Jaśkowski’s Discovery and Various Ways to Master Contradictions
DOI:
https://doi.org/10.12775/LLP.2025.023Słowa kluczowe
ex falso/contradictione quodlibet, explosion, paraconistency, paraconsistent logic, American-Australian school, Canadian school, Belgian school, Brazilian school, Torunian School of Paraconsistent Logic, TSPLAbstrakt
This paper presents the history of the world’s first school of paraconsistency: the Torunian School of Paraconsistent Logic. Its founding father is the Polish logician Stanisław Jaśkowski, who first formulated a system of paraconsistent logic. Both the approach he presented and the work of subsequent generations of logicians allow us to speak of the entire school. In this article, we present the characteristics of this school, comparing them with those of other national, but later, schools of paraconsistency. In addition to extensive factual information concerning Jaskowski’s discovery and subsequent development of this topic, we also refer to and comment on the latest works presented in this volume.
Bibliografia
Anderson, Alan Ross, and Nuel D. Belnap, Jr., 1959, “Modalities in Ackermann’s ‘rigorous implication’”, The Journal of Symbolic Logic, 24 (2): 107–111. DOI: Crossref
Asenjo, Florencio G., 1966, “A calculus of antinomies”, Notre Dame Journal of Formal Logic, 7 (1): 103–105. DOI: Crossref
Asenjo, Florencio G., and J. Tamburino, 1975, “Logic of antinomies”, Notre Dame Journal of Formal Logic, 16 (1): 17–44. DOI: Crossref
Bartolo Alegre, Luis Felipe, 2025, “Can we test inconsistent empirical theories?”, Logic and Logical Philosophy, 34 (4): 501–545. DOI: Crossref
Batens, Diderik, 1980, “Paraconsistent extensional propositional logics”, Logique et Analyse, 23 (90–91): 195–234.
Batens, Diderik, 1986, “Dialectical dynamics within formal logics”, Logique et Analyse, 114: 161–173.
Batens, Diderik, 1989, “Dynamic dialectical logics”, pages 187–217 in Graham Priest, Richard Routley, and Jean Norman (eds), Paraconsistent Logic. Essays on the Inconsistent, Philosophia Verlag, München.
Batens, Diderik, 1995, “Blocks. The clue to dynamic aspects of logic”, Logique et Analyse, 150–152: 285–328.
Batens, Diderik, 1997, “Inconsistencies and beyond. A logical-philosophical discussion”, Revue Internationale de Philosophie, 200: 259–273.
Batens, Diderik, 1998, “A dynamic semantics for inconsistency-adaptive logics”, Bulletin of the Section of Logic, 27: 15–18.
Batens, Diderik, 1999, “Zero logic adding up to classical logic”, Logical Studies, 2. Link
Batens, Diderik, 2000a, “A survey of inconsistency-adaptive logics”, pages 49–73 in Diderik Batens, Chris Mortensen, Graham Priest, and Jean Paul Van Bendegem (eds.), Frontiers of Paraconsistent Logic, Research Studies Press, Baldock.
Batens, Diderik, 2000b, “Towards the unification of inconsistency handling mechanisms”, Logic and Logical Philosophy, 8: 5–31. DOI: Crossref
Batens, Diderik, 2003, “Criteria causing inconsistencies. General gluts as opposed to negation gluts”, Logic and Logical Philosophy, 11/12: 5–37. DOI: Crossref
Batens, Diderik, and Joke Meheus, 2000, “A tableau method for inconsistency-adaptive logics”, pages 127–142 in Roy Dyckhoff (ed.), Automated Reasoning with Analytic Tableaux and Related Methods, Lecture Notes in Artificial Intelligence, vol. 1847, Springer, Berlin, Heidelberg. DOI: Crossref
Belnap, Nuel D., 1960, “Entailment and relevance”, The Journal of Symbolic Logic, 25 (2): 144–146. DOI: Crossref
Belnap, Nuel D., 1976, “How a computer should think”, pages 30–56 in Ryle Gilbert (ed.), Contemporary Aspects of Philosophy, Oriel Press Ltd., Stocksfield, London, Henley-on-Thames and Boston.
