pp 1–21 | Cite as

A Discrete Duality Between Nonmonotonic Consequence Relations and Convex Geometries

  • Johannes MartiEmail author
  • Riccardo Pinosio
Open Access


In this paper we present a duality between nonmonotonic consequence relations and well-founded convex geometries. On one side of the duality we consider nonmonotonic consequence relations satisfying the axioms of an infinitary variant of System P, which is one of the most studied axiomatic systems for nonmonotonic reasoning, conditional logic and belief revision. On the other side of the duality we consider well-founded convex geometries, which are infinite convex geometries that generalize well-founded posets. Since there is a close correspondence between nonmonotonic consequence relations and path independent choice functions one can view our duality as an extension of an existing duality between path independent choice functions and convex geometries that has been developed independently by Koshevoy and by Johnson and Dean.


Convex geometries Antimatroids Nonmonotonic consequence relations Conditional logic Path independent choice functions Duality 



Research partially supported by EPSRC grant EP/N015843/1.


  1. 1.
    Adaricheva, K., Gorbunov, V., Tumanov, V.I.: Join-semidistributive lattices and convex geometries. Adv. Math. 173.1, 1–49 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Adaricheva, K., Nation, J.B.: A class of infinite convex geometries. Electron. J. Comb. 23, 1 (2016)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Adaricheva, K., Nation, J.B.: Convex geometries. In: Grätzer, G., Wehrung, F. (eds.) Lattice Theory: Special Topics and Applications, pp 153–179. Springer (2016)Google Scholar
  4. 4.
    Baltag, A., Smets, S.: Conditional doxastic models: A qualitative approach to dynamic belief revision. Electron. Notes Theor. Comput. Sci. 165, 5–21 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    van Benthem, J., Pacuit, E.: Dynamic logics of evidence-based beliefs. Studia Logica 99.1-3, 61–92 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Board, O.: Dynamic interactive epistemology. Games Econ. Behav. 49.1, 49–80 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Burgess, J.: Quick completeness proofs for some logics of conditionals. Notre Dame J. Formal Logic 22.1, 76–84 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Danilov, V.I., Koshevoy, G.A.: A new characterization of the path independent choice functions. Math. Soc. Sci. 51.2, 238–245 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Danilov, V.I., Koshevoy, G.A., Savaglio, E.: Hyper-relations, choice functions, and orderings of opportunity sets. Social Choice Welfare 45.1, 51–69 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Paul, H.: Edelman: Abstract convexity and meet-distributive lattices. In: Rival, I. (ed.) Combinatorics and Ordered Sets, vol. 57, pp 127–150. Contemporary Mathematics (1986)Google Scholar
  11. 11.
    Edelman, P.H., Jamison, R.E.: The theory of convex geometries. Geometriae Dedicata 19.3, 247–270 (1985)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Edelman, P.H., Saks, M.E.: Combinatorial representation and convex dimension of convex geometries. Order 5.1, 23–32 (1988)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Girard, P.: From onions to broccoli: Generalizing Lewis’ counterfactual logic. J. Appl. Non-Classical Logics 17.2, 213–229 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Girlando, M., et al.: Standard Sequent Calculi for Lewis’ Logics of Counterfactuals. Logics in Artificial Intelligence. In: Michael, L., Kakas, A. (eds.) , pp 272–287. Springer (2016)Google Scholar
  15. 15.
    Grove, A.: Two modellings for theory change. J. Philos. Logic 17.2, 157–170 (1988)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Paul, R.: Halmos: Lectures on Boolean Algebras. Springer (1974)Google Scholar
  17. 17.
    Joseph, Y.: Halpern: Reasoning About Uncertainty. MIT Press (2003)Google Scholar
  18. 18.
