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.

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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.

4) Tables concerning size and coherence • Definitions of (weak) filters, ideals, and coherence conditions, Definition 6.1 • Correspondence between coherence conditions and semantical rules for nonmonotonic logics, Fact 6.32 Generalities Definition 2.1We use P to denote the power set operator, Π{X i : i ∈ I} := {g : g :shall denote the cardinality of X, and V the set-theoretic universe we work in -the class of all sets. Given a set of pairs X , and a set X, we denote by X ⌈X := {< x, i >∈ X : x ∈ X}. When the context is clear, we will sometime simply write X for X ⌈X.A ⊆ B will denote that A is a subset of B or equal to B, and A ⊂ B that A is a proper subset of B, likewise for A ⊇ B and A ⊃ B.Given some fixed set U we work in, and X ⊆ U, then C(X) := U − X .If Y ⊆ P(X) for some X, we say that Y satisfies (∩) iff it is closed under finite intersections, ( ) iff it is closed under arbitrary intersections, (∪) iff it is closed under finite unions, ( ) iff it is closed under arbitrary unions, (C) iff it is closed under complementation.

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We introduce Information Bearing Relation Systems (IBRS) as an abstraction of many logical systems. 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 "break" the strong coherence properties of preferential structures by higher arrows, i.e. arrows, which do not go to points, but to arrows themselves.

Abstract. K.J. Devlin has extended Jensen's construction of a model of ZFC and CH without Souslin trees to a model without Kurepa trees either. We modify the construction again to obtain a model with these properties, but in addition, without Kurepa trees in ccc-generic extensions. We use a partially defined D-sequence, given by a fine structure lemma. We also show that the usual collapse of x Mahlo to ~02 will give a model without Kurepa trees not only in the model itself, but also in ccc-extensions.