Belnap, Nuel D., 1977a, “A useful four-valued logic”, pages 5–37 in J. M. Dunn and G. Epstein (eds), Modern Uses of Multiple-Valued Logic, Episteme, vol. 2., Springer, Dordrecht. DOI: Crossref
Béziau, Jean-Yves, 2005, “Paraconsistent logic from a modal viewpoint”, Journal of Applied Logic, 3 (1): 7–14. DOI: Crossref
Béziau, Jean-Yves, 2006, “The paraconsistent logic Z”, Logic and Logical Philosophy, 15: 99–111. DOI: Crossref
Beziau, Jean-Yves, 2025, “Is p and ¬p a contradiction?”, Logic and Logical Philosophy, 34 (4): 547–560. DOI: Crossref
Błaszczuk, Jerzy J., 1984, “Some paraconsistent sentential calculi”, Studia Logica, 43: 51–61. DOI: Crossref
Błaszczuk, Jerzy J., and Wiesław Dziobiak, 1975, “Remarks on Perzanowski’s modal system”, Bulletin of the Section of Logic, 4 (2): 57–64.
Błaszczuk, Jerzy J., and Wiesław Dziobiak, 1976, “An axiomatization of M n-counterparts for some modal calculi”, Reports on Mathematical Logic, 6: 3–6.
Błaszczuk, Jerzy J., and Wiesław Dziobiak, 1977, “Modal logics connected with systems S4n of Sobociński”, Studia Logica, 36: 151–175. DOI: Crossref
Carnielli, Walter, Marcelo E. Coniglio, and João Marcos, 2007, “Logics of formal inconsistency”, pages 1–93 in D. M. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Springer Netherlands, Dordrecht. DOI: Crossref
Ciuciura, Janusz, 2005, “On the da Costa, Dubikajtis and Kotas’ system of the discursive logic, D∗2”, Logic and Logical Philosophy, 14: 235–252. DOI: Crossref
Ciuciura, Janusz, 2006, “A quasi-discursive system ND+2 ”, Notre Dame Journal of Formal Logic, 47 (3): 371–384. DOI: Crossref
Ciuciura, Janusz, 2008a, “Frontiers of the discursive logic”, Bulletin of the Section of Logic 37, 2: 81–92.
Ciuciura, Janusz, 2008b, “Negations in the adjunctive discursive logic”, Bulletin of the Section of Logic, 37 (3–4): 143–160.
Ciuciura, Janusz, 2025, “Literal and controllable paraconsistency”, Logic and Logical Philosophy, 34 (4): 561–579. DOI: Crossref
da Costa, Newton C. A., 1963, “Calculs propositionnels pour les systems formels inconsistants”, Comptes Rendus Helodomadaires des Seances de l’Academie des Sciences Paris, Series A, 257: 3790–3792.
da Costa, Newton C. A., 1974, “On the theory of inconsistent formal systems”, Notre Dame Journal of Formal Logic, 15 (4): 497–510. DOI: Crossref
da Costa, Newton C. A., 1975, “Remarks on Jaśkowski’s discussive logic”, Reports on Mathematical Logic, 4: 7–16.
da Costa, Newton C. A., and Ayda I. Arruda, 1963, “O paradoxo de Curry–Moh Shaw-Kwei” Boletim da Sociedade de Matemática de S. Paolo, 18: 83–89.
da Costa, Newton C. A., and Lech Dubikajtis, 1968, “Sur la logique discursive de Jaśkowski”, Bulletin de L’Académie Polonaise des Sciences, 16 (7)): 551–557.
da Costa, Newton C. A., and Lech Dubikajtis, 1977, “On Jaśkowski discussive logic”, pages 37–56 in A. Arruda, N. da Costa, and R. Chuaqui (eds.), Non-Classical Logics, Model Theory and Computability. Proceedings of the Third Latin-American Symposium on Mathematical Logic, Campinas, Brazil, July 11-17, 1976, North-Holland Publishing Company.
de Moraes, Lafayette, 1970, “Sobre a lógica discursiva de Jaśkowski”, Master Thesis, USP.
