Paraconsistent Logic: Consistency, Contradiction and Negation (Logic, Epistemology, and the Unity of Science Book 40)

This book is the first in the field of paraconsistency to offer a comprehensive overview of the subject, including connections to other logics and applications in information processing, linguistics, reasoning and argumentation, and philosophy of science. It is recommended reading for anyone interested in the question of reasoning and argumentation in the presence of contradictions, in semantics, in the paradoxes of set theory and in the puzzling properties of negation in logic programming. Paraconsistent logic comprises a major logical theory and offers the broadest possible perspective on the debate of negation in logic and philosophy. It is a powerful tool for reasoning under contradictoriness as it investigates logic systems in which contradictory information does not lead to arbitrary conclusions. Reasoning under contradictions constitutes one of most important and creative achievements in contemporary logic, with deep roots in philosophical questions involving negation and consistency This book offers an invaluable introduction to a topic of central importance in logic and philosophy. It discusses (i) the history of paraconsistent logic; (ii) language, negation, contradiction, consistency and inconsistency; (iii) logics of formal inconsistency (LFIs) and the main paraconsistent propositional systems; (iv) many-valued companions, possible-translations semantics and non-deterministic semantics; (v) paraconsistent modal logics; (vi) first-order paraconsistent logics; (vii) applications to information processing, databases and quantum computation; and (viii) applications to deontic paradoxes, connections to Eastern thought and to dialogical reasoning. Preface 7 Book’s Content: A Road Map 11 References 18 Acknowledgments 19 Contents 20 1 Contradiction and (in)Consistency 24 1.1 Introduction 24 1.2 On the Philosophy of the Logics of Formal Inconsistency 27 1.3 A Historical Sketch: The Forerunners of the Logics of Formal Inconsistency 33 1.4 Paraconsistency and the Nature of Logic 34 1.5 Paraconsistency and the Nature of Contradictions 38 1.6 Contradiction, Consistency and Negation 41 1.6.1 On Contradiction 42 1.6.2 On Consistency 43 1.6.3 On Negation 44 1.7 Varieties of Paraconsistency Involvement 47 References 49 2 A Basic Logic of Formal Inconsistency: mbC 52 2.1 Introducing mbC 52 2.2 A Valuation Semantics for mbC 58 2.3 Applications of mbC-Valuations 62 2.4 Recovering Classical Logic Inside mbC 66 2.5 Reintroducing mbC as an Expansion of CPL 73 2.5.1 The New Presentation mbC of mbC 74 2.5.2 Valuation Semantics for mbC 76 2.5.3 Equivalence Between mbC and mbC 78 References 84 3 Some Extensions of mbC 86 3.1 A Wider Form of Truth-Functionality for Consistency 86 3.2 A Hidden Consistency Operator 91 3.3 Consistency and Inconsistency as Derived Connectives 94 3.4 Some Conceptual Differences Between mbC and mbC 114 3.5 Inconsistency Operators and Double-Negations 118 3.6 Propagating Consistency 127 3.7 da Costa's Hierarchy and Consistency Propagation 134 3.8 A Stronger Consistency Propagation 138 References 142 4 Matrices and Algebraizability 144 4.1 Logical Matrices 144 4.2 Uncharacterizability by Finite Matrices 145 4.3 The Problem of Algebraizability of LFIs 152 4.4 Some 3-Valued LFIs 159 4.4.1 Halldén's Logic of Nonsense (1949) 159 4.4.2 Segerberg's Logic of Nonsense (1965) 161 4.4.3 da Costa and D'Ottaviano's Logic J3 (1970) 163 4.4.4 Sette's Logic P1 (1973) 167 4.4.5 Asenjo-Priest's Logic LP (1966--1979) 172 4.4.6 Ciore and Other