The Theory of Inconsistency has a long lineage, stretching back to Herakleitos, Hegel and Marx. In the late twentieth-century, it was placed on a rigorous footing with the discovery of paraconsistent logic and inconsistent mathematics. Paraconsistent logics, many of which are now known, are "inconsistency tolerant", that is, they lack the rule of Boolean logic that a contradiction implies every proposition. When this constricting rule was seen to be arbitrary, inconsistent mathematical structures were free to be described. This book continues the development of inconsistent mathematics by taking up inconsistent geometry, hitherto largely undeveloped. It has two main goals. First, various geometrical structures are shown to deliver models for paraconsistent logics. Second, the "impossible pictures" of Reutersvaard, Escher, the Penroses and others are addressed. The idea is to derive inconsistent mathematical descriptions of the content of impossible pictures, so as to explain rigorously how they can be impossible and yet classifiable into several basic types. The book will be of interest to logicians, mathematicians, philosophers, psychologists, cognitive scientists, and artists interested in impossible images. It contains a gallery of previously-unseen coloured images, which illustrates the possibilities available in representing impossible geometrical shapes. Chris Mortensen is Emeritus Professor of Philosophy at the University of Adelaide. He is the author of Inconsistent Mathrmatics (Kluwer 1995), and many articles in the Theory of Inconsistency.
Author(s): Chris Mortensen
Series: Studies in Logic 27
Publisher: College Publications
Year: 2010
Language: english
Commentary: better than the first one
Pages: 173
Cover......Page 1
Title page......Page 4
Preface......Page 9
I Paraconsistent Logics and Geometrical Theories......Page 12
1.1 Introduction......Page 14
1.2 Natural Logic is Paraconsistent......Page 16
2.1 Introduction......Page 22
2.2 Separation Principles......Page 24
2.3 Conditions on Identity and Disidentity......Page 27
3.1 Introduction......Page 30
3.2 Lukasiewicz Logic......Page 31
3.3 Extending the Value Space......Page 35
3.4 Abelian Logic......Page 38
4.2 Ulam Games......Page 42
4.3 Two Example Games with One Lie......Page 43
4.4 Logics......Page 46
4.5 The General Case L>1......Page 47
4.6 Paraconsistency......Page 49
5.1 The Idea of Symmetry......Page 52
5.2 An Inconsistent Approach......Page 53
5.3 A FormaI Theory......Page 54
5.4 Recovering the Consistent Theory......Page 57
5.6 The Dihedral Group......Page 59
6.1 General Algebras......Page 62
6.2 Groups......Page 64
7.2 Simplices......Page 68
7.3 The Routley Functor on Groups......Page 72
7.4 An Application to Homology......Page 74
II Inconsistent Images......Page 78
8.1 Introduction......Page 80
8.2 Consistent Mathematical Approaches......Page 82
8.3 Conclusion......Page 85
9.1 Introduction......Page 88
9.2 Level 1 Analysis: Two Dimensions......Page 91
9.3 Level 2 Analysis: Three Dimensions......Page 95
9.4 Level 3: Faces......Page 97
10.2 Primary and Secondary Matrices......Page 100
10.4 Nullity......Page 103
10.5 The Structure of the Null Space......Page 104
10.6 The Unit Equation......Page 106
10.7 Conclusion......Page 107
11.1 Introduction......Page 109
11.2 Elementary Operations on Matrices......Page 110
11.3 The Routley Star on Matrices......Page 111
11.4 Transposes......Page 114
11.6 Conclusion......Page 115
12.1 Introduction......Page 117
12.2 Chained Neckers......Page 118
12.3 Degree of Inconsistency......Page 121
12.4 Elementary Row and Colunm Operations......Page 123
12.5 The Routley Functor......Page 125
12.7 Conclusion......Page 126
13.1 Introduction......Page 128
13.2 The Triangle is a Paradox......Page 129
13.3 The Triangle is an Occlusion Paradox......Page 130
13.4 From Cognitive Science to Logic......Page 134
13.5 From Logic to Mathematics......Page 136
13.6 Conclusion......Page 138
14.1 Introduction......Page 140
14.2 Logic......Page 141
14.4 Mathematics......Page 142
14.5 Conclusion......Page 145
15.1 Introduction......Page 146
15.2 Logic......Page 147
15.3 Cognition......Page 148
15.4 Mathematics......Page 149
15.5 Conclusion......Page 151
Bibliography......Page 153
A Gallery of Inconsistent Images......Page 157
B Mostly B&W Images......Page 167
Index......Page 171
