A Logical Theory of Nonmonotonic Inference and Belief Change

Abstract

The main subject and objective of this book are logical foundations of nonmonotonic reasoning. This bears a presumption that there is such a thing as a general theory of non monotonic reasoning, as opposed to a bunch of systems for such a reasoning existing in the literature. It also presumes that this kind of reasoning can be analyzed by logical tools (broadly understood), just as any other kind of reasoning.In order to achieve our goal, we will provide a common logical basis and semantic representation in which different kinds of non monotonic reasoning can be interpreted and studied. The suggested framework will subsume basic forms of nonmonotonic inference, including not only the usual skeptical one, but also various forms of credulous (brave) and defeasible reasoning, as well as some new kinds such as contraction inference relations that express relative independence of pieces of data. In addition, the same framework will serve as a basis for a general theory of belief change which, among other things, will allow us to unify the main approaches to belief change existing in the literature, as well as to provide a constructive view of the semantic representation used.This book is a monograph rather than a textbook, with all its advantages (mainly for the author) and shortcomings (for the reader). It is primarily intended for specialists in Artificial Intelligence and Knowledge Representation who are interested in having versatile tools for describing commonsense reasoning tasks, as well as in knowing representation capabilities of such tools. It also could be of interest to general logicians; our study demonstrates, after all, that nonmonotonic reasoning is just a special kind of logical reasoning that can and should reach the level of sophistication common to established logical formalisms.

Revision over Partial Pre-orders: A Postulational Study

“…or its weaker version defined when initial epistemic state is represented by a partial preorder [4,5]: if Bel(Φ) |= µ, then Bel(Φ•µ) ≡ Bel(Φ)∧µ. This is not surprising, since in our framework, all units play an equal role in the belief change, while in logic-based revision, only units that determine belief sets play crucial roles 1 .…”

“…In [29,3] a lexicographic strategy, associated with a set of three rationality postulates, have been defined to revise a total pre-order by a new total pre-order. In [5,4], different strategies have been proposed to revise an epistemic state represented by a partial pre-order on the possible worlds.…”

Defeasible Inheritance-Based Description Logics

“…On the other hand, Lehmann and Magidor's rational closure falls into the preferential approach to non-monotonic reasoning; such an approach, since the first formulation by Shoham (1988), has become a main representative of non-monotonic reasoning (Kraus, Lehmann, & Magidor, 1990;Lehmann & Magidor, 1992;Makinson, 1994Makinson, , 2005Freund, 1998;Bochman, 2001;Rott, 2001;Schlechta, 2004), particularly appreciated for the solid logical characterization of the consequence relation.…”

“…Intuitively, in the preferential setting, saying that C ⊢ D is valid is equivalent to saying that its negation is an absurdity (¬(C ⊃ D)| ∼⊥) (Bochman, 2001, sect. 6.5).…”

Knowledge acquisition: Past, present and future

“…This includes formalization of inheritance systems in semantic networks (Gabbay and Schlechta, 2009) and also the development of general principles of rational non-monotonic reasoning (Gabbay, 1985;Lehmann, 1995;Bochman, 2001). Kraus, Lehmann and Magidor (1990) showed that the principles could be represented as preferences over the epistemic states consistent with the available information, and Lehmann (2001) showed that these results could also be interpreted in terms of Arrow's (1959) rational choice functions which are the foundation of social choice theory in economics (Arrow, Sen and Suzumura, 2002).…”