Roadmap for preferential logics

Abstract

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.

Conditionals and Modularity in General Logics

“…The first time that abstract size was related to non-monotonic logics was, to our the best of knowledge, in the second author's [Sch90] and [Sch95-1], and, independently, in [BB94]. More detailed remarks can also be found in [GS08c], [GS09a], and [GS08f]. More detailed remarks can also be found in [GS08c], [GS09a], and [GS08f].…”

“…Most of the material of this chapter (unless marked as "new") was published previously; see [Sch04], [GS08b], [GS08c], [GS09a], and [GS08f]. In addition, we also want to put our work a bit more in perspective, and make it self-contained, for the convenience of the reader.…”

“…All parts of Chapter 2 (page 31) which are not marked as new material were published in some of [Sch04], [GS08b], [GS08c], [GS09a], [GS08f].…”

Constrained Consequence

“…In Artificial Intelligence, there has been a great deal of work done on non-monotonic consequence relations [8,10,11,22,30,35]. Here we give a brief outline of propositional preferential and rational consequence, as initially defined by Kraus et al [30], and analyze how constrained consequence |< stands in comparison to them.…”

Cumulativity Without Closure of the Domain Under Finite Unions

“…The reader will find here more than he really needs to understand the main result, but it will help him to put things into perspective. We also show (without proof) connections and differences, which can be quite subtle -see [GS08c] for details and proofs.…”