A. Tarski [22] proposed the study of infinitary consequence operations as the central topic of mathematical logic. He considered monotonicity to be a property of all such operations. In this paper, we weaken the monotonicity requirement and consider more general operations, inference operations. These operations describe the nonmonotonic logics both humans and machines seem to be using when infering defeasible information from incomplete knowledge. We single out a number of interesting families of inference operations. This study of infinitary inference operations (...) is inspired by the results of [12] on nonmonotonic inference relations, and relies on some of the definitions found there. (shrink)

Since the earliest formalisation of default logic by Reiter many contributions to this appealing approach to nonmonotonic reasoning have been given. The different formalisations are here presented in a general framework that gathers the basic notions, concepts and constructions underlying default logic. Our view is to interpret defaults as special rules that impose a restriction on the juxtaposition of monotonic Hubert-style proofs of a given logicL. We propose to describe default logic as a logic where the juxtaposition of default proofs (...) is subordinate to a restriction condition . Hence a default logic is a pair (L, ) where properties of the logic , like compactness, can be interpreted through the restriction condition . Different default systems are then given a common characterization through a specific condition on the logicL. We also prove cumulativity for any default logic (L, ) by slightly modifying the notion of default proof. We extend, in fact, the language ofL in a way close to that followed by Brewka in the formulation of his cumulative default system. Finally we show the existence of infinitely many intermediary default logics, depending on and called linear logics, which lie between Reiter's and ukaszewicz' versions of default logic. (shrink)

The “preferential entailments” considered in this text are all defined in the same way, by a binary relation, or “preference relation”. This relation can be among interpretations, or sets of interpretations, or among “states” which are copies of interpretations or copies of sets of interpretations. This provides four kinds of preferential entailments. What we do here is to provide a characterization result for these four kinds of preferential entailments. We choose properties as simple and natural as possible, and sometimes we (...) provide various characterizations for the same notion. It appears that the apparently most complicated notion possesses by far the simplest characterization result. A by-product of our results is that case is equivalent to case : we can define directly the relation among sets of interpretations, eliminating the need for “states” in this case. The paper deals mainly with propositional logic, however it describes also the situation in first order logic. Any complete theory has one model in the propositional case and many models in the first order case. This is why the simplest notion in the first order case has exactly the same syntactical characterization as the notion in the propositional case. (shrink)