On Paracomplete Versions of Jaśkowski's Discussive Logic

Krystyna Mruczek-Nasieniewska; Yaroslav Petrukhin; Vasily Shangin

doi:10.18778/0138-0680.2024.01

Authors

Krystyna Mruczek-Nasieniewska Nicolaus Copernicus University in Toruń, Department of Logic, Institute of Philsophy, Faculty of Philosophy and Social Sciences Author https://orcid.org/0000-0001-9125-2446
Yaroslav Petrukhin University of Łódź, Department of Logic, Institute of Philosophy, Faculty of History and Philosophy Author https://orcid.org/0000-0002-7731-1339
Vasily Shangin Lomonosov Moscow State University, Department of Logic, Faculty of Philosophy Author https://orcid.org/0000-0002-7380-3213

DOI:

https://doi.org/10.18778/0138-0680.2024.01

Keywords:

discussive logic, discursive logic, modal logic, paracomplete logic, paraconsistent logic

Abstract

Jaśkowski's discussive (discursive) logic D2 is historically one of the first paraconsistent logics, i.e., logics which 'tolerate' contradictions. Following Jaśkowski's idea to define his discussive logic by means of the modal logic S5 via special translation functions between discussive and modal languages, and supporting at the same time the tradition of paracomplete logics being the counterpart of paraconsistent ones, we present a paracomplete discussive logic D2^p.

References

G. Achtelik, L. Dubikajtis, E. Dudek, J. Konior, On independence of axioms of Jaśkowski’s discussive propositional calculus, Reports on Mathematical Logic, vol. 11 (1981), pp. 3–11.

S. Akama, J. M. Abe, K. Nakamatsu, Constructive discursive logic with strong negation, Logique et Analyse. Nouvelle Série, vol. 54(215) (2011), pp. 395–408, URL: https://www.jstor.org/stable/44085016

S. Akama, N. C. A. da Costa, Why paraconsistent logics?, [in:] S. Akama (ed.), Towards paraconsistent engineering, vol. 110 of Intelligent Systems Reference Library, Springer, Cham (2016), pp. 7–24, DOI: https://doi.org/10.1007/978-3-319-40418-9_2 DOI: https://doi.org/10.1007/978-3-319-40418-9_2

A. Almukdad, D. Nelson, Constructible falsity and inexact predicates, The Journal of Symbolic Logic, vol. 49(1) (1984), pp. 231–233, DOI: https://doi.org/10.2307/2274105 DOI: https://doi.org/10.2307/2274105

A. R. Anderson, N. D. Belnap, Jr., Entailment: The Logic of Relevance and Necessity, Volume 1, Princeton University Press, Princeton, N.J.-London (1975).

O. Arieli, A. Avron, Four-valued paradefinite logics, Studia Logica, vol. 105(6) (2017), pp. 1087–1122, DOI: https://doi.org/10.1007/s11225-017-9721-4 DOI: https://doi.org/10.1007/s11225-017-9721-4

A. Avron, Paraconsistent fuzzy logic preserving non-falsity, Fuzzy Sets and Systems, vol. 292 (2016), pp. 75–84, DOI: https://doi.org/10.1016/j.fss.2014.07.001 DOI: https://doi.org/10.1016/j.fss.2014.07.001

R. Ballarin, Modern Origins of Modal Logic, [in:] E. N. Zalta, U. Nodelman (eds.), The Stanford Encyclopedia of Philosophy, Fall 2023 ed., Metaphysics Research Lab, Stanford University (2023), URL: https://plato.stanford.edu/archives/fall2023/entries/logic-modal-origins/

J.-Y. Béziau, The Future of Paraconsistent Logics, Logical Studies, vol. 2 (1999), pp. 1–28, URL: http://wwwa.unine.ch/unilog/jyb/future-pl.pdf

D. A. Bochvar, On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus, History and Philosophy of Logic, vol. 2 (1981), pp. 87–112, DOI: https://doi.org/10.1080/01445348108837023 translated from the Russian by Merrie Bergmann. DOI: https://doi.org/10.1080/01445348108837023

