Modal logics for qualitative possibility theory

Possibilistic logic has been proposed as a numerical formalism for reasoning with uncertainty. There has been interest in developing qualitative accounts of possibility, as well as an explanation of the relationship between possibility and modal logics. We present two modal logics that can be used to represent and reason with qualitative statements of possibility and necessity. These logics have a natural semantics based on a qualitative abstraction of possibility distributions. Within this modal framework, we are able to identify interesting relationships among possibilistic logic, beliefs, and conditionals. In particular, we demonstrate that possibilistic logic natural~ induces a notion of belief identical to that of the widely used epistemic logic weak $5, and that current approaches to conditional default reasoning and belief revision can be mapped into possibilistic logic, including the means of conditional reasoning based on high probabilities investigated by Adams [1] and Pearl [2, 3]. K E Y W O R D S : Possibility theory, modal logic, belief logics, conditionals, default reasoning, belief revision, high probability 1. I N T R O D U C T I O N There has been a great deal o f interest in the relationship between numerical and nonnumer ica l approaches to uncertain reasoning in artificial intelligence (AI). While numerical methods, for instance, those based on classical probabili ty theory, tend to be more general than their qualitaAddress correspondence to Craig Boutilier, Department of Computer Science, University of British Columbia, Vancouver, BC V6T 1Z4 Canada. Some parts of this paper appeared in preliminary form as "Modal Logics for Qualitative Possibility and Beliefs," Proceedings of the 8th Conference on Uncertainty in AI (UAI-02), Stanford, CA, 17-24, 1992. Received May 1, 1993; accepted September 22, 1993. International Journal of Approximate Reasoning 1994; 10:173-201 © 1994 Elsevier Science Inc. 655 Avenue of the Americas, New York, NY 10010 0888-613X/94/$7.00 173