Many systems that exhibit nonmonotonic behavior have been described and studied already in the literature. The general notion of nonmonotonic reasoning, though, has almost always been described only negatively, by the property it does not enjoy, i.e. monotonicity. We study here general patterns of nonmonotonic reasoning and try to isolate properties that could help us map the field of nonmonotonic reasoning by reference to positive properties. We concentrate on a number of families of nonmonotonic consequence relations, defined in the style of Gentzen [13]. Both proof-theoretic and semantic points of view are developed in parallel. The former point of view was pioneered by D. Gabbay in [10], while the latter has been advocated by Y. Shoham in [38]. Five such families are defined and characterized by representation theorems, relating the two points of view. One of the families of interest, that of preferential relations, turns out to have been studied by E. Adams in [2]. The pr...

This paper presents a logical approach to nonmonotonic reasoning based on the notion of a nonmonotonic consequence relation. A conditional knowledge base, consisting of a set of conditional assertions of the type if . . . then . . . , represents the explicit defeasible knowledge an agent has about the way the world generally behaves. We look for a plausible definition of the set of all conditional assertions entailed by a conditional knowledge base. In a previous paper [17], S. Kraus and the authors defined and studied preferential consequence relations. They noticed that not all preferential relations could be considered as reasonable inference procedures. This paper studies a more restricted class of consequence relations, rational relations. It is argued that any reasonable nonmonotonic inference procedure should define a rational relation. It is shown that the rational relations are exactly those that may be represented by a ranked preferential model, or by a (non-standard) probabilistic model.

The notion of the rational closure of a positive knowledge base K of conditional assertions # i # i (standing for if # i then normally # i ) was first introduced by Lehmann in [2] and developed by Lehmann and Magidor in [3]. Following those authors we would also argue that the rational closure is, in a strong sense, the minimal information, or simplest, rational consequence relation satisfying K. In practice however one might expect a knowledge base to consist not just of positive conditional assertions, # i # i , but also negative conditional assertions, # i (standing for not {if # i then normally # i }).

In the last years many logics of nonmonotonicity have been developed using various different formalisms and axiomatizations which makes them very difficult to compare. We develop a classification scheme for these logics using only a few simple concepts and axioms based on conditional logics, properties of partial pre-orders of possible worlds and centering assumptions. Our framework (the P-Systems) allows us to discuss the similarities, main differences and possible extensions of these logics in a simple and natural way.

We define a first-order conditional logic in which conditionals, such as ff ! fi are interpreted as saying that most/normal/common/typical objects which satisfy ff satisfy fi as well. This qualitative "statistical" interpretation is achieved by imposing additional structure on the domain of a single first-order model in the form of an ordering over domain elements and tuples. ff ! fi then holds if all objects with property ff whose ranking is minimal satisfy fi as well. These minimally ranked objects represent the typical or common objects having the property ff. This semantics differs from that of the more common subjective interpretation of conditionals, in which conditionals are interpreted over sets of standard first-order structures. Our semantics provides a more natural way of modeling qualitative statistical statements, such as "most birds fly," or "normal birds fly." We provide a sound and complete axiomatization of this logic, and we show that it can be given probabilistic sem...

A first-order conditional logic is defined in which conditionals such as ff ! fi are interpreted as saying that most/normal/typical objects which satisfy ff satisfy fi as well. This qualitative statistical interpretation is achieved by imposing additional structure on the domain of a single first-order model in the form of an ordering over domain elements and tuples. ff ! fi then holds if all objects with property ff whose ranking is minimal satisfy fi as well. These minimally ranked objects represent the typical or common objects having the property ff. This semantics differs from that of the more common subjective interpretation of conditionals over sets of standard first-order structures, and it provides a more natural way of modeling qualitative statistical statements, such as "most birds fly," or "normal birds fly." We provide a sound and complete axiomatization of this logic as well as a probabilistic semantics for it. 1 INTRODUCTION Conditional logics have been the focus of muc...

The SCAN algorithm has been proposed for second order quantifier elimination. In particular it can be applied to find correspondence axioms for systems of modal logic. Up to now, what has been studied are systems with unary modal operators. In this paper we study how SCAN can be applied to various systems of conditional logic, which are logical systems with binary modal operators. Keywords conditional logic, correspondence theory iii 1 Introduction A conditional is an expressions of the form if : : : then : : : . There are various kinds of conditionals that fit into that pattern, such as counterfactual conditionals ("if it were the case that A then it would be the case that B"), causal conditionals ("if A then causally B"), action conditionals ("if A then B is obtained (can be performed)"), conditional obligations ("if A then B should be brought about"), generic conditionals ("if A then normally B") etc. What is commom to all these constructions is that the antecedent is connec...