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.

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.

The volume analyses and develops David Makinson’s efforts to make classical logic useful outside its most obvious application areas. The book contains chapters that analyse, appraise, or reshape Makinson’s work and chapters that develop themes emerging from his contributions. These are grouped into major areas to which Makinsons has made highly influential contributions and the volume in its entirety is divided into four sections, each devoted to a particular area of logic: belief change, uncertain reasoning, normative systems and the resources (...) of classical logic. Among the contributions included in the volume, one chapter focuses on the “inferential preferential method”, i.e. the combined use of classical logic and mechanisms of preference and choice and provides examples from Makinson’s work in non-monotonic and defeasible reasoning and belief revision. One chapter offers a short autobiography by Makinson which details his discovery of modern logic, his travels across continents and reveals his intellectual encounters and inspirations. The chapter also contains an unusually explicit statement on his views on the role of logic in philosophy. (shrink)

In this paper we address the modelling of reasoning processes in natural agents. We focus on a very basic kind of non-monotonic inference for which we identify a simple and plausible underlying process, and we develop a family of logical models that allow to match this process. Partial worlds models, as we call them, are a variant of Kraus, Lehmann and Magidor’s cumulative models. We show that the inference relations they induce form a strict subclass of cumulative relations and tackle (...) the issue of providing sound and complete sets of rules to characterise them. Taking inspiration from the work of Gabbay and Schlechta, we analyse the question in terms of definable sets of partial worlds and conclude that completeness is probably unreachable using a standard propositional language. This brings us to enrich our language with an additional connective, which makes it possible to distinguish between two kinds of disjunction in the partial worlds context. Within this renewed framework, we provide two representation theorems: one for inference relations induced by precisification-free smooth models, and the other for inference relations induced by precisification-free ranked models. Finally we give an aperçu of how partial worlds models lend themselves to the modelling of learning. (shrink)

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)

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)

There are various contexts in which it is not pertinent to generate and attend to all the classical consequences of a given premiss—or to trace all the premisses which classically entail a given consequence. Such contexts may involve limited resources of an agent or inferential engine, contextual relevance or irrelevance of certain consequences or premisses, modelling everyday human reasoning, the search for plausible abduced hypotheses or potential causes, etc. In this paper we propose and explicate one formal framework for a (...) whole spectrum of consequence relations, flexible enough to be tailored for choices from a variety of contexts. We do so by investigating semantic constraints on classical entailment which give rise to a family of infra-classical logics with appealing properties. More specifically, our infra-classical reasoning demands (beyond ${\alpha\models\beta}$ ) that Mod(β) does not run wild, but lies within the scope (whatever that may mean in some specific context) of Mod(α), and which can be described by a sentence ${\bullet\alpha}$ with ${\beta\models\bullet\alpha}$ . Besides being infra-classical, the resulting logic is also non-monotonic and allows for non-trivial reasoning in the presence of inconsistencies. (shrink)