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A Discrete Duality Between Nonmonotonic Consequence Relations and Convex Geometries


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.


  1. Adaricheva, K., Gorbunov, V., Tumanov, V.I.: Join-semidistributive lattices and convex geometries. Adv. Math. 173.1, 1–49 (2003)

    MathSciNet  Article  Google Scholar 

  2. Adaricheva, K., Nation, J.B.: A class of infinite convex geometries. Electron. J. Comb. 23, 1 (2016)

    MathSciNet  MATH  Google Scholar 

  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)

  4. Baltag, A., Smets, S.: Conditional doxastic models: A qualitative approach to dynamic belief revision. Electron. Notes Theor. Comput. Sci. 165, 5–21 (2006)

    MathSciNet  Article  Google Scholar 

  5. van Benthem, J., Pacuit, E.: Dynamic logics of evidence-based beliefs. Studia Logica 99.1-3, 61–92 (2011)

    MathSciNet  Article  Google Scholar 

  6. Board, O.: Dynamic interactive epistemology. Games Econ. Behav. 49.1, 49–80 (2004)

    MathSciNet  Article  Google Scholar 

  7. Burgess, J.: Quick completeness proofs for some logics of conditionals. Notre Dame J. Formal Logic 22.1, 76–84 (1981)

    MathSciNet  Article  Google Scholar 

  8. Danilov, V.I., Koshevoy, G.A.: A new characterization of the path independent choice functions. Math. Soc. Sci. 51.2, 238–245 (2006)

    MathSciNet  Article  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

  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)

  11. Edelman, P.H., Jamison, R.E.: The theory of convex geometries. Geometriae Dedicata 19.3, 247–270 (1985)

    MathSciNet  MATH  Google Scholar 

  12. Edelman, P.H., Saks, M.E.: Combinatorial representation and convex dimension of convex geometries. Order 5.1, 23–32 (1988)

    MathSciNet  MATH  Google Scholar 

  13. Girard, P.: From onions to broccoli: Generalizing Lewis’ counterfactual logic. J. Appl. Non-Classical Logics 17.2, 213–229 (2007)

    MathSciNet  Article  Google Scholar 

  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)

  15. Grove, A.: Two modellings for theory change. J. Philos. Logic 17.2, 157–170 (1988)

    MathSciNet  MATH  Google Scholar 

  16. Paul, R.: Halmos: Lectures on Boolean Algebras. Springer (1974)

  17. Joseph, Y.: Halpern: Reasoning About Uncertainty. MIT Press (2003)

  18. Johnson, M.R., Dean, R.A.: Locally complete path independent choice functions and their lattices. Math. Soc. Sci. 42.1, 53–87 (2001)

    MathSciNet  Article  Google Scholar 

  19. Kashiwabara, K., Nakamura, M., Okamoto, Y.: The affine representation theorem for abstract convex geometries. Comput. Geom. 30.2, 129–144 (2005)

    MathSciNet  Article  Google Scholar 

  20. Korte, B., Lovász, L.: Homomorphisms and Ramsey properties of antimatroids. Discret. Appl. Math. 15.2, 283–290 (1986)

    MathSciNet  Article  Google Scholar 

  21. Korte, B., Lovász, L., Schrader, R.: Greedoids. Springer (1991)

  22. Koshevoy, G.A.: Choice functions and abstract convex geometries. Math. Soc. Sci. 38.1, 35–44 (1999)

    MathSciNet  Article  Google Scholar 

  23. Kratzer, A.: Partition and revision: The semantics of counterfactuals. J. Philos. Logic 10.2, 201–216 (1981)

    MathSciNet  MATH  Google Scholar 

  24. Kraus, S., Lehmann, D., Magidor, M.: Nonmonotonic reasoning, preferential models and cumulative logics. Artif. Intell. 44.1–2, 167–207 (1990)

    MathSciNet  Article  Google Scholar 

  25. Lewis, D.: Counterfactuals. Blackwell (1973)

  26. Marti, J., Pinosio, R.: A game semantics for system. Studia Logica 104.6, 1119–1144 (2016)

    MathSciNet  Article  Google Scholar 

  27. Marti, J., Pinosio, R.: Topological semantics for conditionals. In: Punčochár, V., Švarný, P. (eds.) The Logica Yearbook 2013. College Publications (2014)

  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)

    MathSciNet  Article  Google Scholar 

  29. Monjardet, B.: A use for frequently rediscovering a concept. Order 1.4, 415–417 (1985)

    MathSciNet  Article  Google Scholar 

  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)

  31. Nute, D.: Topics in Conditional Logic. Reidel (1980)

  32. Plott, C.R.: Path independence, rationality, and social choice. In: Econometrica, pp. 1075–1091 (1973)

  33. Pozzato, G.L.: Conditional and preferential logics: Proof methods and theorem proving, vol. 208. Frontiers in Artificial Intelligence and Applications. IOS Press (2010)

  34. Richter, M., Rogers, L.G.: Embedding convex geometries and a bound on convex dimension. Discret. Math. 340.5, 1059–1063 (2017)

    MathSciNet  Article  Google Scholar 

  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)

  36. Robert, C.: Stalnaker: A theory of conditionals. In: Rescher, N. (ed.) Studies in Logical Theory, pp 98–112. Blackwell (1968)

  37. Touazi, F, Cayrol, C., Dubois, D.: Possibilistic reasoning with partially ordered beliefs. J. Appl. Logic 13.4, 770–798 (2015)

    MathSciNet  Article  Google Scholar 

  38. van de Vel, M.L.J.: Theory of Convex Structures. Elsevier (1993)

  39. Veltman, F.: Logics for Conditionals. PhD thesis University of Amsterdam (1985)

  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)

  41. Wahl, N.: Antimatroids of finite character. J. Geom. 70.1, 168–175 (2001)

    MathSciNet  Article  Google Scholar 

  42. Wolter, F.: The algebraic face of minimality. Logic Logical Philos. 6.0, 225–240 (2004)

    MathSciNet  Article  Google Scholar 

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

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Correspondence to Johannes Marti.

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Marti, J., Pinosio, R. A Discrete Duality Between Nonmonotonic Consequence Relations and Convex Geometries. Order 37, 151–171 (2020).

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