Dunn, Jon Michael, 1966, “The algebra of intensional logics”, Ph.D. thesis, University of Pittsburgh.
Estrada-González, Luis, Marek Nasieniewski, and Ricardo Arturo Nicolás-Francisco, 2025, “Discussive connexivity” (submitted).
Furmanowski, Tomasz, 1975, “Remarks on discussive propositional calculus”, Studia Logica, 34: 39–43. DOI: Crossref
Grigoriev, Oleg, Marek Nasieniewski, Krystyna Mruczek-Nasieniewska, Yaroslav Petrukhin, and Vasily Shangin, 2023, ‘Axiomatizing a minimal discussive logic”, Studia Logica, 111: 855–895. DOI: Crossref
Jarmużek, Tomasz, and Marek Nasieniewski, 2025, “On discussive-like logics obtained by means of certain types of tableau systems” (submitted).
Jarmużek, Tomasz, Marek Nasieniewski, and Kacper Wąsiak, 2025, “A tableau system for Béziau’s logic Z” (submitted).
Jaśkowski, Stanisław, 1948, “Rachunek zdań dla systemów dedukcyjnych sprzecznych”, Studia Societatis Scientiarum Torunensis, Sect. A, I (5): 57–77.
Jaśkowski, Stanisław, 1949, “O koniunkcji dyskusyjnej w rachunku zdań dla systemów dedukcyjnych sprzecznych”, Studia Societatis Scientiarum Torunensis, Sect. A, I (8): 171–172.
Jaśkowski, Stanisław, 1999a, “A propositional calculus for inconsistent deductive systems”, Logic and Logical Philosophy, 7: 35–56. The second English translation of (Jaśkowski, 1948). DOI: Crossref
Jaśkowski, Stanisław, 1999b, “On the discussive conjunction in the propositional calculus for inconsistent deductive systems”, Logic and Logical Philosophy, 7: 57–59. The English translation of (Jaśkowski, 1949). DOI: Crossref
Kotas, Jerzy, 1971 “On the algebra of classes of formulae of Jaśkowski’s discussive system”, Studia Logica, 27: 81–90. DOI: Crossref
Kotas, Jerzy, 1974a, “The axiomatization of S. Jaśkowski’s discussive system”, Studia Logica, 33 (2): 195–200. DOI: Crossref
Kotas, Jerzy, 1974b, “On quantity of logical values in the discussive D2 system and in modular logic”, Studia Logica, 33: 273–275. DOI: Crossref
Kotas, Jerzy, and Newton C. A. da Costa, 1979, “A new formulation of discussive logic”, Studia Logica, 38: 429–445. DOI: Crossref
Marcos, João, 2005, “Nearly every normal modal logic is paranormal”, Logique et Analyse, 48 (189/192): 279–300.
Meheus, Joke, 1999, “Erotetic arguments from inconsistent premises”, Logique et Analyse, 165–166: 49–80.
Meheus, Joke, 2006, “An adaptive logic based on Jaśkowski’s approach to paraconsistency”, Journal of Philosophical Logic, 35: 539–567. DOI: Crossref
Mruczek-Nasieniewska, Krystyna, and Marek Nasieniewski, 2004, “A modal formulation of some paraconsistent logics. Part II”, presented at International Workshop on Problem Solving in the Sciences: Adaptive and Interrogative Perspectives – 9th VlaPolo, Ghent University.
Mruczek-Nasieniewska, Krystyna, and Marek Nasieniewski, 2005, “Syntactical and semantical characterization of a class of paraconsistent logics”, Bulletin of the Section of Logic, 34 (4): 229–248.
