Preferential Accessibility and Preferred Worlds

Modal accounts of normality in non-monotonic reasoning traditionally have an underlying semantics based on a notion of preference amongst worlds. In this paper, we motivate and investigate an alternative semantics, based on ordered accessibility relations in Kripke frames. The underlying intuition is that some world tuples may be seen as more normal, while others may be seen as more exceptional. We show that this delivers an elegant and intuitive semantic construction, which gives a new perspective on defeasible necessity. Technically, the revisited logic does not change the expressive power of our previously defined preferential modalities. This conclusion follows from an analysis of both semantic constructions via a generalisation of bisimulations to the preferential case. Reasoners based on the previous semantics therefore also suffice for reasoning over the new semantics. We complete the picture by investigating different notions of defeasible conditionals in modal logic that can also be captured within our framework.

[1]  Philippe Lamarre,et al.  On the Connection between Non-monotonic Inference Systems and Conditional Logics , 1992, KR.

[2]  Richard Booth,et al.  PTL: A Propositional Typicality Logic , 2012, JELIA.

[3]  Thomas Andreas Meyer,et al.  Preferential Role Restrictions , 2013, Description Logics.

[4]  Y. Shoham.  Reasoning About Change: Time and Causation from the Standpoint of Artificial Intelligence , 1987 .

[5]  Daniel Lehmann,et al.  Another perspective on default reasoning , 1995, Annals of Mathematics and Artificial Intelligence.

[6]  Peter Gärdenfors,et al.  Nonmonotonic Inference Based on Expectations , 1994, Artif. Intell..

[7]  Sarit Kraus,et al.  Nonmonotonic Reasoning, Preferential Models and Cumulative Logics , 1990, Artif. Intell..

[8]  Joseph Y. Halpern,et al.  Plausibility measures and default reasoning , 1996, JACM.

[9]  Richard Booth,et al.  A propositional typicality logic for extending rational consequence , 2013 .

[10]  Peter Gärdenfors,et al.  On the logic of theory change: Partial meet contraction and revision functions , 1985, Journal of Symbolic Logic.

[11]  Laura Giordano,et al.  Semantic characterization of rational closure: From propositional logic to description logics , 2015, Artif. Intell..

[12]  Laura Giordano,et al.  Preferential Description Logics , 2007, LPAR.

[13]  David Makinson,et al.  Five faces of minimality , 1993, Stud Logica.

[14]  B. Hansson.  An analysis of some deontic logics , 1969 .

[15]  Richard Booth,et al.  On the Entailment Problem for a Logic of Typicality , 2015, IJCAI.

[16]  David Lewis,et al.  Semantic Analyses for Dyadic Deontic Logic , 1974 .

[17]  Thomas Andreas Meyer,et al.  Preferential Reasoning for Modal Logics , 2011, M4M/LAMAS.

[18]  Perlindström.  First Order Predicate Logic with Generalized Quantifiers , 1966 .

[19]  Mario R. F. Benevides,et al.  On vague notions and modalities: a modular approach , 2010, Log. J. IGPL.

[20]  H. Prade,et al.  Possibilistic logic , 1994 .

[21]  Robert Stalnaker.  A Theory of Conditionals , 2019, Knowledge and Conditionals.

[22]  Thomas Andreas Meyer,et al.  Normal Modal Preferential Consequence , 2012, Australasian Conference on Artificial Intelligence.

[23]  Thomas Andreas Meyer,et al.  Semantic Foundation for Preferential Description Logics , 2011, Australasian Conference on Artificial Intelligence.

[24]  Sven Ove Hansson,et al.  A textbook of belief dynamics - theory change and database updating , 1999, Applied logic series.

[25]  Katarina Britz,et al.  Introducing Role Defeasibility in Description Logics , 2016, JELIA.

[26]  Dimitris Askounis,et al.  Knowledge Means 'All', Belief Means 'Most' , 2012, JELIA.

[27]  Katarina Britz,et al.  From KLM-style conditionals to defeasible modalities, and back , 2018, J. Appl. Non Class. Logics.

[28]  James Hawthorne,et al.  Nonmonotonic conditionals that behave like conditional probabilities above a threshold , 2007, J. Appl. Log..

[29]  Wilfrid Hodges,et al.  Model Theory: The existential case , 1993 .

[30]  Daniel Lehmann,et al.  What does a Conditional Knowledge Base Entail? , 1989, Artif. Intell..

[31]  Frank Wolter,et al.  Handbook of Modal Logic , 2007, Studies in logic and practical reasoning.

[32]  Alexandru Baltag,et al.  A qualitative theory of dynamic interactive belief revision , 2008 .

[33]  Churn-Jung Liau,et al.  On the possibility theory-based semantics for logics of preference , 1999, Int. J. Approx. Reason..

[34]  Hirofumi Katsuno,et al.  Propositional Knowledge Base Revision and Minimal Change , 1991, Artif. Intell..

[35]  Thomas Andreas Meyer,et al.  Introducing Defeasibility into OWL Ontologies , 2015, International Semantic Web Conference.

[36]  Umberto Straccia,et al.  Rational Closure for Defeasible Description Logics , 2010, JELIA.

[37]  Katarina Britz,et al.  Defeasible Modalities , 2013, TARK.

[38]  Didier Dubois,et al.  Possibilistic and Standard Probabilistic Semantics of Conditional Knowledge Bases , 1997, J. Log. Comput..

[39]  Laura Giordano,et al.  A non-monotonic Description Logic for reasoning about typicality , 2013, Artif. Intell..

[40]  Craig Boutilier,et al.  Conditional Logics of Normality: A Modal Approach , 1994, Artif. Intell..

[41]  Churn-Jung Liau,et al.  Possibilistic Reasoning - A Mini-Survey and Uniform Semantics , 1996, Artif. Intell..

[42]  Thomas Andreas Meyer,et al.  Semantic Preferential Subsumption , 2008, KR.