Studia Logica

, Volume 101, Issue 4, pp 713–747 | Cite as

The Proof by Cases Property and its Variants in Structural Consequence Relations

  • Petr Cintula
  • Carles NogueraEmail author


This paper is a contribution to the study of the rôle of disjunction in Abstract Algebraic Logic. Several kinds of (generalized) disjunctions, usually defined using a suitable variant of the proof by cases property, were introduced and extensively studied in the literature mainly in the context of finitary logics. The goals of this paper are to extend these results to all logics, to systematize the multitude of notions of disjunction (both those already considered in the literature and those introduced in this paper), and to show several interesting applications allowed by the presence of a suitable disjunction in a given logic.


Abstract Algebraic Logic Generalized disjunction Proof by cases properties Consequence relations Filter-distributive logics Protoalgebraic logics 


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  1. 1.
    Baaz, Matthias, Infinite-valued Gödel logic with 0-1-projections and relativisations, in Petr Hájek, (ed.), Gödel'96: Logical Foundations of Mathematics, Computer Science, and Physics, vol. 6 of Lecture Notes in Logic, Springer-Verlag, Brno, 1996, pp. 23–33.Google Scholar
  2. 2.
    Blok, Willem J., and Don L. Pigozzi, Algebraizable Logics, vol. 396 of Memoirs of the American Mathematical Society, American Mathematical Society, Providence, RI, 1989. Freely downloadable from
  3. 3.
    Blok, Willem J., and Don L. Pigozzi, Local deduction theorems in algebraic logic, in Algebraic logic (Budapest, 1988), vol. 54 of Colloq. Math. Soc. János Bolyai, North-Holland, Amsterdam, 1991, pp. 75–109.Google Scholar
  4. 4.
    Cintula Petr, Noguera Carles: Implicational (semilinear) logics I: A new hierarchy. Archive for Mathematical Logic 49(4), 417–446 (2010)CrossRefGoogle Scholar
  5. 5.
    Czelakowski Janusz: Logical matrices, primitive satisfaction and finitely based logics. Studia Logica 42(1), 89–104 (1983)CrossRefGoogle Scholar
  6. 6.
    Czelakowski, Janusz, Remarks on finitely based logics, in Proceedings of the Logic Colloquium 1983. Vol. 1. Models and Sets, vol. 1103 of Lecture Notes in Mathematics, Springer, Berlin, 1984, pp. 147–168.Google Scholar
  7. 7.
    Czelakowski Janusz: Local deductions theorems. Studia Logica 45(4), 377–391 (1986)CrossRefGoogle Scholar
  8. 8.
    Czelakowski Janusz: Protoalgebraic Logics, vol. 10 of Trends in Logic. Kluwer, Dordrecht (2001)CrossRefGoogle Scholar
  9. 9.
    Diego Antonio: Sur les algèbres de Hilbert. Collection de Logique Mathématique. Ser. A. (Ed Hermann) 21, 1–52 (1966)Google Scholar
  10. 10.
    Dummett Michael: A propositional calculus with denumerable matrix. Journal of Symbolic Logic 24(2), 97–106 (1959)CrossRefGoogle Scholar
  11. 11.
    Dummett Michael: The Logical Basis of Metaphysics. Harvard University Press, Cambridge, MA (1991)Google Scholar
  12. 12.
    Dzik Wojciech: On the content of lattices of logics part 1: The representation theorem for lattices of logics. Reports on Mathematical Logic 13, 17–28 (1981)Google Scholar
  13. 13.
    Font, Josep Maria, and Ramon Jansana, A General Algebraic Semantics for Sentential Logics, vol. 7 of Lecture Notes in Logic, 2 edn., Association for Symbolic Logic, Ithaca, NY, 2009. Freely downloadable from
  14. 14.
    Font Josep Maria, Jansana Ramon, Pigozzi Don L.: A Survey of Abstract Algebraic Logic. Studia Logica Special Issue on Abstract Algebraic Logic II 74(1– 2), 13–97 (2003)Google Scholar
  15. 15.
    Font Josep Maria, Jansana Ramon, Pigozzi Don L.: Update to “A Survey of Abstract Algebraic Logic". Studia Logica 91(1), 125–130 (2009)CrossRefGoogle Scholar
  16. 16.
    Font Josep Maria, Verdú Ventura: Algebraic logic for classical conjunction and disjunction. Studia Logica 50(3–4), 391–419 (1991)CrossRefGoogle Scholar
  17. 17.
    Galatos Nikolaos: Equational bases for joins of residuated-lattice varieties. Studia Logica 76(2), 227–240 (2004)CrossRefGoogle Scholar
  18. 18.
    Galatos, Nikolaos, Peter Jipsen, Tomasz Kowalski, and Hiroakira Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, vol. 151 of Studies in Logic and the Foundations of Mathematics, Elsevier, Amsterdam, 2007.Google Scholar
  19. 19.
    Hájek Petr: Metamathematics of Fuzzy Logic, vol. 4 of Trends in Logic. Kluwer, Dordrecht (1998)CrossRefGoogle Scholar
  20. 20.
    Hart James B., Rafter Lori, Tsinakis Constantine: The structure of commutative residuated lattices. International Journal of Algebra and Computation 12, 509–524 (2002)CrossRefGoogle Scholar
  21. 21.
    Jansana, Ramon, Selfextensional logics with implication, in Jean–Yves Béziau, (ed.),Logica Universalis, Birkhäuser, Basel, 2005, pp. 65–88.Google Scholar
  22. 22.
    Jansana Ramon: Selfextensional logics with conjunction. Studia Logica 84(1), 63–104 (2006)CrossRefGoogle Scholar
  23. 23.
    Łukasiewicz Jan, Tarski Alfred: Untersuchungen über den Aussagenkalkül. Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, cl. III 23(iii), 30–50 (1930)Google Scholar
  24. 24.
    McKinsey J.C.C.: Proof of the independence of the primitive symbols of Heyting’s calculus of propositions. Journal of Symbolic Logic 4(4), 155–158 (1939)CrossRefGoogle Scholar
  25. 25.
    Sato Kentaro: Proper semantics for substructural logics, from a stalker theoretic point of view. Studia Logica 88(2), 295–324 (2008)CrossRefGoogle Scholar
  26. 26.
    Torrens, Antoni, and Ventura Verdú, Distributivity and irreducibility in closure systems, Tech. rep., Faculty of Mathematics, University of Barcelona, Barcelona, 1982.Google Scholar
  27. 27.
    Verdú Ventura: Lògiques distributives i booleanes. Stochastica 3, 97–108 (1979)Google Scholar
  28. 28.
    Wang San-Min, Cintula Petr: Logics with disjunction and proof by cases. Archive for Mathematical Logic 47(5), 435–446 (2008)CrossRefGoogle Scholar

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© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Institute of Computer Science Academy of Sciences of the Czech RepublicPragueCzech Republic
  2. 2.Faculty of Mathematics and GeoinformaticsVienna University of TechnologyViennaAustria
  3. 3.Artificial Intelligence Research Institute, IIIASpanish National Research Council, CSIC Campus de la Universitat Autònoma de BarcelonaBellaterraSpain
  4. 4.Institute of Information Theory and Automation Academy of Sciences of the Czech RepublicPrague 8Czech Republic

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