Contents. Introduction. Part One: The History of Paraconsistent Logic

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1 Contents Introduction xix Part One: The History of Paraconsistent Logic I. First Historical Introduction: A Preliminary History of Paraconsistent and Dialethic Approaches G. Priest and R. Routley 3 1. Western thought until the fall of the Roman Empire: paraconsistent elements 5 2. Elements of paraconsistent thinking in Eastern philosophy Paraconsistent thought in Christendom The modern revival: paraconsistent approaches through idealism and common sense 21 5 Contemporary paraconsistent development and approaches A theory for contradictory objects: Meinong The overturning of conventional logical wisdom: Lukasiewicz The Russian forerunners: Vasil'ev and Bochvar An isolated figure in contemporary history: Wittgenstein The Polish continuation: Jaskowski The Latin American development: da Costa's theories The position of relevant logics and the contrasting attitudes of their proponents The Australian movement 55 Appendix: note on recent activity elsewhere 57 Notes 58 References 70 vii

2 II. An Outline of the History of (Logical) Dialectic G. Priest and R. Routley The origins of dialectic in Western philosophy The development of dialectic in Greek philosophy Dialectic in later Greek and Medieval philosophy Kant and Fichte Hegel Marx and the Marxists Summary and prospects 92 Notes 92 References 96 III. Aspects of the Historical Development of Paraconsistent Logic A. I. Arruda Introduction The origins of paraconsistent logic Jan Lukasiewicz ( ) Nikolaj A. Vasil'ev ( ) Stanislaw Jaskowski ( ) Newton C. A. da Costa The present-day development of paraconsistent logic Paraconsistent logic in Australia Paraconsistent logic in Brazil The elaboration of paraconsistent logics other than da Costa's system which would be adequate for the construction of paraconsistent set theories The problem of constructing paraconsistent set theories with strong forms of the axiom schema of separation Discussive logic and dialectical logic The construction of different types of paraconsistent logic as the possible formalizations of the imaginary logic of Vasil'ev and some kinds of logic of vagueness The generalization of the classical method of valuations to paraconsistent logic, and the use of it in order to prove completeness theorems for these logics and to obtain decision methods for them 114 viii

3 3.2.6 The development of higher-order logics corresponding to da Costa's systems C n The algebraization of the systems C n Many-valued paraconsistent logic The relation between fuzzy logic and paraconsistent logic Paraconsistent model theory Paraconsistent logic in Poland Paraconsistent logic in the United States Paraconsistent logic in Argentina Paraconsistent logic in Belgium Paraconsistent logic in Ecuador Paraconsistent logic in Italy Paraconsistent logic in Peru 127 References 127 IV. Classical Logic aufgehoben G. Priest Nineteenth century logic Classical logic: the progressive phase Classical logic: the degenerating phase The shape of things to come The solution of classical anomalies Preservation of problem-solving ability Providing the basis for a new and fruitful research program Conclusion 145 Notes 145 References 147 Part Two: Systems of Paraconsistent Logic V. Systems of Paraconsistent Logic G. Priest and R. Routley Paraconsistency: characterization and motivation The proof theoretic reason The semantical reason 153 ix

4 2. Approaches to paraconsistent logical theory: Initial systemic taxonomy of paraconsistent logics; zero degree formulas Non-adjunctive systems: Jaskowski's system Positive-plus systems: da Costa's main system The relevant approach Approaches to paraconsistent logical theory: implication Non-adjunctive systems, such as Jaskowski's system Positive-plus systems, such as da Costa's main system The relevant approach 177 Notes 180 References 184 VI. Dynamic Dialectical Logics D. Batens Introduction Regular paraconsistent extensional propositional logics Enters dynamics The dynamic dialectical logic DPI* The (decent) dynamic dialectical logic DPI Some metatheory Theorems, axiomatizations, elegance Semantics Some other dynamic dialectical logics of the type of DPI Final comments and some open problems 213 Notes 216 References 217 VII. Paraconsistent and Combinatory Logic M. W. Bunder Introduction Combinatory Logic Paraconsistent Logics 220 x

5 4. 5. Paraconsistent Systems based on "n 0 X = X => HHJ 0 Paraconsistent Systems based on ~i X = X => EJI References VIII. Problems of Modal and Discussive Logics J. Kotas and N. C. A. da Costa Introduction Some modal and discussive systems Structural completeness Modal systems with the DSF-property Generalized Jaskowski logics Strong implication Higher-order discussive logic Discussive set theory References IX. Abelian Logic (from A to Z) R. K. Meyer and J. K. Slaney Notes References , 2. 3, 4 5 X. Paraconsistency and C! C. Mortensen Introduction C, and some related systems Does Q have a reasonable conditional and biconditional? Some extensions of Q More about extending C, Notes References xi

6 XI. On Detonating P. K. Schotch and R. E. Jennings Introduction A general theory Some variations Conclusion 325 Notes 326 References 327 XII. Consistency, Completeness and Negation D. Vakarelov Distributive logics Syntax Theories, co-theories and prime theories A characterization of distributive logics Semantics Negation and consistency Regular negation Some familiar laws of negation Semantics for some regular logics Standard logics Paraconsistency Filtrations Negation and completeness Co-regular negation Semantics for some co-regular logics Paraconsistency and paracompleteness Concluding remarks and open problems 361 References 362 Part Three: Applications of Paraconsistent Logic XIII. Applications of Paraconsistent Logic G. Priest and R. Routley Introduction: the variety and types of applications 367 xii

