this paper: default reasoning. In recent years, a number of different semantics for defaults have been proposed, such as preferential structures, ffl-semantics, possibilistic structures, and -rankings, that have been shown to be characterized by the same set of axioms, known as the KLM properties. While this was viewed as a surprise, we show here that it is almost inevitable. In the framework of plausibility measures, we can give a necessary condition for the KLM axioms to be sound, and an additional condition necessary and sufficient to ensure that the KLM axioms are complete. This additional condition is so weak that it is almost always met whenever the axioms are sound. In particular, it is easily seen to hold for all the proposals made in the literature. Categories and Subject Descriptors: F.4.1 [Mathematical Logic and Formal Languages]:

Tarski gave a general semantics for deductive reasoning: a formula a may be deduced from a set A of formulas i a holds in all models in which each of the elements of A holds. A more liberal semantics has been considered: a formula a may be deduced from a set A of formulas i a holds in all of the preferred models in which all the elements of A hold. Shoham proposed that the notion of preferred models be de ned by a partial ordering on the models of the underlying language. A more general semantics is described in this paper, based on a set of natural properties of choice functions. This semantics is here shown to be equivalent to a semantics based on comparing the relative importance of sets of models, by what amounts to a qualitative probability measure. The consequence operations de ned by the equivalent semantics are then characterized by a weakening of Tarski's properties in which the monotonicity requirement is replaced by three weaker conditions. Classical propositional connectives are characterized by natural introduction-elimination rules in a nonmonotonic setting. Even in the nonmonotonic setting, one obtains classical propositional logic, thus showing that monotonicity is not required to justify classical propositional connectives.

tions beliefs. We say that an agent believes ' if she acts as though ' is true. As time passes and new evidence is observed, changes in an agent's defeasible assumptions lead to changes in her beliefs. Thus, the question of belief change---that is, how beliefs change over time---is a central one for understanding systems that can make and modify defeasible assumptions. In this dissertation, we propose a new approach to the question of belief change. This approach is based on developing a semantics for beliefs. This semantics is embedded in a framework that models agents' knowledge (or information) as well as their beliefs, and how these change in time. We argue, and demonstrate by examples, that this framework can naturally model any dynamic system (e.g., agents and their environment). Moreover, the framework allows us to consider what the properties of well-behaved belief change should be. As we show, such a framework can g

Semantics has been many times a problem in connection to both nonmonotonic and paraconsistent logics. In previous works, a system of logics called IDL & LEI was developed to formalize practical reasoning, i.e., reasoning under incomplete knowledge. This system combines the features of nonmonotonicity to model inferences on the basis of partial evidence, with paraconsistency to deal with the inconsistencies introduced by extended inferences. The aim of this paper is to present a semantics that is, as much as possible, uniform with respect to both deductive and nonmonotonic part of this system. This semantics is presented in a possible worlds style and is intended to reect the basic intuitions assumed in designing IDL & LEI. A possible worlds semantics to the paraconsistent logic LEI is rst given and it is extended to encompass the nonmonotonic features formalized by the logic IDL. The resulting semantics is shown to be sound and complete with respect to the whole IDL & LEI system.

Abstract. We present here a view on abstraction based on the relation between sentences in a partially ordered language L and truth values of these sentences on a set of instances W. In Formal Concept Analysis, this relation is materialized as a lattice denoted as G that relates L and the powerset P(W)). We show here that projections on a lattice (here either L or the powerset P(W)) that are known to ensure structure-preserving reductions of G, are equivalent to abstractions, defined here as sets of subsets closed under union (regarding P(W)) or under minimal specialization (regarding L) and order them in a lattice of abstractions. We then discuss specifically abstractions A of P(W) and discuss the properties, of abstract implications. We then exhibit the class of (non normal) monotonic modal logics the semantic basis of which relies on such abstractions, and discuss how reasoning may be performed at variable abstraction levels. 1

Abstract. We present a view of abstraction based on a structure preserving reduction of the Galois connection between a language L of terms and the powerset of a set of instances O. Such a relation is materialized as an extension-intension lattice, namely a concept lattice when L is the powerset of a set P of attributes. We define and characterize an abstraction A as some part of either the language or the powerset of O, defined in such a way that the extension-intension latticial structure is preserved. Such a structure is denoted for short as an abstract lattice. We discuss the extensional abstract lattices obtained by so reducing the powerset of O, together together with the corresponding abstract implications, and discuss alpha lattices as particular abstract lattices. Finally we give formal framework allowing to define a generalized abstract lattice whose language is made of terms mixing abstract and non abstract conjunctions of properties. 1

: In this paper we propose a new semantics of the '2` operator of modal logic, inspired by its interpretation as expectation instead of the more traditional necessity interpretation. The semantics is based on a weakening of the traditional logical characterization of a majority in terms of a Filter, in that closure under (finite) intersection is replaced by the finite intersection property (namely, finite intersections are not empty). The semantics is accompanied by a sound and complete axiomatization, in which the regularity property / ` 1in E / i ' ! : E : 1in / i ! is replaced by ( E / E OE) ! : E :(/ OE). We see the latter as a major attribute of expectation. The paper provides additional results about the logic, such as a finitemodel property, and a PSPACE characterization of the complexity of satisfiability. shai@cs.technion.ac.il y brezner@csc.cs.technion.ac.il z francez@cs.technion.ac.il 1 1

In this paper we investigate nonmonotonic ‘modes of inference’. Our approach uses modal (conditional) logic to establish a uniform framework in which to study nonmonotonic consequence. We consider a particular mode of inference which employs a majority-based account of default reasoning—one which differs from the more familiar preferential accounts—and show how modal logic supplies a framework which facilitates analysis of, and comparison with more traditional formulations of nonmonotonic consequence.

Abstract. We present a view of abstraction based on a structure preserving reduction of the Galois connection between a language L of terms and the powerset of a set of instances O. Such a relation is materialized as an extension-intension lattice, namely a concept lattice when L is the powerset of a set P of attributes. We define and characterize an abstraction A as some part of either the language or the powerset of O, defined in such a way that the extension-intension latticial structure is preserved. Such a structure is denoted for short as an abstract lattice. We discuss the extensional abstract lattices obtained by so reducing the powerset of O, together together with the corresponding abstract implications, and discuss alpha lattices as particular abstract lattices. Finally we give formal framework allowing to define a generalized abstract lattice whose language is made of terms mixing abstract and non abstract conjunctions of properties. 1

We give a conceptual analysis of (defeasible or nonmonotonic) inheritance diagrams, and compare our analysis to the ”small/big sets ” of preferential and related reasoning. In our analysis, we consider nodes as information sources and truth values, direct links as information, and valid paths as information channels and comparisons of truth values. This results in an upward chaining, split validity, off-path preclusion inheritance formalism. We show that the small/big sets of preferential reasoning have to be relativized if we want them to conform to inheritance theory, resulting in a more cautious approach, perhaps closer to actual human reasoning. Finally, we interpret inheritance diagrams as theories of prototypical reasoning, based on two distances: set difference, and information difference. We will also see that some of the major distinctions between inheritance formalisms are consequences of deeper and more general problems of treating conflicting