J. Ciuciura, A quasi-discursive system ND₂⁺, Notre Dame Journal of Formal Logic, vol. 47(3) (2006), pp. 371–384, DOI: https://doi.org/10.1305/ndjfl/1163775444 DOI: https://doi.org/10.1305/ndjfl/1163775444

J. Ciuciura, Frontiers of the discursive logic, Bulletin of the Section of Logic, vol. 37(2) (2008), pp. 81–92.

J. Ciuciura, Non-adjunctive discursive logic, Bulletin of the Section of Logic, vol. 42(3-4) (2013), pp. 169–181, DOI: https://doi.org/10.1007/bf03322857 DOI: https://doi.org/10.1007/BF03322857

N. da Costa, L. Dubikajtis, On Jaśkowski’s Discussive Logic, [in:] A. I. Arruda, N. C. A. da Costa, R. Chuaqui (eds.), Non-Classical Logics, Model Theory, And Computability, vol. 89 of Studies in Logic and the Foundations of Mathematics, Elsevier (1977), pp. 37–56, DOI: https://doi.org/10.1016/S0049-237X(08)70644-X DOI: https://doi.org/10.1016/S0049-237X(08)70644-X

R. Ertola, F. Esteva, T. Flaminio, L. Godo, C. Noguera, Paraconsistency properties in degree-preserving fuzzy logics, Soft Computing, vol. 19(3) (2015), pp. 531–546, DOI: https://doi.org/10.1007/s00500-014-1489-0 DOI: https://doi.org/10.1007/s00500-014-1489-0

T. Furmanowski, Remarks on discussive propositional calculus, Studia Logica, vol. 34(1) (1975), pp. 39–43, DOI: https://doi.org/10.1007/BF02314422 DOI: https://doi.org/10.1007/BF02314422

J. W. Garson, Modal logic for philosophers, Cambridge University Press, Cambridge (2006), DOI: https://doi.org/10.1017/CBO9780511617737 DOI: https://doi.org/10.1017/CBO9780511617737

K. Gödel, Zum Intuitionistischen Aussagenkalkül, Anzeiger der Akademie der Wissenschaften in Wien, vol. 69 (1932), pp. 65–66.

O. Grigoriev, M. Nasieniewski, K. Mruczek-Nasieniewska, Y. Petrukhin, V. Shangin, Axiomatizing a minimal discussive logic, Studia Logica, vol. 111(5) (2023), pp. 855–895, DOI: https://doi.org/10.1007/s11225-023-10042-9 DOI: https://doi.org/10.1007/s11225-023-10042-9

A. Heyting, Die formalen Regeln der intuitionistischen Logik, Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin (1930).

S. Jaśkowski, Recherches sur le système de la logique intuitioniste, [in:] Actes du Congrès International de Philosophie Scientifique, VI Philosophie des mathématiques, Hermann & Cie, Paris (1936), pp. 58–61, (in: Actualités scientifiques et industrielles, vol. 393).

S. Jaśkowski, On the discussive conjunction in the propositional calculus for inconsistent deductive systems, Logic and Logical Philosophy, vol. 7 (1999), pp. 57–59, DOI: https://doi.org/10.12775/llp.1999.004 translated by Jerzy Perzanowski from the Polish O koniunkcji dyskusyjnej w rachunku zdań dla systemów dedukcyjnych sprzecznych, Studia Societatis Scientiarum Torunensis. Sectio A, Mathematica-Physica, vol. 1(8) (1949), no. 8, 171–172.

S. Jaśkowski, A propositional calculus for inconsistent deductive systems, Logic and Logical Philosophy, vol. 7 (1999), pp. 35–56, DOI: https://doi.org/10.12775/llp.1999.003 translated by Jerzy Perzanowski from the Polish Rachunek zdań dla systemow dedukcyjnych sprzecznych, Studia Societatis Scientiarum Torunensis. Sectio A, Mathematica-Physica, vol. 1(5) (1948), pp. 57–77; another English translation, by Olgierd Wojtasiewicz: Propositional calculus for contradictory deductive systems, Studia Logica, vol. 24 (1969), pp. 143–157.