Mruczek-Nasieniewska, Krystyna, and Marek Nasieniewski, 2008, “Paraconsitent logics obtained by J.-Y. Béziau’s method by means of some non-normal modal logics”, Bulletin of the Section of Logic, 37 (3/4): 185–196.
Mruczek-Nasieniewska, Krystyna, and Marek Nasieniewski, 2017, “Logics with impossibility as the negation and regular extensions of the deontic logic D2”, Bulletin of the Section of Logic, 46 (3/4): 263–282. DOI: Crossref
Mruczek-Nasieniewska, Krystyna, and Marek Nasieniewski, 2018, “A characterisation of some-like logics”, Logica Universalis, 12: 207–219. DOI: Crossref
Mruczek-Nasieniewska, Krystyna, and Marek Nasieniewski, 2019, “A Kotas-style characterisation of minimal discussive logic”, Axioms, 8 (4), 108: 1–17. DOI: Crossref
Mruczek-Nasieniewska, Krystyna, and Marek Nasieniewski, 2020, “On correspondence of standard modalities and negative ones on the basis of regular and quasi-regular logics”, Studia Logica, 108: 1087–1123. DOI: Crossref
Mruczek-Nasieniewska, Krystyna, and Marek Nasieniewski, “On a modal discussive logic corresponding to an extended model of discussion”, in Filip Grgić, Kordula Świętorzecka, Anna Brożek (eds.), Logic, Knowledge, and Tradition. Essays in Honor of Srećko Kovač (accepted).
Mruczek-Nasieniewska, Krystyna, Marek Nasieniewski, and Andrzej Pietruszczak, 2019, “A modal extension of Jaśkowski’s discussive logic D2”, Logic Journal of IGPL, 27: 451–477. DOI: Crossref
Mruczek-Nasieniewska, Krystyna, Marek Nasieniewski, and Andrzej Pietruszczak, 2025, “Modal discussive logics”, Journal of Philosophical Logic, 54: 99–142. DOI: Crossref
Nasieniewski, Marek, 2002, “A comparison of two approaches to parainconsistency: Flemish and Polish”, Logic and Logical Philosophy, 9: 47–74. DOI: Crossref
Nasieniewski, Marek, 2003, “A modal formulation of some paraconsistent logics”, presented at International Workshop on Problem Solving in the Sciences: Adaptive and Interrogative Perspectives – 8th VlaPolo (Zielona Góra, Poland).
Nasieniewski, Marek, Ricardo Arturo Nicolás-Francisco, 2025, “On some concepts constituting discussive logic”, Studia z Historii Filozofii (submitted).
Nasieniewski, Marek, and Andrzej Pietruszczak, 2008, “The weakest regular modal logic defining Jaśkowski,’s logic D2”, Bulletin of the Section of Logic, 37 (3/4): 197–210.
Nasieniewski, Marek, and Andrzej Pietruszczak, 2009a, “New axiomatizations of the weakest regular modal logic defining Jaśkowski’s logic D2”, Bulletin of the Section of Logic, 38 (1–2): 45–50.
Nasieniewski, Marek, and Andrzej Pietruszczak, 2009b, “Semantics for regular logics connected with Jaśkowski’s discussive logic D2”, Bulletin of the Section of Logic, 38 (3/4): 173–187.
Nasieniewski, Marek, and Andrzej Pietruszczak, 2011, “A method of generating modal logics defining Jaśkowski’s discussive logic D2”, Studia Logica, 97 (1): 161–182. DOI: Crossref
Nasieniewski, Marek, and Andrzej Pietruszczak, 2012, “On the weakest modal logics defining Jaśkowski’s logic D2 and the D2-consequence”, Bulletin of the Section of Logic, 41 (3/4): 215–232.
