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Disjunctive and Conjunctive Multiple-Conclusion Consequence Relations

  • Marek NowakEmail author
Article

Abstract

Two different kinds of multiple-conclusion consequence relations taken from Shoesmith and Smiley (Multiple-conclusion logic, Cambridge University Press, Cambridge, 1978) and Galatos and Tsinakis (J Symb Logic 74:780–810, 2009) or Nowak (Bull Sect Logic 46:219–232, 2017), called here disjunctive and conjunctive, respectively, defined on a formal language, are considered. They are transferred into a bounded lattice and a complete lattice, respectively. The properties of such abstract consequence relations are presented.

Keywords

Closure operation Closure system Multiple-conclusion consequence relation Galois connection 

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Notes

Acknowledgements

I am very indebted to both anonymous referees for their useful hints changing essentially the first version of the text.

References

  1. 1.
    Blyth, T.S., Lattices and Ordered Algebraic Structures, Springer, New York, 2005.Google Scholar
  2. 2.
    Carnap, R., Formalization of Logic, Harvard University Press, Cambridge, 1943.Google Scholar
  3. 3.
    Czelakowski, J., and G. Malinowski, Key notions of Tarski’s methodology of deductive systems, Studia Logica 44:321–351, 1985.CrossRefGoogle Scholar
  4. 4.
    Denecke, K., M. Erné, and S.L. Wismath (eds.), Galois Connections and Applications, Kluwer, Alphen aan den Rijn, 2004.Google Scholar
  5. 5.
    Domenach, F., and B. Leclerc, Biclosed binary relations and Galois connections, Order 18:89–104, 2001.CrossRefGoogle Scholar
  6. 6.
    Erné, M., J. Koslowski, A. Melton, and G.E. Strecker, A Primer on Galois Connections, Annals of the New York Academy of Sciences 704:103–125, 1993.CrossRefGoogle Scholar
  7. 7.
    Galatos, N., and C. Tsinakis, Equivalence of consequence relations: an order-theoretic and categorical perspective, Journal of Symbolic Logic 74:780–810, 2009.CrossRefGoogle Scholar
  8. 8.
    Gentzen, G.K.E., Untersuchungen uber das logische Schliesen. I, Mathematische Zeitschrift 39:176–210, 1934. [English translation: Investigation into Logical Deduction, in M.E. Szabo, The collected Works of Gerhard Gentzen, North Holland, 1969, pp. 68–131.]Google Scholar
  9. 9.
    Hacking, I., What is logic?, The Journal of Philosophy 76:285–319, 1979.CrossRefGoogle Scholar
  10. 10.
    Kneale, W., The province of logic, in H.P. Lewis (ed.), Contemporary British Philosophy, Allen and Unwin, Crows Nest, 1956, pp. 237–261.Google Scholar
  11. 11.
    Nowak, M., A syntactic approach to closure operation, Bulletin of the Section of Logic 46:219–232, 2017.Google Scholar
  12. 12.
    Scott, D., Completeness and axiomatizability in many-valued logic, in Proceedings of Symposia in Pure Mathematics, vol. 25 (Proceedings of the Tarski Symposium), American Mathematical Society, Providence, 1974, pp. 411–435.Google Scholar
  13. 13.
    Shoesmith, D.J., and T.J. Smiley, Multiple-Conclusion Logic, Cambridge University Press, Cambridge, 1978.CrossRefGoogle Scholar
  14. 14.
    Skura T., and A. Wiśniewski, A system for proper multiple-conclusion entailment, Logic and Logical Philosophy 24:241–253, 2015.Google Scholar
  15. 15.
    Wójcicki, R., Theory of Logical Calculi. Basic Theory of Consequence Operations, Kluwer, Alphen aan den Rijn, 1988.CrossRefGoogle Scholar
  16. 16.
    Zygmunt J., An Essay in Matrix Semantics for Consequence Relations, Wydawnictwo Uniwersytetu Wrocławskiego, Wroclaw, 1984.Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of LogicUniversity of LodzLodzPoland

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