S. C. Kleene, An addendum, The Journal of Symbolic Logic, vol. 28 (1963), pp. 154–156, DOI: https://doi.org/10.2307/2271596 DOI: https://doi.org/10.2307/2271596

A. Kolmogoroff, Zur Deutung der intuitionistischen Logik, Mathematische Zeitschrift, vol. 35(1) (1932), pp. 58–65, DOI: https://doi.org/10.1007/BF01186549 DOI: https://doi.org/10.1007/BF01186549

J. Kotas, The axiomatization of S. Jaśkowski’s discussive system, Studia Logica, vol. 33 (1974), pp. 195–200, DOI: https://doi.org/10.1007/BF02120494 DOI: https://doi.org/10.1007/BF02120494

J. Kotas, N. C. A. da Costa, On some Modal Logical Systems Defined in Connexion with Jaśakowski’s Problem, [in:] A. I. Arruda, N. C. A. da Costa, R. Chuaqui (eds.), Non-Classical Logics, Model Theory, And Computability, vol. 89 of Studies in Logic and the Foundations of Mathematics, Elsevier (1977), pp. 57–73, DOI: https://doi.org/10.1016/S0049-237X(08)70645-1 DOI: https://doi.org/10.1016/S0049-237X(08)70645-1

S. Kovač, In what sense is Kantian principle of contradiction non-classical?, Logic and Logical Philosophy, vol. 17(3) (2008), pp. 251–274, DOI: https://doi.org/10.12775/LLP.2008.013 DOI: https://doi.org/10.12775/LLP.2008.013

A. Loparić, N. C. A. da Costa, Paraconsistency, paracompleteness, and valuations, Logique et Analyse. Nouvelle Série, vol. 27(106) (1984), pp. 119–131, URL: https://www.jstor.org/stable/44084079

A. Loparić, N. C. A. da Costa, Paraconsistency, paracompleteness, and induction, Logique et Analyse. Nouvelle Série, vol. 29(113) (1986), pp. 73–80.

J. Łukasiewicz, On three-valued logic (in Polish), Ruch Filozoficzny, vol. 5 (1920), pp. 170–171, English translation: J. Łukasiewicz: Selected Works, L. Borkowski (ed.), Amsterdam, North-Holland Publishing Company (1970), pp. 87–88.

K. Mruczek-Nasieniewska, M. Nasieniewski, A. Pietruszczak, A modal extension of Jaśkowski’s discussive logic D₂, Logic Journal of the IGPL, vol. 27(4) (2019), pp. 451–477, DOI: https://doi.org/10.1093/jigpal/jzz014 DOI: https://doi.org/10.1093/jigpal/jzz014

M. Nasieniewski, A. Pietruszczak, A method of generating modal logics defining Jaśkowski’s discussive logic D₂, Studia Logica, vol. 97(1) (2011), pp. 161–182, DOI: https://doi.org/10.1007/s11225-010-9302-2 DOI: https://doi.org/10.1007/s11225-010-9302-2

M. Nasieniewski, A. Pietruszczak, On the weakest modal logics defining Jaśkowski’s logic D₂ and the D₂-consequence, Bulletin of the Section of Logic, vol. 41(3–4) (2012), pp. 215–232.

