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A Method of Generating Modal Logics Defining Jaśkowski’s Discussive D2 Consequence

  • Marek Nasieniewski
  • Andrzej Pietruszczak
Chapter
Part of the Logic, Argumentation & Reasoning book series (LARI, volume 5)

Abstract

Jaśkowski’s logic D 2 is usually understood as a set of discussive formulae. Studying Jaśkowski’s paper one can also find a consequence relation (the D 2 -consequence). The logic D 2 was meant to express this consequence relation. Since the logic D 2 was formulated with the help of a modal logic, the consequence relation is also defined in the modal language. It is known that the logic D 2 can be defined by other modal logics than S5. A similar question arises as regards the consequence relation. In Nasieniewski and Pietruszczak (On modal logics defining Jaśkowski’s D2-consequence. In: Tanaka K, Berto F, Mares E, Paoli F (eds) Paraconsistency: logic and applications. Logic, epistemology and the unity of science, chap 8, vol 26. Springer, Dordrecht/New York, pp 141–161, 2013) there are given modal logics other than S5 which define exactly the same consequence relation. In the present paper we try to develop a more general method of defining modal logics which also allow to define the D 2 -consequence.

Keywords

Jaśkowski’s logic D2 D2-consequence Deducibility in D2 

Notes

Acknowledgements

We would like to thank both anonymous referees for their valuable remarks on an earlier version of this paper.

References

  1. Bull, R. A., & Segerberg, K. (1984). Basic modal logic. In D. Gabbay & F. Guenthner (Eds.), Handbook of philosophical logic (Vol. II, pp. 1–88). Dordrecht: Reidel.CrossRefGoogle Scholar
  2. Chellas, B. F. (1980). Modal logic: An introduction. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  3. Jaśkowski, S. (1948). Rachunek zdań dla systemów dedukcyjnych sprzecznych. Studia Societatis Scientiarum Torunensis, Sect. A, I(5), 57–77. The first English version: Propositional calculus for contradictory deductive systems. Studia Logica, 24, 143–157 (1969).Google Scholar
  4. Jaśkowski, S. (1999a). A propositional calculus for inconsistent deductive systems. Logic and Logical Philosophy, 7, 35–56. The second English version of Rachunek zdań dla systemów dedukcyjnych sprzecznych.Google Scholar
  5. Jaśkowski, S. (1999b). On the discussive conjunction in the propositional calculus for inconsistent deductive systems. Logic and Logical Philosophy, 7, 57–59. The English version of O koniunkcji dyskusyjnej w rachunku zdań dla systemów dedukcyjnych sprzecznych, Studia Societatis Scientiarum Torunensis, Sect. A, Vol. I, no. 8, 171–172 (1949).Google Scholar
  6. Lemmon, E. J., & Scott, D. (1977). “Lemmon Notes”: An introduction to modal logic (American philosophical quarterly monograph series). Oxford: Basil Blackwell.Google Scholar
  7. Nasieniewski, M., & Pietruszczak, A. (2008). The weakest regular modal logic defining Jaśkowski’s logic D2. Bulletin of the Section of Logic, 37(3/4), 197–210.Google Scholar
  8. Nasieniewski, M., & Pietruszczak, A. (2009). New axiomatizations of the weakest regular modal logic defining Jaśkowski’s logic D2. Bulletin of the Section of Logic, 38(1/2), 45–50.Google Scholar
  9. Nasieniewski, M., & Pietruszczak, A. (2011). A method of generating modal logics defining Jaśkowski’s discussive logic D2. Studia Logica, 97(1), 161–182.CrossRefGoogle Scholar
  10. Nasieniewski, M., & Pietruszczak, A. (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.Google Scholar
  11. Nasieniewski, M., & Pietruszczak, A. (2013). On modal logics defining Jaśkowski’s D2-consequence. In K. Tanaka, F. Berto, E. Mares, & F. Paoli (Eds.), Paraconsistency: Logic and applications (Logic, epistemology and the unity of science, chap. 8, Vol. 26, pp. 141–161). Dordrecht/New York: Springer.Google Scholar
  12. Perzanowski, J. (1975). On M-fragments and L-fragments of normal modal propositional logics. Reports on Mathematical Logic, 5, 63–72.Google Scholar
  13. Segerberg, K. (1971). An essay in classical modal logic (Vols. 1 and 2). Uppsala: Department of Philosophy, Uppsala University.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of LogicNicolaus Copernicus UniversityToruńPoland

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