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CONDITIONAL LOGIC IS COMPLETE FOR CONVEXITY IN THE PLANE

Part of: General logic

Published online by Cambridge University Press:  08 July 2021

JOHANNES MARTI
JOHANNES MARTI*
Affiliation:
ILLC, UNIVERSITY OF AMSTERDAM SCIENCE PARK 107 1098 XG AMSTERDAM, NETEHERLANDS
*
E-mail: johannes.marti@gmail.com
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Abstract

We prove completeness of preferential conditional logic with respect to convexity over finite sets of points in the Euclidean plane. A conditional is defined to be true in a finite set of points if all extreme points of the set interpreting the antecedent satisfy the consequent. Equivalently, a conditional is true if the antecedent is contained in the convex hull of the points that satisfy both the antecedent and consequent. Our result is then that every consistent formula without nested conditionals is satisfiable in a model based on a finite set of points in the plane. The proof relies on a result by Richter and Rogers showing that every finite abstract convex geometry can be represented by convex polygons in the plane.

Keywords

conditional logicconvex geometrycompletenessgeometric semantics

MSC classification

Primary: 03B45: Modal logic (including the logic of norms)
Type
Research Article
Information
The Review of Symbolic Logic , Volume 16 , Issue 2 , June 2023 , pp. 529 - 552
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

