On Preserving

Paraconsistent logic is a theory of reasoning in philosophy that studies inconsistent data. The discipline has several different schools of thought, including preservationism, which responds to the problems that arise when human beings continue to reason when faced with inconsistent data. On Preserving is the first complete account of the Preservationist School, which developed in Canada out of the early work of Raymond Jennings, Peter Schotch, and their students.Assembling the previously scattered works of the Preservationist School, this collection contains all of the most significant works on the basic theory of the preservationist approach to paraconsistent logic. With essays both written and rewritten specifically for this volume, the contributors cover topics that include the motivation for the preservationist approach, as well as more technical results of their research. Concise and unified, On Preserving is the ideal introduction to a distinct philosophical field.

Acknowledgments	p. ix
Introduction to the Essays	p. 3
The Origins of Preservationism	p. 3
How Paraconsistent Inference Fell out of Modal Semantics	p. 7
The Concept of Level	p. 11
Preservation	p. 11
The Essays	p. 13
Paraconsistency: Who Needs It?	p. 17
Introduction	p. 17
The Strange Case of C.I. Lewis	p. 20
The Inconsistency of Belief	p. 23
Inconsistency and Ethics	p. 27
Summing Up	p. 30
Weakly Additive Algebras and a Completeness Problem	p. 33
Introduction and Brief History	p. 33
Weakly Additive Operators	p. 37
A Further Generalization	p. 40
Duality Theory for Weakly Additive Operators	p. 41
A Dualization of Neighbourhood Structures	p. 49
Introduction	p. 49
Hyperframes	p. 50
Normal Hyperframes	p. 54
Weakly Aggregative Modal Logic	p. 57
Polyadic Modal Logics and Their Monadic Fragments	p. 61
Introduction	p. 61
Polyadic Modal Languages and Multi-ary Relational Frames	p. 64
Normal Polyadic Modal Logics	p. 67
Frame Definability	p. 73
Soundness and Completeness	p. 75
Diagonal Fragments of Normal Polyadic Modal Logics	p. 81
Preserving What?	p. 85
Introduction	p. 85
Making a Few Things Precise	p. 88
What's Wrong with This Picture?	p. 92
Speak of the Level	p. 93
Level Preservation	p. 96
Yes, But Is It Inference?	p. 101
Forcing in Comparison with Other Level-Preserving Relations	p. 103
Preserving Logical Structure	p. 105
Introduction	p. 105
Definitions	p. 108
Substitutions	p. 118
Algebraic Concerns	p. 121
Denials	p. 125
Forcing and Preservationism	p. 127
Structural Rules and Compactness	p. 129
First-Order Logic and Forcing	p. 134
Conclusion	p. 142
Representation of Forcing	p. 145
Introduction	p. 145
Background	p. 146
Representation	p. 148
The Representation Theorem	p. 151
More Definitions and Motivation	p. 152
	p. 156
	p. 158
The Representation of Forcing	p. 158
Forcing and Practical Inference	p. 161
Introduction	p. 161
Dirty Hands	p. 164
¿-Forcing	p. 165
A-Forcing	p. 170
Wrap-up	p. 172
Ambiguity Games and Preserving Ambiguity Measures	p. 175
Ambiguity	p. 175
The Logic of Paradox	p. 177
Restoring Symmetry	p. 180
Ambiguity and Quantification	p. 186
Echoes of Supervaluation	p. 187
Final Remarks on Preservation	p. 188
Nomenclature	p. 189
References	p. 195
Index	p. 199
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