M. Nasieniewski, A. Pietruszczak, Axiomatisations of minimal modal logics defining Jaśkowski-like discussive logics, [in:] A. Indrzejczak, J. Kaczmarek, M. Zawidzki (eds.), Trends in Logic XIII, Wydawnictwo Uniwersytetu Łódzkiego, Łódź (2014), pp. 149–163. DOI: https://doi.org/10.1007/978-81-322-2719-9_9

D. Nelson, Constructible falsity, The Journal of Symbolic Logic, vol. 14 (1949), pp. 16–26, DOI: https://doi.org/10.2307/2268973 DOI: https://doi.org/10.2307/2268973

H. Omori, Sette’s logics, revisited, [in:] Logic, rationality, and interaction, vol. 10455 of Lecture Notes in Computer Science, Springer, Berlin (2017), pp. 451–465, DOI: https://doi.org/10.1007/978-3-662-55665-8 DOI: https://doi.org/10.1007/978-3-662-55665-8_31

H. Omori, J. Alama, Axiomatizing Jaśkowski’s discussive logic ${bf D_2}$, Studia Logica, vol. 106(6) (2018), pp. 1163–1180, DOI: https://doi.org/10.1007/s11225-017-9780-6 DOI: https://doi.org/10.1007/s11225-017-9780-6

J. Perzanowski, On M-fragments and L-fragments of normal modal propositional logics, Reports on Mathematical Logic, vol. 5 (1975), pp. 63–72.

Y. Petrukhin, Generalized correspondence analysis for three-valued logics, Logica Universalis, vol. 12(3-4) (2018), pp. 423–460, DOI: https://doi.org/10.1007/s11787-018-0212-9 DOI: https://doi.org/10.1007/s11787-018-0212-9

A. Pietz, U. Rivieccio, Nothing but the truth, Journal of Philosophical Logic, vol. 42(1) (2013), pp. 125–135, DOI: https://doi.org/10.1007/s10992-011-9215-1 DOI: https://doi.org/10.1007/s10992-011-9215-1

V. M. Popov, Between the logic Par and the set of all formulae, [in:] The Proceeding of the 6th Smirnov Readings in Logic, Contemporary Notebooks, Moscow (2009), pp. 93–95, (in Russian).

V. M. Popov, Between Int<ω,ω> and intuitionistic propositional logic, Logical investigations, vol. 19 (2013), pp. 197–199, URL: https://iphras.ru/uplfile/logic/log19/LI19_Popov.pdf DOI: https://doi.org/10.21146/2074-1472-2013-19-0-197-199

J. B. Rosser, Logic for mathematicians, 2nd ed., Chelsea Publishing Co., New York (1978).

J. B. Rosser, A. R. Turquette, Many-valued logics, Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam (1951).

A. M. Sette, On the propositional calculus P¹, Mathematica Japonica, vol. 18 (1973), pp. 173–180.

A. M. Sette, W. A. Carnielli, Maximal weakly-intuitionistic logics, Studia Logica, vol. 55(1) (1995), pp. 181–203, DOI: https://doi.org/10.1007/BF01053037 DOI: https://doi.org/10.1007/BF01053037

D. J. Shoesmith, T. J. Smiley, Multiple-conclusion logic, Cambridge University Press, Cambridge-New York (1978), DOI: https://doi.org/10.1017/CBO9780511565687 DOI: https://doi.org/10.1017/CBO9780511565687

J. Słupecki, G. Bryll, T. Prucnal, Some remarks on three-valued logic of J. Łukasiewicz, Studia Logica, vol. 21 (1967), pp. 45–70, DOI: https://doi.org/10.1007/BF02123418 DOI: https://doi.org/10.1007/BF02123418

B. C. van Fraassen, Singular Terms, Truth-Value Gaps, and Free Logic, The Journal of Philosophy, vol. 63(17) (1966), pp. 481–495, URL: http://www.jstor.org/stable/2024549 DOI: https://doi.org/10.2307/2024549

V. L. Vasyukov, A new axiomatization of Jaśkowski’s discussive logic, Logic and Logical Philosophy, vol. 9 (2001), pp. 35–46, DOI: https://doi.org/10.12775/llp.2001.003 DOI: https://doi.org/10.12775/LLP.2001.003

H. Wansing, Negation, [in:] L. Goble (ed.), The Blackwell Guide to Philosophical Logic, Blackwell Philosophy Guides, Blackwell, Oxford (2001), pp. 415–436, DOI: https://doi.org/10.1002/9781405164801.ch18 DOI: https://doi.org/10.1111/b.9780631206934.2001.00021.x