Abstract A vast and interesting family of natural semantics lor belief revision is defined. Suppose one is given a distance d between any two models. One may then define the revision of a theory K by a formula α as the theory defined by the set of all those models of α that are closest, by d. to the set of models of K. This family is characterized by a set of rationality postulates that extends the AGM postulates. The new postulates describe properties of iterated revisions.

This paper deals with some fundamental concepts and questions of preferential structures. Traditionally, a model for preferential reasoning is a strict partial order on the set of classical models of the language; in this article it will be a total order on the classical models. Instead of representing non-monotonic inference relations by individual partial orders, we represent them by sets of total orders. We thus stay close to the way completeness proofs are done in classical logic. Our new approach will also justify multiple copies (or labelling functions) present in most work on preferential structures. A representation result for the finite case is proven; for the infinite case it remains an open question.

Abstract Preferential structures are probably the best examined semantics for nonmonotonic and deontic logics: in a wider sense, they also provide semantical approaches to theory revision and update, and other fields where a preference relation between models is a natural approach. They have been widely used to differentiate the various systems of such logics, and their construction is one of the main subjects in the formal investigation of these logics. We introduce new techniques to construct preferential structures for completeness proofs. Since our main interest is to provide general techniques, which can be applied in various situations and for various base logics (propositional and other), we take a purely algebraic approach, which can be translated into logics by easy lemmata, in particular, we give a clean construction via indexing by trees for transitive structures, this allows us to simplify the proofs of earlier work by the author, and to extend the results given there.

One of the major open problems of preferential structures was perhaps to give an unrestricted representation result. We give a solution to the problem in this short, technical paper. Up to now - to the author's knowledge - all representation results for preferential structures were subject to some restriction: definability preservation (the author's terminology, fullness in Lehmann's terminology) or some kind of finiteness. The results presented here are valid without any restrictions.

We give a semantics to iterated update by a preference relation on possible developments. An iterated update is a sequence of formulas, giving (incomplete) information about successive states of the world. A development is a sequence of models, describing a possible trajectory through time. We assume a principle of inertia and prefer those developments which are compatible with the information and avoid unnecessary changes. The logical properties of the updates defined in this way are considered, and a representation result is proved.

To solve a problem posed by Bezzazi, Makinson and Pérez, we construct an injective, non-smooth preferential structure validating cumulativity and weak determinacy, in which negation rationality fails. Essential use is made of infinite sequences of models approaching sets of models. To the author's knowledge, this is the first time that such topological constructions have been used in the context of preferential structures.

Our subject is the representation and analysis of simple first-order default statements of ordinary language, such as “normally, birds fly”. There are, among other approaches, two kinds of analysis, both semantic in style. One interprets “normally, b

We extend earlier work to stoppered (or smooth) and ranked classical preferential models, giving several soundness and completeness results for these structures. In addition, we discuss the number of copies of models needed to represent arbitrary logics defined by preferential structures.

We interpret (open normal) defaults as generalized first-order logic-quantifiers, and give a semantics and a corresponding sound and complete axiom system. Nested and negated defaults are admissible and have a clear meaning. Moreover, the logic provides a notion of consistency for default theories, which is used for a theory revision approach in an order-sorted language.

The idea of preferential choice is applied here to dynamic structures in two directions: (1) we show that a deontic choice function of ‘good’ developments can be represented by a ranked, stoppered preferential relation on all developmentθ (2) we generalize the Katsuno/Mendelzon Update Semantics to preferences between developments and obtain a representation theorem for arbitrarily many time points.

We show that, under some very weak assumptions about the definitions of sceptical and extension-based defeasible inheritance, directly sceptical inheritance cannot capture the intersection of extensions.

We first show that a result of Kraus, Lehmann and Magidor on classical preferential models does not carry over to the general infinite case. We further show that—in the absence of all restrictions on finiteness—‘logically nice’ (definability preserving) classical preferential models correspond essentially to infinite conditionalization.