Nasieniewski, Marek, and Andrzej Pietruszczak, 2013, “On modal logics defining Jaśkowski’s D2-consequence”, pages 141–161, chapter 9, in K. Tanaka, F. Berto, E. Mares, and F. Paoli (eds.), Paraconsistency: Logic and Applications, series “Logic, Epistemology and the Unity of Science”, Vol. 26, Springer. DOI: Crossref
Nasteniewski, Marek, and Andrzej Pietruszczak, 2014, “Axiomatisations of minimal modal logics Defining Jaśkowski-like Discussive Logics”, pages 149–163 in Andrzej Indrzejczak, Janusz Kaczmarek, and Michał Zawidzki (eds.), Trends in Logic XIII, Wydawnictwo Uniwersytetu Łódzkiego.
Nicolás-Francisco, Ricardo Arturo, 2022, “On the Polish ‘via modalization’ approach to paraconsistency”, Edukacja Filozoficzna Edukacja Filozoficzna, 74: 161–182. DOI: Crossref
Omori, Hitoshi, and Jesse Alama, 2018, “Axiomatizing Jaśkowski’s discussive logic D2”, Studia Logica, 106 (6): 1163–1180. DOI: Crossref
Perzanowski, Jerzy, 1975, “On M-fragments i L-fragments of normal modal propositional logics”, Reports on Mathematical Logic, 5: 63–72.
Priest, Graham, 1979, “The logic of paradox”, Journal of Philosophical Logic, 8 (1): 219–241. DOI: Crossref
Priest, Graham, Koji Tanaka, and Zach Weber, 2025, “Paraconsistent logic”, in Edward N. Zalta and Uri Nodelman (eds.), The Stanford Encyclopedia of Philosophy, Spring 2025 Edition. Link
Routley, Richard, and Robert K. Meyer, 1973, “The semantics of entailment”, pages 199–243 in H. Leblanc (ed.), Truth, Syntax and Modality, Studies in Logic and the Foundations of Mathematics, vol. 68, Elsevier. DOI: Crossref
Schotch, Peter K., and Raymond E. Jennings, 1980, “Inference and necessity”, Journal of Philosophical Logic, 9 (3): 327–340. DOI: Crossref
Schotch, Peter K., and Raymond E. Jennings, 1989, “On detonating”, pages 306–327 in G. Priest, R. Routley, and J. Norman (eds.), Paraconsistent Logic: Essays on the Inconsistent, Philosophia Verlag, München–Hamden–Wien.
Schotch, Peter K., Bryson Brown, and Raymond Jennings (eds.), , 2009, On Preserving: Essays on Preservationism and Paraconsistent Logic, University of Toronto Press.
Trybus, Adam, 2025, “What is the principle of (non-)nontradiction, precisely? The struggle at the dawn of formal logic”, Logic and Logical Philosophy, 34 (4): 581–602. DOI: Crossref
Urchs, Max, 1986, “On two systems of Stanisław Jaśkowski”, The Journal of Non-Classical Logic, 3 (1): 25–32. Link
Urchs, Max, 1993, “Powerful paraconsistent logic”, Acta Universitatis Lodziensis. Folia Philosophica, 9: 109–114.
Urchs, Max, 1995, “Discursive logic towards a logic of rational discourse”, Studia Logica, 54: 231–249. DOI: Crossref
Urchs, Max, 2025, “The beginnings of Toruń logic”, Logic and Logical Philosophy, 34 (4): 603–630. DOI: Crossref
Verdée, Peter, 2009, “Adaptive logics using the minimal abnormality strategy are Π11-complex”, Synthese, 167 (1): 93–104. DOI: Crossref
Walicki, Michał, 2025, “Paradoxes versus contradictions in logic of sentential operators”, Logic and Logical Philosophy, 34 (4): 631–675. DOI: Crossref
Wansing, Heinrich, 2025, “Constructive logic is connexive and contradictory”, Logic and Logical Philosophy, 34 (4): 677–703. DOI: Crossref
Weber, Zach, 2025, “True, untrue, valid, invalid, provable, unprovable”, Logic and Logical Philosophy, 34 (4): 705–733. DOI: Crossref
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Prawa autorskie (c) 2025 Davide Fazio, Tomasz Jarmużek, Jacek Malinowski, Krystyna Mruczek-Nasieniewska

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