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...

Abstract: Logical theory traditionally assumes the following: (1) Logical inference is a relation between sentences (or propositions), not between thoughts (or anything cognitive). (2) The validity of an argument is only dependent on the logical structure of the sentences and independent of their meaning. In practical reasoning, however, these assumptions are not valid. In this paper I want to show that by taking expectations into account, one can achieve a much better understanding of how logic is put to work by humans. In particular, one obtains a very simple analysis of nonmonotonic reasoning. I will also discuss the cognitive origins of expectations. Then a man said: Speak to us of Expectations. He then said: If a man does not see or hear the waters of the Jordan, then he should not taste the pomegranate or ply his wares in an open market. If a man would not labour in the salt and rock quarries, then he should not accept of the Earth that which he refuses to give of himself. Such a man would expect a pear of a peach tree. Such a man would expect a stone to lay an egg.

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 Hilbert-style proofs of a given logic L. We propose to describe default logic as a logic where the juxtaposition of default proofs is subordinate to a restriction condition \Psi. Hence a default logic is a pair (L; \Psi) where properties of the logic L, like compactness, can be interpreted through the restriction condition \Psi. Different default systems are then given a common characterization through a specific condition on the logic L. We also prove cumulativity for any default logic (L; \Psi) by slightly modifying the notion of default proof. We extend, in fact, the language of L 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 \Psi and called linear logics, which lie between Reiter's and / Lukaszewicz' versions of default logic.

In the present paper an approach to nonmonotonic reasoning is discussed which is inspired by the paradigm of abstract model theory. The aim is to analyse nonmonotonic reasoning within the framework of inference operations based on model operators. This is done by using the concept of a deductive frame and its semantical counterpart, a semantical frame. These are considered as a basis for nonmonotonic model theory. Then several representation theorems are proved and compactness properties are analysed in detail. These concepts are applied to classical nonmonotonic systems, in particular to minimal reasoning in propositional calculus and predicate calculus, and to propositional Poole systems. keywords : nonmonotonic inference, compactness, model theory 1 Introduction Given a language L a logic can be described on an abstract level by a functor C : 2 L ! 2 L such that for every set X ` L, C(X) is understood to be the set of all the consequences of the set X of premises. If ...

Logical theory traditionally assumes the following: (1) Logical inference is a relation between sentences (or propositions), not between thoughts (or anything cognitive). (2) The validity of an argument is only dependent on the logical structure of the sentences and independent of their meaning.

It is a great pleasure for me to present those reflections to the Festschrift of Boaz (Boris) Trakhtenbrot, who, with great constancy, manifested his interest in my explorations in quantic and other exotic logics and whose encouragements have been most appreciated. 1 Abstract. Cumulative logics are studied in an abstract setting, i.e., without connectives, very much in the spirit of Makinson’s [11] early work. A powerful representation theorem characterizes those logics by choice functions that satisfy a weakening of Sen’s property α, in the spirit of the author’s [9]. The representation results obtained are surprisingly smooth: in the completeness part the choice function may be defined on any set of worlds, not only definable sets and no definability-preservation property is required in the soundness part. For abstract cumulative logics, proper conjunction and negation may be defined. Contrary to the situation studied in [9] no proper disjunction seems to be definable in general. The cumulative relations of [8] that satisfy some weakening of the consistency preservation property all define cumulative logics with a proper negation. Quantum Logics, as defined by [3] are such cumulative logics but the negation defined by orthogonal complement does not provide a proper negation. 1

Acknowledgements vi