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. bya 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 similarconstructions 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. Key features:· provides a coherent picture of several formalisms of nonmonotonic logics.· gives completeness and incompleteness results for many variants of preferential, distance based, and other semantics.· gives probably the first systematic investigation of definability preservation and its consequences.· gives new proof techniques for completeness results.· is centered on semantics
Author(s): Schlechta K.
Series: Studies in logic and practical reasoning 2
Publisher: Elsevier
Year: 2004
Language: English
Pages: 467
Title ......Page 2
Copyright ......Page 3
Foreword ......Page 4
Contents ......Page 8
1.1 The main topics of the book ......Page 20
1.1.1 Conceptual analysis ......Page 21
1.1.2 Generalized modal logic and integration ......Page 22
1.1.3 Formal results ......Page 26
1.1.4 The role of semantics ......Page 28
1.1.5 Various remarks ......Page 30
1.2 Historical remarks ......Page 31
1.4 Overview of the chapters ......Page 34
1.4.2 The formal part ( Chapters 3-7) ......Page 35
1.4.4 Problems, ideas, and techniques ......Page 39
1.5 Specific remarks on propositional logic ......Page 43
1.6.1 The algebraic part ......Page 45
1.6.2 The logical part ......Page 47
2.1 Introduction ......Page 56
2.2 Reasoning types ......Page 58
2.2.1 Traditional nonmonotonic logics ......Page 59
2.2.2 Prototypical and ideal cases ......Page 67
2.2.3 Extreme cases and interpolation ......Page 68
2.2.4 Clustering ......Page 69
2.2.5 Certainty ......Page 70
2.2.6 Quality of an answer, approximation, and complexity ......Page 71
2.2.7 Useful reasoning ......Page 73
2.2.8 Inheritance and argumentation ......Page 74
2.2.9 Dynamic systems ......Page 79
2.2.10 Theory revision ......Page 81
2.2.12 Counterfactual conditionals ......Page 88
2.3 Basic semantical concepts ......Page 89
2.3.1 Preference ......Page 90
2.3.2 Distance ......Page 100
2.3.3 Size ......Page 102
2.4 Coherence ......Page 111
3.1.1 General discussion ......Page 120
3.1.2 The basic results ......Page 128
3.2 General preferential structures Ill 3.2.1 General minimal preferential structures ......Page 132
3.2.2 Transitive minimal preferential structures ......Page 135
3.2.3 One copy version ......Page 140
3.2.4 A very short remark on X-logics ......Page 141
3.3.1 Smooth minimal preferential structures with arbi-trarily many copies ......Page 144
3.3.2 Smooth and transitive minimal preferential struc-tures ......Page 151
3.4 The logical characterization of general and smooth preferential models ......Page 157
3.4.1 Simplifications of the general transitive limit case ......Page 160
3.5 A counterexample to the KLM-system ......Page 165
3.5.1 The formal results ......Page 167
3.6.1 The formal results ......Page 170
3.7.1 Introduction ......Page 176
3.7.2 Completeness and incompleteness results for plausi-bility logic ......Page 181
3.8 The role of copies in preferential structures ......Page 193
3.9.1 Introduction ......Page 195
3.9.2 Validity in traditional and in our preferential struc-tures ......Page 200
3.9.3 The disjoint union of models and the problem of mul-tiple copies ......Page 202
3.9.4 Representation in the finite case ......Page 206
3.10.1 Introduction ......Page 210
3.10.2 The minimal variant ......Page 220
3.10.3 The limit variant without copies ......Page 230
4.1 Introduction ......Page 242
4.1.1 Theory revision ......Page 244
4.1.2 Counterfactuals ......Page 249
4.1.3 Summary ......Page 250
4.2.1 Introduction ......Page 251
4.2.2 The algebraic results ......Page 252
4.2.3 The logical results ......Page 267
4.2.4 There is no finite characterization ......Page 273
4.2.5 The limit case ......Page 277
4.3.1 Introduction ......Page 280
4.3.2 The results ......Page 282
5.1.1 The problem ......Page 290
5.1.2 The remedy ......Page 294
5.1.3 Basic definitions and results ......Page 298
5.1.4 A remark on definability preservation and modal logic ......Page 301
5.2.1 The algebraic results ......Page 303
5.2.2 The logical results ......Page 313
5.2.3 The general case and the limit version cannot be characterized ......Page 318
5.3.1 The algebraic result ......Page 331
5.3.2 The logical result ......Page 334
6.1 Introduction ......Page 338
6.1.1 The general situation and the Farkas algorithm ......Page 339
6.1.2 Update by minimal sums ......Page 340
6.1.4 "Between" and "behind" ......Page 344
6.1.5 Summary ......Page 345
6.2 The Farkas algorithm ......Page 346
6.3.1 Introduction ......Page 348
6.3.2 An abstract result ......Page 350
6.3.3 Representation ......Page 354
6.3.4 There is no finite representation for our type of up-date possible ......Page 361
6.4.1 Introduction ......Page 368
6.4.2 A characterization of Markov systems in the finite case ......Page 374
6.4.3 There is no finite representation possible ......Page 378
6.5.1 There is no finite representation for "between" and "behind" ......Page 380
7.1 Introduction ......Page 386
7.1.1 The details ......Page 387
7.2.1 Introduction ......Page 391
7.2.2 Results ......Page 392
7.3.1 Introduction ......Page 398
7.3.2 Presentation of the three systems ......Page 400
7.3.3 Comparison of the systems of Ben-David/Ben-Eliyahu and the author ......Page 405
7.3.4 Comparison of the systems of Ben-David/Ben-Eliyahu and of Friedman/Halpern ......Page 409
7.4.1 Introduction ......Page 415
7.4.2 Results ......Page 417
8.1 Introduction ......Page 430
8.1.1 Rules or object language? ......Page 432
8.1.2 Various levels of reasoning ......Page 433
8.2 Reasoning types and concepts ......Page 435
8.3.1 Classical modal logic ......Page 441
8.3.2 Classical propositional operators have no unique interpretation ......Page 443
8.3.3 Combining individual completeness results ......Page 448
9 Conclusion and outlook ......Page 452
Bibliography ......Page 454
Index ......Page 459
