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...

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.

This paper is a contribution to the reconstruction of Tarski's semantic background and to the history of metalanguage and truth. Although in his 1933 monograph Tarski credits his master, Stanisl/aw Le#niewski, with crucial negative results on the semantics of natural language, the conceptual relationship between the two logicians has never been investigated in a thorough manner. This paper shows that it was not Tarski, but Le#niewski who first avowed the impossibility of giving a satisfactory theory of truth for ordinary language, and the necessity of sanitation of the latter for scientific purposes. In an early article (1913) Le#niewski gave an interesting solution to the Liar Paradox, which, although different from Tarski's in detail, is nevertheless important to Tarski's semantic background. To illustrate this I give an analysis of Le#niewski's solution and of some related aspects of Le#niewski's later thought.