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Coherent Systems

1st Edition - September 21, 2004

Author: Karl Schlechta

Hardback ISBN:

9 7 8 - 0 - 4 4 4 - 5 1 7 8 9 - 0

eBook ISBN:

9 7 8 - 0 - 0 8 - 0 5 0 2 1 9 - 9

One aspect of common sense reasoning is reasoning about normal cases, e.g. a physician will first try to interpret symptoms by a common disease, and will take more exotic… Read more

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One aspect of common sense reasoning is reasoning about normal cases, e.g. a physician will first try to interpret symptoms by a common disease, and will take more exotic possibilities only later into account. Such "normality" can be encoded, e.g. by a relation, where case A is considered more normal than case B. This gives a standard semantics or interpretation to nonmonotonic reasoning (a branch of common sense reasoning), or, more formally, to nonmonotonic logics. We consider in this book the repercussions such normality relations and similar constructions have on the resulting nonmonotonic logics, i.e. which types of logic are adequate for which kind of relation, etc.

We show in this book that some semantics correspond nicely to some logics, but also that other semantics do not correspond to any logics of the usual form.

Foreword (by David Makinson)SummaryAcknowledgementsCHAPTER 1 : INTRODUCTION1.1 The main topics of the book1.1.1 Conceptual analysis1.1.2 Generalized modal logic and integration1.1.3 Formal results1.1.4 The role of semantics1.1.5 Various remarks1.2 Historical remarks1.3 Organization of the book1.4 Overview of the chapters1.5 Specific remarks on propositional logic1.6 Basic definitionsCHAPTER 2 : CONCEPTS2.1 Introduction2.2 Reasoning types2.2.1 Traditional nonmonotonic logics2.2.2 Prototypical and ideal cases2.2.3 Extreme cases and interpolation2.2.4 Clustering2.2.5 Certainty2.2.6 Quality of an answer, approximation, and complexity2.2.7 Useful reasoning2.2.8 Inheritance and argumentation2.2.9 Dynamic systems2.2.10 Theory revision2.2.11 Update2.2.12 Counterfactual conditionals2.3 Basic semantical concepts2.3.1 Preference2.3.2 Distance2.3.3 Size2.4 CoherenceCHAPTER 3 : PREFERENCES3.1 Introduction3.2 General preferential structures3.2.1 General minimal preferential structures3.2.2 Transitive minimal preferential structures3.2.3 One copy version3.2.4 A very short remark on X-logics3.3 Smooth minimal preferential structures3.3.1 Smooth minimal preferential structures with arbitrarily many copies3.3.2 Smooth and transitive minimal preferential structures3.4 The logical characterization of general and smooth preferential models3.4.1 Simplifications of the general transitive limit case3.5 A counterexample to the KLM-system3.5.1 The formal results3.6 A nonsmooth model of cumulativity3.6.1 The formal results3.7 Plausibility logic - problems without closure under finite union3.7.1 Introduction3.7.2 Completeness and incompleteness results for plausibility logic3.8 The role of copies in preferential structures3.9 A new approach to preferential structures3.9.1 Introduction3.9.2 Validity in traditional and in our preferential structures3.9.3 The disjoint union of models and the problem of multiple copies3.9.4 Representation in the finite case3.10 Ranked preferential structures3.10.1 Introduction3.10.2 The minimal variant3.10.3 The limit variant without copiesCHAPTER 4 : DISTANCES4.1 Introduction4.1.1 Theory revision4.1.2 Counterfactuals4.1.3 Summary4.2 Revision by distance4.2.1 Introduction4.2.2 The algebraic results4.2.3 The logical results4.2.4 There is no finite characterization4.2.5 The limit case4.3 Local and global metrics for the semantics of counterfactuals4.3.1 Introduction4.3.2 The resultsCHAPTER 5 : DEFINABILITY PRESERVATION5.1 Introduction5.1.1 The problem5.1.2 The remedy5.1.3 Basic definitions and results5.1.4 A remark on definability preservation and modal logic5.2 Preferential structures5.2.1 The algebraic results5.2.2 The logical results5.2.3 The general case and the limit version cannot be characterized5.3 Revision5.3.1 The algebraic result5.3.2 The logical resultCHAPTER 6 : SUMS6.1 Introduction6.1.1 The general situation and the Farkas algorithm6.1.2 Update by minimal sums6.1.3 Comments on "Belief revision with unreliable observations"6.1.4 "Between" and "behind"6.1.5 Summary6.2 The Farkas algorithm6.3 Representation for update by minimal sums6.3.1 Introduction6.3.2 An abstract result6.3.3 Representation6.3.4 There is no finite representation for our type of update possible6.4 Comments on "Belief revision with unreliable observations"6.4.1 Introduction6.4.2 A characterization of Markov systems (in the finite case)6.4.3 There is no finite representation possible6.5 "Between" and "Behind"6.5.1 There is no finite representation for "between" and "behind"CHAPTER 7 : SIZE7.1 Introduction7.1.1 The details7.2 Generalized quantifiers7.2.1 Introduction7.2.2 Results7.3 Comparison of three abstract coherent systems based on size7.3.1 Introduction7.3.2 Presentation of the three systems7.3.3 Comparison of the systems of Ben-David/Ben-Eliyahu and the author7.3.4 Comparison of the systems of Ben-David/Ben-Eliyahu and of Friedman/Halpern7.4 Theory revision based on model size7.4.1 Introduction7.4.2 ResultsCHAPTER 8 : INTEGRATION8.1 Introduction8.1.1 Rules or object language?8.1.2 Various levels of reasoning8.2 Reasoning types and concepts8.3 Formal aspects8.3.1 Classical modal logic8.3.2 Classical propositional operators have no unique interpretation8.3.3 Combining individual completeness resultsCHAPTER 9 : CONCLUSION AND OUTLOOKBibliography.Index.