Logica Universalis

, Volume 11, Issue 3, pp 345–382 | Cite as

Sequent Systems for Negative Modalities

  • Ori LahavEmail author
  • João Marcos
  • Yoni Zohar
Open Access


Non-classical negations may fail to be contradictory-forming operators in more than one way, and they often fail also to respect fundamental meta-logical properties such as the replacement property. Such drawbacks are witnessed by intricate semantics and proof systems, whose philosophical interpretations and computational properties are found wanting. In this paper we investigate congruential non-classical negations that live inside very natural systems of normal modal logics over complete distributive lattices; these logics are further enriched by adjustment connectives that may be used for handling reasoning under uncertainty caused by inconsistency or undeterminedness. Using such straightforward semantics, we study the classes of frames characterized by seriality, reflexivity, functionality, symmetry, transitivity, and some combinations thereof, and discuss what they reveal about sub-classical properties of negation. To the logics thereby characterized we apply a general mechanism that allows one to endow them with analytic ordinary sequent systems, most of which are even cut-free. We also investigate the exact circumstances that allow for classical negation to be explicitly defined inside our logics.


Negative modalities sequent systems cut-admissibility Analyticity 

Mathematics Subject Classification

03B53 03B60 03B45 03F05 



Open access funding provided by Max Planck Society. The authors acknowledge partial support by CNPq, by The Israel Science Foundation (Grant No. 817-15), by the Marie Curie project GeTFun (PIRSES-GA-2012-318986) funded by EU-FP7, and by the Humboldt Foundation. They also take the chance to thank Elaine Pimentel, Heinrich Wansing, Alessandra Palmigiano, Dorota Leszczyńska-Jasion, and two anonymous referees for their comments on an earlier incarnation of this manuscript.


