 3, pp. 133–138.
Arruda, Alves, 1979b – 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. 139144.
Avron et all, 2018 – Avron, A., Arieli, O., Zamansky, A. “Theory of Eﬀective Propositional Paraconsistent Logics”, in: Studies in Logic, Mathematical Logic and Foundations, Vol. 75. College Publications, 2018.
Batens, 1980 – Batens, D. “Paraconsistent extensional propositional logics”, Logique et Analys, 1980, Vol. 23, No. 90–91, pp. 195–234.
Batens, De Clercq, 2004 – Batens D., De Clercq, K. “A Rich Paraconsistent Extension of Full Positive Logic”, Logique et Analys, 2004, Vol. 47, No. 185–188, pp. 227–257.
Carnielli, Coniglio, 2016 – Carnielli, W., Coniglio, M.E. “Paraconsistent Logic: Consistency, Contradiction and Negation”, in: Logic, Epistemology, and the Unity of Science, Vol. 40, Springer International Publishing, 2016. 398 pp.
Carnielli et all, 2007 – Carnielli, W., Coniglio, M.E., Marcos, J. “Logics of Formal Inconsistency”, in D.M. Gabbay, F. Guenthner (eds.) Handbook of Philosophical Logic, 2nd edition, Vol. 14. Springer, 2007, pp. 1–93.
Ciuciura, 2018 – Ciuciura, J. Hierarchies of the paraconsistent calculi, Wydawnictwo Uniwersytetu L´odzkiego, L´od´z, 2018. (In Polish)
Ciuciura, 2014 – Ciuciura, J. “Paraconsistent heap. A Hierarchy of mbCn-systems”, Bulletin of the Section of Logic, 2014, Vol. 43, No. 3–4, pp. 173–182.
Ciuciura, 2015a – Ciuciura, J. “Paraconsistency and Sette’s calculus P1”, Logic and Logical Philosophy, 2015, Vol. 24, No. 2, pp. 265–273.
Ciuciura, 2015b – Ciuciura, J. “A Weakly-Intuitionistic Logic I1”, Logical Investigations, 2015, Vol. 21, No. 2, pp. 53–60.
da Costa, 1974 – da Costa, N.C.A. “On the theory of inconsistent formal systems”, Notre Dame Journal of Formal Logic, 1974, Vol. 15, No. 4, pp. 497–510.
da Costa, Marconi, 1986 – da Costa, N.C.A., Marconi, D. “A note on paracomplete logic”, Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali. 1986. Rendiconti, Serie 8, Vol. 80, No. 7–12, pp. 504–509.
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, Paracompleteness, and Valuations”, Logique et Analyse, 1984, Vol. 27, No. 106, pp. 119–131.
Petrukhin, 2018 – Petrukhin, Y. “Generalized Correspondence Analysis for ThreeValued 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. Basel: Birkh¨auser, 2008. 178 pp.
Popov, 2002 – Popov, V.M. “On a three-valued paracomplete logic”, Logical Investigations, 2002, Vol. 9, pp. 175–178. (In Russian)
Post, 1921 – Post, E.L. “Introduction to a general theory of elementary propositions”, American Journal of Mathematics, 1921, Vol. 43, No. 3, pp. 163–185.
Sette, 1973 – Sette, A.M. “On the propositional calculus P1”, Mathematica Japonicae, 1973, Vol. 18, No. 3, pp. 173–180.
Sette, Carnielli, 1995 – Sette, A.M., Carnielli, W.A. “Maximal weakly-intuitionistic logics”, Studia Logica, 1995, Vol. 55, No. 1, pp. 181–203.
Slupecki, 1939 – Slupecki, J. Dow´od aksjomatyzowalno´sci pelnych system´ow wielowarto´sciowych rachunku zda´n, Sprawozdania z pos. Tow. Nauk. Warsz, 32, wydz. III, 1939. (In Polish)
W´ojcicki, 1988 – W´ojcicki, R. Theory of Logical Calculi. Basic Theory of Consequence Operations, Synthese Library vol. 199, Netherlands: Springer, 1988. 474 pp.