    Johnson, M.R., Dean, R.A.: Locally complete path independent choice functions and their lattices. Math. Soc. Sci. 42.1, 53–87 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kashiwabara, K., Nakamura, M., Okamoto, Y.: The affine representation theorem for abstract convex geometries. Comput. Geom. 30.2, 129–144 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Korte, B., Lovász, L.: Homomorphisms and Ramsey properties of antimatroids. Discret. Appl. Math. 15.2, 283–290 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Korte, B., Lovász, L., Schrader, R.: Greedoids. Springer (1991)Google Scholar
  22. 22.
    Koshevoy, G.A.: Choice functions and abstract convex geometries. Math. Soc. Sci. 38.1, 35–44 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kratzer, A.: Partition and revision: The semantics of counterfactuals. J. Philos. Logic 10.2, 201–216 (1981)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Kraus, S., Lehmann, D., Magidor, M.: Nonmonotonic reasoning, preferential models and cumulative logics. Artif. Intell. 44.1–2, 167–207 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lewis, D.: Counterfactuals. Blackwell (1973)Google Scholar
  26. 26.
    Marti, J., Pinosio, R.: A game semantics for system. Studia Logica 104.6, 1119–1144 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Marti, J., Pinosio, R.: Topological semantics for conditionals. In: Punčochár, V., Švarný, P. (eds.) The Logica Yearbook 2013. College Publications (2014)Google Scholar
  28. 28.
    Monjardet, B., Raderanirina, V.: The duality between the anti-exchange closure operators and the path independent choice operators on a finite Set. Math. Soc. Sci. 41.2, 131–150 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Monjardet, B.: A use for frequently rediscovering a concept. Order 1.4, 415–417 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Negri, S., Olivetti, N.: A sequent calculus for preferential conditional logic based on neighbourhood semantics. In: De Nivelle, H. (ed.) Automated Reasoning with Analytic Tableaux and Related Methods, pp 115–134. Springer (2015)Google Scholar
  31. 31.
    Nute, D.: Topics in Conditional Logic. Reidel (1980)Google Scholar
  32. 32.
    Plott, C.R.: Path independence, rationality, and social choice. In: Econometrica, pp. 1075–1091 (1973)Google Scholar
  33. 33.
    Pozzato, G.L.: Conditional and preferential logics: Proof methods and theorem proving, vol. 208. Frontiers in Artificial Intelligence and Applications. IOS Press (2010)Google Scholar
  34. 34.
    Richter, M., Rogers, L.G.: Embedding convex geometries and a bound on convex dimension. Discret. Math. 340.5, 1059–1063 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Schröder, L., Pattinson, D., Hausmann, D.: Optimal tableaux for conditional logics with cautious monotonicity. In: Coelho, H., Studer, R., Wooldridge, M. (eds.) Proceedings ECAI 2010. Frontiers in Artificial Intelligence and Applications, vol. 215, pp 707–712. IOS Press (2010)Google Scholar
  36. 36.
    Robert, C.: Stalnaker: A theory of conditionals. In: Rescher, N. (ed.) Studies in Logical Theory, pp 98–112. Blackwell (1968)Google Scholar
  37. 37.
    Touazi, F, Cayrol, C., Dubois, D.: Possibilistic reasoning with partially ordered beliefs. J. Appl. Logic 13.4, 770–798 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    van de Vel, M.L.J.: Theory of Convex Structures. Elsevier (1993)Google Scholar
  39. 39.
    Veltman, F.: Logics for Conditionals. PhD thesis University of Amsterdam (1985)Google Scholar
  40. 40.
    Veltman, F.: Prejudices, presuppositions, and the theory of counterfactuals. In: Groenendijk, J., Stokhof, M. (eds.) Amsterdam Papers in Formal Grammar, vol. 1, pp 248–282 (1976)Google Scholar
  41. 41.
    Wahl, N.: Antimatroids of finite character. J. Geom. 70.1, 168–175 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Wolter, F.: The algebraic face of minimality. Logic Logical Philos. 6.0, 225–240 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Open Access This 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.University of StrathclydeGlasgowUK
  2. 2.University of AmsterdamAmsterdamNetherlands

Personalised recommendations