  1. 1.
    Avron, A., Ciabattoni, A., Zamansky, A.: Canonical calculi: invertibility, axiom expansion and (non)-determinism. In: Proceedings of the Fourth International Computer Science Symposium in Russia on Computer Science—Theory and Applications, CSR’09, pp. 26–37. Springer, Berlin (2009)Google Scholar
  2. 2.
    Avron, A., Konikowska, B., Zamansky, A.: Efficient reasoning with inconsistent information using C-systems. Inf. Sci. 296, 219–236 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Avron, A., Zamansky, A.: A paraconsistent view on \(B\) and \(S5\). In: Beklemishev, L., Demri, S., Máté, A. (eds.) Advances in Modal Logic, vol. 11, pp. 21–37. College Publications, London (2016)Google Scholar
  4. 4.
    Béziau, J.-Y.: Paraconsistent logic from a modal viewpoint. J. Appl. Log. 3, 7–14 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Carnielli, W.A., Marcos, J.: A taxonomy of C-systems. In: Carnielli, W.A., Coniglio, M.E., D’Ottaviano, I.M.L. (eds.) Paraconsistency: The Logical Way to the Inconsistent, Vol. 228 of Lecture Notes in Pure and Applied Mathematics, pp. 1–94. Marcel Dekker, New York (2002)CrossRefGoogle Scholar
  6. 6.
    Celani, S., Jansana, R.: Priestley duality, a Sahlqvist theorem and a Goldblatt–Thomason theorem for positive modal logic. Log. J. IGPL 7(6), 683–715 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    da Costa, N.C.A.: Calculs propositionnels pour les systèmes formels inconsistants. Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, Séries A-B 257, 3790–3793 (1963)zbMATHGoogle Scholar
  8. 8.
    Dodó, A., Marcos, J.: Negative modalities, consistency and determinedness. Electron. Notes Theor. Comput. Sci. 300, 21–45 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Došen, K.: Negative modal operators in intuitionistic logic. Publications de L’Institut Mathématique (Beograd) (N.S.) 35(49), 3–14 (1984)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Dunn, J.M.: Positive modal logic. Studia Log. 55(2), 301–317 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dunn, J.M., Zhou, C.: Negation in the context of Gaggle Theory. Studia Log. 80(2/3), 235–264 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fitting, M.: Proof Methods for Modal and Intuitionistic Logics. Springer, Berlin (1983)CrossRefzbMATHGoogle Scholar
  13. 13.
    Gentzen, G.: Untersuchungen über das logische Schließen I, 1934 An English translation appears in ‘The Collected Works of Gerhard Gentzen, edited by E. Szabo, North-Holland (1969)Google Scholar
  14. 14.
    Greco, G., Ma, M., Palmigiano, A., Tzimoulis, A., Zhao, Z.: Unified correspondence as a proof-theoretic tool. J. Log. Comput. exw022 (2016). doi: 10.1093/logcom/exw022
  15. 15.
    Hartonas, C.: Modal and temporal extensions of non-distributive propositional logics. Log. J. IGPL 24, 156–185 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Humberstone, L.: Contrariety and subcontrariety: the anatomy of negation (with special reference to an example of J.-Y. Béziau). Theoria 71(3), 241–262 (2005)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Humberstone, L.: Sentence connectives in formal logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy, 2015th edn. Fall, Cambridge (2015)Google Scholar
  18. 18.
    Indrzejczak, A.: Natural Deduction, Hybrid Systems and Modal Logics. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  19. 19.
    Jaśkowski, S.: A propositional calculus for inconsistent deductive systems (in Polish). Studia Societatis Scientiarum Torunensis, Sectio A, 5, 57–77 (1948). Translated into English in Studia Logica, 24, 143–157 (1967), and in Logic and Logical Philosophy 7, 35–56 (1999)Google Scholar
  20. 20.
    Kawai, H.: Sequential calculus for a first order infinitary temporal logic. Math. Log. Q. 33(5), 423–432 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lahav, O., Avron, A.: A unified semantic framework for fully structural propositional sequent systems. ACM Trans. Comput. Log. 14(4), 27:1–27:33 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lahav, O., Marcos, J., Zohar, Y.: It ain’t necessarily so: basic sequent systems for negative modalities. In: Beklemishev, L., Demri, S., Máté, A. (eds.) Advances in Modal Logic, vol. 11, pp. 449–468. College Publications, London (2016)Google Scholar
  23. 23.
    Lahav, O., Zohar, Y.: SAT-based decision procedure for analytic pure sequent calculi. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) Automated Reasoning, Volume 8562 of Lecture Notes in Computer Science, pp. 76–90. Springer, Berlin (2014)Google Scholar
  24. 24.
    Lavendhomme, R., Lucas, T.: Sequent calculi and decision procedures for weak modal systems. Studia Log. 66(1), 121–145 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Marcos, J.: Modality and paraconsistency. In: Bilkova, M., Behounek, L. (eds.) The Logica Yearbook 2004, pp. 213–222. Filosofia, Vaudreuil-Dorion (2005)Google Scholar
  26. 26.
    Marcos, J.: Nearly every normal modal logic is paranormal. Logique et Analyse (N.S.) 48(189/192), 279–300 (2005)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Marcos, J.: On negation: pure local rules. J. Appl. Log. 3(1), 185–219 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Nelson, D.: Negation and separation of concepts in constructive systems. In: Heyting, A. (ed.) Constructivity in Mathematics, Studies in Logic and the Foundations of Mathematics, pp. 208–225. North-Holland, Amsterdam (1959)Google Scholar
  29. 29.
    Onishi, T.: Substructural negations. Australas. J. Log. 12(4), 177–203 (2015) Article no. 1Google Scholar
  30. 30.
    Poggiolesi, F.: Gentzen Calculi for Modal Propositional Logic. Springer, Berlin (2010)zbMATHGoogle Scholar
  31. 31.
    Prior, A.: Past, Present and Future. Oxford University Press, Oxford (1967)CrossRefzbMATHGoogle Scholar
  32. 32.
    Rauszer, C.: An algebraic and Kripke-style approach to a certain extension of intuitionistic logic. Dissertationes Mathematicae, vol. CLXVII. Instytut Matematyczny Polskiej Akademii Nauk, Warsaw (1980)Google Scholar
  33. 33.
    Restall, G.: Combining possibilities and negations. Studia Log. 59(1), 121–141 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Ripley, D.W.: Negation in Natural Language. Ph.D. thesis, University of North Carolina at Chapel Hill (2009)Google Scholar
  35. 35.
    Schütte, K.: Beweistheorie. Springer, Berlin (1960)zbMATHGoogle Scholar
  36. 36.
    Segerberg, K.: An Essay in Classical Modal Logic. Uppsala University, Uppsala (1971)zbMATHGoogle Scholar
  37. 37.
    Takano, M.: A modified subformula property for the modal logics \(K5\) and \(K5D\). Bull. Sect. Log. 30(2), 115–123 (2001)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Vakarelov, D.: Consistency, completeness and negation. In: Priest, G., Sylvan, R., Norman, J. (eds.) Paraconsistent Logic: Essays on the Inconsistent, pp. 328–363. Philosophia Verlag, Munchen (1989)Google Scholar
  39. 39.
    Wansing, H.: Sequent systems for modal logics. In: Gabbay, D.M., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 8, 2nd edn, pp. 61–145. Springer, Berlin (2002)CrossRefGoogle Scholar
  40. 40.
    Wójcicki, R.: Theory of Logical Calculi. Kluwer, Dordrecht (1988)CrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Max Planck Institute for Software Systems (MPI-SWS)KaiserslauternGermany
  2. 2.Federal University of Rio Grande do NorteNatalBrazil
  3. 3.Tel Aviv UniversityTel AvivIsrael

Personalised recommendations