Related Systems 174 4.4.7 LFI1, MPT and J3 181 References 191 5 LFIs Based on Other Logics 194 5.1 LFIs Based on Positive Intuitionistic Logic 194 5.1.1 Basic Features of Positive Intuitionistic Logic 194 5.1.2 Johansson's Minimal Logic 198 5.1.3 Nelson's Paraconsistent Logic N4 202 5.1.4 An Intuitionistic Version of mbC 210 5.2 LFIs Based on Fuzzy Logics 214 5.2.1 Preliminaries on MFL 214 5.2.2 Fuzzy Logics with a Consistency Operator 220 5.2.3 Propagation of Consistency and DAT 228 5.2.4 Fuzzy Logics with an Inconsistency Operator 231 5.3 A Modal LFI Based on Belnap and Dunn's Logic mathcalBD 234 5.3.1 The Logic M4m of Tetravalent Modal Algebras 236 5.3.2 M4m as an LFI 241 5.3.3 M4m as a dC-System 245 5.3.4 The Contrapositive Implication 247 5.3.5 A Hilbert-Style Axiomatization of M4mc 249 5.4 Paraconsistent Modalities, Consistency and Determinedness 252 References 256 6 Semantics of Non-deterministic Character for LFIs 260 6.1 Fidel Structures for mbC 261 6.2 Fidel Structures for Some Extensions of mbC 265 6.3 Non-deterministic Matrices 274 6.4 Swap Structures for mbC 276 6.5 Swap Structures for Some Extensions of mbC 283 6.6 Axiom (cl) and Uncharacterizability by Finite Nmatrices 295 6.7 Some Remarks on Fidel Structures and Swap Structures 302 6.8 The Possible-Translations Semantics 303 6.8.1 Possible-Translations Semantics for Some LFIs 303 6.8.2 A 3-Valued Possible-Translations Semantics for Cila 305 6.8.3 Some Remarks on Possible-Translations Semantics 311 References 313 7 First-Order LFIs 315 7.1 The Logic QmbC 316 7.2 Basic Properties of QmbC 318 7.3 Tarskian Paraconsistent Structures 324 7.4 Soundness Theorem for QmbC 330 7.5 Completeness Theorem for QmbC 332 7.5.1 Henkin Theories 333 7.5.2 Canonical Interpretations 334 7.6 Compactness and Lowenhëim-Skolem Theorems for QmbC 338 7.7 QmbC with Equality 340 7.8 First-Order Characterization of Other Quantified LFIs 344 7.9 First-Order LFI1 and the Logic of Quasi-truth 346 7.9.1 Semantics of Partial Structures 346 7.9.2 The Logic QLFI1° 354 7.10 First-Order P1 and Partial Structures 357 References 365 8 Paraconsistent Set Theory 366 8.1 Antinomic Sets and Paraconsistency 367 8.2 LFIs Predicating on Consistency 370 8.3 Some Extensions of ZFmbC 375 8.4 Inconsistent Sets and Proper Classes 381 8.5 On Models 384 References 386 9 Paraconsistency and Philosophy of Science: Foundations and Perspectives 389 9.1 An Epistemological Understanding of Paraconsistency, and Its Significance for Science 389 9.2 Consistency and Contradiction in Scientific Theories 391 9.2.1 The Heritage of Kant 391 9.2.2 Some Historical Examples 393 9.2.3 The Beginning of Quantum Theory and Paraconsistency 394 9.2.4 Mercury's Orbit and a Non-existent Planet 394 9.2.5 Contradictions in Phlogiston, the Imponderable 396 9.2.6 The Special Theory of Relativity 397 9.2.7 Mathematics, and the Meaning of Objects that Mean Nothing 399 9.3 Quasi-truth and the Reconciliation of Science and Rationality 402 9.4 An Evidence-Based Approach to Paraconsistency 404 9.5 Summing Up 405 References 407 Index 410 Index of Logic Systems 416 Front Matter....Pages i-xxiv Contradiction and (in)Consistency....Pages 1-28 A Basic Logic of Formal Inconsistency: mbC ....Pages 29-62 Some Extensions of mbC....Pages 63-120 Matrices and Algebraizability....Pages 121-170 LFIs Based on Other Logics....Pages 171-236 Semantics of Non-deterministic Character for LFIs....Pages 237-291 First-Order LFIs....Pages 293-343 Paraconsistent Set Theory....Pages 345-367 Paraconsistency and Philosophy of Science: Foundations and Perspectives....Pages 369-389 Back Matter....Pages 391-398