Abstract
We introduce and explore the notion of duality for entailment relations induced by preference orderings on states. We discuss the relationship between these preferential entailment relations from the perspectives of Boolean algebra, inference rules, and modal axiomatisation. Interpreting the preference relations as accessibility relations establishes modular Gödel-Löb logic as a suitable modal framework for rational preferential reasoning.
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Amati, G., Carlucci, L., Gabbay, D., & Pirri, F. (1996). A structural property on modal frames characterizing default logic. Logic Journal of the IGPL, 4(1), 7–22.
Blackburn, P., de Rijke, M., & Venema, Y. (2002). Modal logic: Cambridge tracts in theoretical computer science (Vol. 53). Cambridge: Cambridge University Press.
Boolos, G. (1993). The logic of provability. Cambridge: Cambridge University Press.
Boutilier, C. (1994). Conditional logics of normality: A modal approach. Artificial Intelligence, 68(1), 87–154.
Britz, K., Heidema, J., & Labuschagne, W. (2007). Entailment, duality, and the forms of reasoning. http://www.cs.otago.ac.nz/research/publications/OUCS-2007-01.pdf.
Crocco, G., & Lamarre, P. (1992). On the connections between nonmonotonic inference systems and conditional logics. In R. Nebel, C. Rich, & W. Swartout (Eds.), Principles of knowledge representation and reasoning (KR’92) (pp. 565–571). San Mateo: Morgan Kaufmann.
Davis, E. (1990). Representations of commonsense knowledge. San Mateo: Morgan Kaufmann.
Delgrande, J., Schaub, T., Tourpits, H., & Wang, K. (2004). A classification and survey of preference handling approaches in non-monotonic reasoning. Computational Intelligence, 20(2), 308–334.
Dickman, M. (1975). Large infinitary languages. Amsterdam: North Holland.
Freund, M., & Lehmann, D. (1994). Belief revision and rational inference. Tech. Rep. TR–94–16, Leibniz Center for Research in Computer Science, Hebrew University, Jerusalem. arXiv:cs/0204032v1 [cs.AI].
Freund, M., Lehmann, D., & Morris, P. (1991). Rationality, transitivity and contraposition. Artificial Intelligence, 52(2), 191–203.
Gabbay, D., & Woods, J. (Eds.) (2007). The many valued and nonmonotonic turn in logic: Handbook of the history of logic (Vol. 8). Amsterdam: North-Holland/Elsevier.
Giordano, L., Gliozzi, V., Olivetti, N., & Pozzato, G. (2005). Analytic tableaux for KLM preferential and cumulative logics. In G. Sutcliffe & A. Voronkov (Eds.), LPAR 2005: Lecture notes in artificial intelligence (Vol. 3835, pp. 666–681). Berlin: Springer.
Hansson, S., & Grüne-Yanoff, T. (2006). Preferences. In E. Zalta (Ed.), The Stanford encyclopedia of philosophy. Center for Logic and Information, Stanford University, winter 2006 ed. plato.stanford.edu/archives/win2006/entries/preferences.
Karp, C. (1964). Languages with expressions of infinite length. Amsterdam: North Holland.
Kraus, S., Lehmann, D., & Magidor, M. (1990). Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44, 167–207.
Lehmann, D., & Magidor, M. (1992). What does a conditional knowledge base entail? Artificial Intelligence, 55, 1–60.
Makinson, D. (2005). How to go nonmonotonic. In Handbook of philosophical logic, 2nd ed. (Vol. 12, pp. 175–278). Heidelberg: Springer.
Meier, B., & Robinson, M. (2004). Why the sunny side is up: Associations between affect and vertical position. Psychological Science, 15, 243–247.
Moinard, Y., & Rolland, R. (2002). Characterizations of preferential entailments. Logic Journal of the IGPL, 10(3), 245–272.
Segerberg, K. (1971). An essay in classical modal logic: Filosofiska Studier (Vol. 13). Uppsala: Uppsala Universitet.
Shoham, Y. (1988). Reasoning about change: Time and causation from the standpoint of artificial intelligence. Cambridge, MA: The MIT Press.
Zhu, Z. (2006). Similarity between preferential models. Theoretical Computer Science, 353, 26–52.
Zhu, Z., & Zhang, R. (2007). An algebraic characterization of equivalent preferential models. Journal of Symbolic Logic, 72(3), 803–833.
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Britz, K., Heidema, J. & Labuschagne, W. Semantics for Dual Preferential Entailment. J Philos Logic 38, 433–446 (2009). https://doi.org/10.1007/s10992-008-9097-z
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DOI: https://doi.org/10.1007/s10992-008-9097-z