A vast and interesting family of natural semantics for 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. (shrink)

Nonmonotonic logics were created as an abstraction of some types of common sense reasoning, analogous to the way classical logic serves to formalize ideal reasoning about mathematical objects. These logics are nonmonotonic in the sense that enlarging the set of axioms does not necessarily imply an enlargement of the set of formulas deducible from these axioms. Such situations arise naturally, for example, in the use of information of different degrees of reliability. This book emphasizes basic concepts by outlining connections between (...) different formalisms of nonmonotonic logic, and gives a coherent presentation of recent research results and reasoning techniques. It provides a self-contained state-of-the-art survey of the area addressing researchers in AI lo. (shrink)

A vast and interesting family of natural semantics for 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 $\alpha$ as the theory defined by the set of all those models of $\alpha$ 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. (shrink)

Plausibility Logic was introduced by Daniel Lehmann. We show—among some other results—completeness of a subset of Plausibility Logic for Preferential Models, and incompleteness of full Plausibility Logic for smooth Preferential Models.

No aConsiders the question of how far the different ‘closeness’ relations, indexed by worlds, in a given model for counterfactual conditionals may be derived from a common source. Counterbalancing some well-known negative observations, we show that there is also a strong positive answer.

The two volumes in this advanced textbook present results, proof methods, and translations of motivational and philosophical considerations to formal constructions. In this Vol. I the author explains preferential structures and abstract size. In the associated Vol. II he presents chapters on theory revision and sums, defeasible inheritance theory, interpolation, neighbourhood semantics and deontic logic, abstract independence, and various aspects of nonmonotonic and other logics. In both volumes the text contains many exercises and some solutions, and the author limits the (...) discussion of motivation and general context throughout, offering this only when it aids understanding of the formal material, in particular to illustrate the path from intuition to formalisation. Together these books are a suitable compendium for graduate students and researchers in the area of computer science and mathematical logic. (shrink)

The two volumes in this advanced textbook present results, proof methods, and translations of motivational and philosophical considerations to formal constructions. In the associated Vol. I the author explains preferential structures and abstract size. In this Vol. II he presents chapters on theory revision and sums, defeasible inheritance theory, interpolation, neighbourhood semantics and deontic logic, abstract independence, and various aspects of nonmonotonic and other logics. In both volumes the text contains many exercises and some solutions, and the author limits the (...) discussion of motivation and general context throughout, offering this only when it aids understanding of the formal material, in particular to illustrate the path from intuition to formalisation. Together these books are a suitable compendium for graduate students and researchers in the area of computer science and mathematical logic. (shrink)

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. (shrink)

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. (shrink)

We consider three different measures of distance between classical propositional models, and provide sound and complete axiomatisations for the ensuing conditional semantics, by translating conditional formulas into equivalent classical ones.

We give a systematic overview of semantical and logical rules in non monotonic and related logics. We show connections and sometimes subtle differences, and also compare such rules to uses of the notion of size.

We introduce Information Bearing Relation Systems (IBRS) as an abstraction of many logical systems. These are networks with arrows recursively leading to other arrows etc. We then define a general semantics for IBRS, and show that a special case of IBRS generalizes in a very natural way preferential semantics and solves open representation problems for weak logical systems. This is possible, as we can the strong coherence properties of preferential structures by higher arrows, that is, arrows, which do not go (...) to points, but to arrows themselves. (shrink)

Agents act on the basis of their beliefs and these beliefs change as they interact with other agents. In this book the authors propose and explain general logical tools for handling change. These tools include preferential reasoning, theory revision, and reasoning in inheritance systems, and the authors use these tools to examine nonmonotonic logic, deontic logic, counterfactuals, modal logic, intuitionistic logic, and temporal logic. This book will be of benefit to researchers engaged with artificial intelligence, and in particular agents, multiagent (...) systems and nonmonotonic logic. (shrink)

We show how to develop a multitude of rules of nonmonotonic logic from very simple and natural notions of size, using them as building blocks.

For nonmonotonic logics, Cumulativity is an important logical rule. We show here that Cumulativity fans out into an infinity of different conditions, if the domain is not closed under finite unions.

The problem of interpolation is a classical problem in logic. Given a consequence relation |~ and two formulas φ and ψ with φ |~ ψ we try to find a “simple" formula α such that φ |~ α |~ ψ. “Simple" is defined here as “expressed in the common language of φ and ψ". Non-monotonic logics like preferential logics are often a mixture of a non-monotonic part with classical logic. In such cases, it is natural examine also variants of the (...) interpolation problem, like: is there “simple" α such that φ ⊢ α |~ ψ where ⊢ is classical consequence? We translate the interpolation problem from the syntactic level to the semantic level. For example, the classical interpolation problem is now the question whether there is some “simple" model set X such that M(φ) ⫅ X ⫅ M(ψ). We can show that such X always exist for monotonic and antitonic logics. The case of non-monotonic logics is more complicated, there are several variants to consider, and we mostly have only partial results. (shrink)

-- 9. Bd. Der handschriftliche Nachlass ab Frühjahr 1885 in differenzierter Transkription nach Marie-Luise Haase und Michael Kohlenbach. pt. <9>. Arbeitshefte W II 6 und W II 7.

We investigate different aspects of independence here, in the context of theory revision, generalizing slightly work by Chopra, Parikh, and Rodrigues, and in the context of preferential reasoning.

Booth and his co-authors have shown in [2], that many new approaches to theory revision (with fixed K ) can be represented by two relations, , where is a sub-relation of < . They have, however, left open a characterization of the infinite case, which we treat here.

We introduce -ranked preferential structures and combine them with an accessibility relation. -ranked preferential structures are intermediate between simple preferential structures and ranked structures. The additional accessibility relation allows us to consider only parts of the overall -ranked structure. This framework allows us to formalize contrary to duty obligations, and other pictures where we have a hierarchy of situations, and maybe not all are accessible to all possible worlds. Representation results are proved.