Adams, E. (1975). The Logic of Conditionals: An Application of Probability to Deductive Logic. Dordrecht: Springer.CrossRefGoogle Scholar
Adaricheva, K., & Bolat, M. (2019). Representation of convex geometries by circles on the plane. Discrete Mathematics, 342(3), 726–746.CrossRefGoogle Scholar
Adaricheva, K., & Nation, J. B. (2016). Convex geometries. In Grätzer, G., and Wehrung, F., editors. Lattice Theory: Special Topics and Applications. Basel: Birkhäuser, pp. 153–179.CrossRefGoogle Scholar
Aiello, M., & van Benthem, J. (2002). A modal walk through space. Journal of Applied Non-Classical Logics, 12(3–4), 319–363.CrossRefGoogle Scholar
Arrow, K. J. (1959). Rational choice functions and orderings. Economica, 26(102), 121–127.CrossRefGoogle Scholar
Balbiani, P. (1998). The modal multilogic of geometry. Journal of Applied Non-Classical Logics, 8(3), 259–281.CrossRefGoogle Scholar
Baltag, A., & Smets, S. (2006). Conditional doxastic models: A qualitative approach to dynamic belief revision. Electronic Notes in Theoretical Computer Science, 165, 5–21.CrossRefGoogle Scholar
Bezhanishvili, G., & Gehrke, M. (2005). Completeness of S4 with respect to the real line: Revisited. Annals of Pure and Applied Logic, 131(1–3), 287–301.CrossRefGoogle Scholar
Burgess, J. (1981). Quick completeness proofs for some logics of conditionals. Notre Dame Journal of Formal Logic, 22(1), 76–84.CrossRefGoogle Scholar
Chellas, B. F. (1975). Basic conditional logic. Journal of Philosophical Logic, 4(2), 133–153.CrossRefGoogle Scholar
Czédli, G. (2014). Finite convex geometries of circles. Discrete Mathematics, 330, 61–75.CrossRefGoogle Scholar
Czédli, G., & Kincses, J. (2017). Representing convex geometries by almost-circles. Acta Scientiarum Mathematicarum, 83, 393–414.CrossRefGoogle Scholar
Edelman, P. H. (1980). Meet-distributive lattices and the anti-exchange closure. Algebra Universalis, 10(1), 290–299.CrossRefGoogle Scholar
Edelman, P. H., & Jamison, R. E. (1985). The theory of convex geometries. Geometriae Dedicata, 19(3), 247–270.CrossRefGoogle Scholar
Geffner, H. (1992). High-probabilities, model-preference and default arguments. Minds and Machines, 2(1), 51–70.CrossRefGoogle Scholar
Girard, P. (2007). From onions to broccoli: Generalizing Lewis' counterfactual logic. Journal of Applied Non-Classical Logics, 17(2), 213–229.CrossRefGoogle Scholar
Girlando, M., Negri, S., & Olivetti, N. (2021). Uniform labelled calculi for preferential conditional logics based on neighbourhood semantics. Journal of Logic and Computation. https://doi.org/10.1093/logcom/exab019.CrossRefGoogle Scholar
Grove, A. (1988). Two modellings for theory change. Journal of Philosophical Logic, 17(2), 157–170.CrossRefGoogle Scholar
Hansson, B. (1969). An analysis of some deontic logics. Noûs, 3(4), 373–398.CrossRefGoogle Scholar
Johnson, M. R., & Dean, R. A. (2001). Locally complete path independent choice functions and their lattices. Mathematical Social Sciences, 42(1), 53–87.CrossRefGoogle Scholar
Kashiwabara, K., Nakamura, M., & Okamoto, Y. (2005). The affine representation theorem for abstract convex geometries. Computational Geometry, 30(2), 129–144.CrossRefGoogle Scholar
Korte, B., Lovász, L., & Schrader, R. (1991). Greedoids. Berlin: Springer.CrossRefGoogle Scholar
Koshevoy, G. A. (1999). Choice functions and abstract convex geometries. Mathematical Social Sciences, 38(1), 35–44.CrossRefGoogle Scholar
Kratzer, A. (1981). Partition and revision: The semantics of counterfactuals. Journal of Philosophical Logic, 10(2), 201–216.CrossRefGoogle Scholar
Kraus, S., Lehmann, D., & Magidor, M. (1990). Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44(1–2), 167–207.CrossRefGoogle Scholar
Kupke, C., & Pattinson, D. (2011). Coalgebraic semantics of modal logics: An overview. Theoretical Computer Science, 412(38), 5070–5094.CrossRefGoogle Scholar
Lewis, D. (1973). Counterfactuals. Malden, MA: Blackwell.Google Scholar
Lewis, D. (1981). Ordering semantics and premise semantics for counterfactuals. Journal of Philosophical Logic, 10(2), 217–234.CrossRefGoogle Scholar
Marti, J., & Pinosio, R. (2014). Topological semantics for conditionals. In Punčochář, V., and Švarný, P., editors. The Logica Yearbook 2013. London: College Publications, pp. 115–128.Google Scholar
Marti, J., & Pinosio, R. (2020). A discrete duality between nonmonotonic consequence relations and convex geometries. Order, 37, 151–171.CrossRefGoogle Scholar
McKinsey, J. C. C., & Tarski, A. (1944). The algebra of topology. Annals of Mathematics, 45(1), 141–191.CrossRefGoogle Scholar
Negri, S., & Olivetti, N. (2015). A sequent calculus for preferential conditional logic based on neighbourhood semantics. In De Nivelle, H., editor. Automated Reasoning with Analytic Tableaux and Related Methods, TABLEAUX 2015. Cham: Springer, pp. 115–134.CrossRefGoogle Scholar
Richter, M., & Rogers, L. G. (2017). Embedding convex geometries and a bound on convex dimension. Discrete Mathematics, 340(5), 1059–1063.CrossRefGoogle Scholar
Rott, H. (2009). Shifting priorities: Simple representations for twenty-seven iterated theory change operators. In Makinson, D., Malinowski, J., and Wansing, H., editors. Towards Mathematical Philosophy. Berlin: Springer, pp. 269–296.CrossRefGoogle Scholar
Sen, A. (1971). Choice functions and revealed preference. The Review of Economic Studies, 38(3), 307–317.CrossRefGoogle Scholar
Shoham, Y. (1988). Reasoning About Change: Time and Causation from the Standpoint of Artificial Intelligence. Cambridge, MA: MIT Press.Google Scholar
van Benthem, J. (2007). Dynamic logic for belief revision. Journal of Applied Non-Classical Logics, 17(2), 129–155.CrossRefGoogle Scholar
van Benthem, J., & Bezhanishvili, G. (2007). Modal logics of space. In Aiello, M., and Van Benthem Ian Pratt-Hartmann, J., editors. Handbook of Spatial Logics. Dordrecht: Springer, pp. 217–298.CrossRefGoogle Scholar
van Benthem, J., & Pacuit, E. (2011). Dynamic logics of evidence-based beliefs. Studia Logica, 99(1–3), 61–92.CrossRefGoogle Scholar
Veltman, F. (1976). Prejudices, presuppositions, and the theory of counterfactuals. In Groenendijk, J., and Stokhof, M., editors. Proceedings of the Amsterdam Colloquium on Montague Grammar and Related Topics. Amsterdam Papers in Formal Grammar, Vol. 1. Amsterdam: Centrale Interfaculteit, Universiteit van Amsterdam, pp. 248–282.Google Scholar
Veltman, F. (1985). Logics for Conditionals. Ph.D. Thesis, University of Amsterdam.Google Scholar
Venema, Y. (1999). Points, lines and diamonds: A two-sorted modal logic for projective planes. Journal of Logic and Computation, 9(5), 601–621.CrossRefGoogle Scholar
Zhu, Z. (2006). Similarity between preferential models. Theoretical Computer Science, 353(1), 26–52.CrossRefGoogle Scholar