7 2. Historical and extant inconsistent theories Philosophy and theology Natural and social sciences Logic and mathematics A more detailed look at some of these theories Naive semantics Naive property theory, set theory and category theory The infinitesimal calculus Quantum mechanics Paraconsistency and wider logical notions Reason, inference, fallacies and the inconsistent Extending paraconsistent logic by intensional functors: modal and tense operators Moral dilemmas: deontic logic Belief systems: doxastic logic Probability and inductive reasoning Information content and data processing Vagueness Conclusion 390 Notes 390 References 392 XIV. Toward an Antinomic Mathematics F. G. Asenjo Rationale A first calculus of antinomies A second calculus of antinomies "Logic of antinomies" Parenthesis: is dialectic logic necessarily antinomic? The comprehension axiom in antinomic logic An antinomic number theory Equality as an antinomic predicate Antinomies and intuitionism Open problems Forecast 412 xiii

8 Notes References XV. The Non-Triviality of Extensional Dialectical Set Theory R. T. Brady and R. Routley Scene setting and systemic preliminaries: DKQ and DST; RM3Q and EDST The modelling for the extensional general comprehension axiom The validity of the extensionality axiom in the structure M a 426 Notes 435 References 435 XVI. The Non-Triviality of Dialectical Set Theory R. T. Brady The logic DSQ and the dialectical set theory DST The determination of the model structure MD for dialectical set theory DST The soundness of DSQ with respect to MD The validity of the GCA in MD and invalidity of some formulae in MD The validity of the EA in MD Some concluding remarks The alternative definition for A-» B in the construction of(s) The restriction of connectives in the GCA The addition of "classical" provability, The incompleteness of the dialectical set theory DST' The total number of logical values 470 xiv Notes 470 References 471 XVII. RWX is not Curry Paraconsistent J. K. Slaney 472 Notes 479 References 480

9 Part Four: The Philosophical Significance of Paraconsistency XVIII. The Philosophical Significance and Inevitability of Paraconsistency G. Priest and R. Routley Introduction Reasons for paraconsistency There are natural inconsistent but non-trivial theories Inconsistent bodies of law and the like Inconsistent theories in philosophy and the history of ideas Inconsistent theories in science and the history of science The matter of non-triviality Naive set theory Naive semantics The truth of some contradictions Multicriterial terms Logical and semantical paradoxes Paradoxes in set theory Paradoxes in semantics Ramifications and consequences of paraconsistency, and further reasons for paraconsistency Pragmatics Assertion: the question of content Criticism and the change of belief Consistency and the metatheory Metaphysics Meinong's theory of objects Dialectic Contradictions in knowledge and the corpus of science Contradictions in the natural world The philosophy of mathematics Godel's first incompleteness theorem Logicism Hilbert's program Conclusion: the ideology of consistency 528 Notes 529 References 537 xv

10 XIX. Wittgenstein and Paraconsistency L. Goldstein Introduction Contradiction as castration Early non-classicism Logical proof and proof in logic Transition Calculus and language-game A new perspective Wittgenstein as dialethist Postscript-conventionalism 557 Notes and References 559 XX. Verum et ens convertuntur The Identity between Truth and Existence within the Framework of a Contradictorial Modal Set Theory L. Pena 563 Introduction Methodological and preliminary remarks Identity truth-existence in ordinary speech The identity between any thing's existence and the thing itself Existence as a redundance predicate Degrees of reality and the need for an infinite-valued logic Non-existence statements Naming and stating Existence = the absolutely real thing Existence and the existential quantifier Why knowing all that exists is being omniscient 606 xvi

11 11. Relations, combinatory approaches, and existence as the relation of being belonged to 608 References 611 XXI. Reductio ad absurdum et modus tollendo ponens G. Priest Introduction Logic as an organon of criticism Plato's Euthypro: an illustration The general unacceptability of contradictions and the particular acceptability of some An important point Reductio ad absurdum as proof Modus tollendo ponens Conclusion: Quasi-valid inferences 625 Note 626 References 626 XXII. Paraconsistent Logic: Some Philosophical Issues F. Miro Quesada Introduction Paraconsistency and heterodoxy The conception of heterodoxy Semantics Bivalent Semantics Other types of semantics Dialectic Paraconsistency and dialectic Dialectic, paraconsistency and relevance Applications Thetic logic Athetic logic 639 xvii

12 6. Paraconsistency and the rationality of logic Classical logic, heterodoxy and rationality Considerations as to the rationality of logic 642 Notes 647 References 651 XXIII. Moral Dilemmas and the Logic of Deontic Notions R. Routley and V. Plumwood The paradoxes of deontic logic and the consistency and modal requirements The simple relevant resolution versus classical epicycling Documenting moral dilemmas Paraconsistent deontic logic; the relevant development Resultant impact on other puzzles and paradoxes of deontic theory of the relevant/paraconsistent shift Upsetting the moral consensus: unrealizable obligations, and inevitable wrongs Resolving moral dilemmas, so far as can be or need be done 678 Notes 683 References 688 Bibliography Index List of Contributors