Logic, Epistemology, and the Unity of Science 40 Walter Carnielli Marcelo Esteban Coniglio Paraconsistent Logic: Consistency, Contradiction and Negation Logic, Epistemology, and the Unity of Science Volume 40 Series editors Shahid Rahman, University of Lille III, France John Symons, University of Texas at El Paso, USA Editorial Board Jean Paul van Bendegem, Free University of Brussels, Belgium Johan van Benthem, University of Amsterdam, The Netherlands Jacques Dubucs, CNRS/Paris IV, France Anne Fagot-Largeault, Collège de France, France Göran Sundholm, Universiteit Leiden, The Netherlands Bas van Fraassen, Princeton University, USA Dov Gabbay, King’s College London, UK Jaakko Hintikka, Boston University, USA Karel Lambert, University of California, Irvine, USA Graham Priest, University of Melbourne, Australia Gabriel Sandu, University of Helsinki, Finland Heinrich Wansing, Ruhr-University Bochum, Germany Timothy Williamson, Oxford University, UK Logic, Epistemology, and the Unity of Science aims to reconsider the question of the unity of science in light of recent developments in logic At present, no single logical, semantical or methodological framework dominates the philosophy of science However, the editors of this series believe that formal techniques like, for example, independence friendly logic, dialogical logics, multimodal logics, game theoretic semantics and linear logics, have the potential to cast new light on basic issues in the discussion of the unity of science This series provides a venue where philosophers and logicians can apply specific technical insights to fundamental philosophical problems While the series is open to a wide variety of perspectives, including the study and analysis of argumentation and the critical discussion of the relationship between logic and the philosophy of science, the aim is to provide an integrated picture of the scientific enterprise in all its diversity More information about this series at http://www.springer.com/series/6936 Walter Carnielli Marcelo Esteban Coniglio • Paraconsistent Logic: Consistency, Contradiction and Negation 123 Walter Carnielli Department of Philosophy and Centre for Logic, Epistemology and the History of Science (CLE) University of Campinas (UNICAMP) Campinas, São Paulo Brazil Marcelo Esteban Coniglio Department of Philosophy and Centre for Logic, Epistemology and the History of Science (CLE) University of Campinas (UNICAMP) Campinas, São Paulo Brazil ISSN 2214-9775 ISSN 2214-9783 (electronic) Logic, Epistemology, and the Unity of Science ISBN 978-3-319-33203-1 ISBN 978-3-319-33205-5 (eBook) DOI 10.1007/978-3-319-33205-5 Library of Congress Control Number: 2016936981 © Springer International Publishing Switzerland 2016 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland To absent friends, and to Elias Alves, in memoriam To our children: Bá, Juju, Matheus, Paolo, Gabriela, Vittorio… and to the kids we would have had, and to Juli and Tati Sine qua non Campinas, February 29, 2016 Preface I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible Innity is merely a faỗon de parler, the real meaning being a limit which certain ratios approach indefinitely near, while others are permitted to increase without restriction C.F Gauss, Brief an Schumacher (1831); Werke 8, 216 (1831) In a letter to the astronomer H.C Schumacher in 1831, Gauss was rebuking mathematicians for their use of the infinite as a number, and even for their use of the symbol for the infinite It would be difficult to sustain that kind of finitism, regardless of any epistemological considerations: a good part of mathematics simply cannot survive with only the potential infinite The reaction against the infinite, as well as against complex or imaginary numbers, and against negative numbers before, are interesting examples of the difficulties faced, even by great minds, in accepting certain abstractions Aristotle in Chaps 4–8 of Book III of Physics argued against the actual infinite, advocating for the potential infinite His idea was that natural numbers could never be conceived as a whole Euclid in a certain sense never proved that there exist infinitely many prime numbers What was actually stated in Proposition 20 of Book IX, carefully avoiding the term infinite, was that “prime numbers are more than any previously thought (total) number of primes”, which agrees with his tradition It was only in the nineteenth century that G Cantor dispelled all those accepted views by showing that an infinite set can be treated as a totality, as a full-fledged mathematical object with honorable properties, no less than the natural numbers Imaginary numbers were introduced to mathematics in the sixteenth century (through Girolamo Cardano, though others had already used them in different guises) These numbers caused an embarrassment among mathematicians for centuries, since they faced astonishing difficulties in accepting an extension of the concept of number, especially in light of the problem of computing the square root of −1 Only after the fundamental works of L Euler and Gauss did the complex vii viii Preface numbers rid themselves of the label “imaginary” given them by Descartes in 1637, and even then not without difficulty The mathematics of the infinite and of complex numbers, and all they represent in contemporary science, are triumphant cases of amplified concepts, but are not the only ones A notable case of expansion of concepts, with deep implications for the development of contemporary logic, can be traced back to Frege and his famous article of 1891, Funktion und Begriff (see [1]) In this seminal paper, Frege recalls how the meaning of the term ‘function’ has changed in the history of mathematics, and how the mathematical operations used to define functions have been extended by, as he says, ‘the progress of science’: basically, through passages (or transitions) to the limit, as in the process of defining a new function y0 ¼ f xị from a function y ẳ f xị (provided that the limits involved in the calculus exist), and through accepting complex numbers in domains and images of functions Starting from this point, Frege goes further into adding expressions that now we call predicates, such as ‘=’, ‘’ Leaving aside his philosophical motivations for seeing arithmetic as a “further development of logic”, what Frege started was a real revolution, that made possible the development of quantifiers and an unprecedented unification of propositional and predicate logic into a far more powerful system than any that preceded it Not only could the truth-values, True and False, be taken as outputs of a function, but any object whatsoever could be similarly treated To rephrase an example from Frege himself, if we suppose ‘the capital of x’ expresses a function, of which ‘the German Empire’ is the argument, Berlin is returned as the value of the function In this way, Frege’s system could represent non-mathematical thoughts and predications, and founded the basis of the modern predicate calculus Frege’s idea of defining an independent notion of ‘concept’ as a function which maps every argument to one of the truth-values True or False was instrumental in the development of a strict understanding of the notions of ‘proof’, ‘derivation’, and ‘semantics’ as parts of the same logic mechanism Regarding ‘concept’ as a wide and independent notion based on an amplification of the idea of function was an essential step for Frege’s fundamental break between the older Aristotelian tradition and the contemporary approach to logic Paraconsistency is the study of logical systems in which the presence of a contradiction does not imply triviality, that is, logical systems with a non-explosive negation : such that a pair of propositions A and :A does not (always) trivialize the system However, it is not only the syntactic and semantic properties of these systems that are worth studying Some questions arise that are perennial philosophical problems The question of the nature of the contradictions allowed by paraconsistent logics has been a focus of debate on the philosophical significance of paraconsistency Although this book is primarily focused on the logico-mathematical development of paraconsistency, the technical results emphasized here aim to help, and hopefully to guide, the study of some of those philosophical problems Preface ix Paraconsistent logics are able to deal with contradictory scenarios, avoiding triviality by means of the rejection of the Principle of Explosion, in the sense that these theories not trivialize in the presence of (at least some) contradictory sentences Different from traditional logic, in paraconsistent logics triviality is not necessarily connected to contradictoriness; in intuitive terms (a more formal account in given in Sect 1.2) the situation could be described by the pictorial equation: contradictoriness þ consistency ¼ triviality The Logics of Formal Inconsistency, from now on LFIs, introduced in [2] and additionally developed in [3], are a family of paraconsistent logics that encompasses a great number of paraconsistent systems, including the majority of systems developed within the Brazilian tradition An important characteristic of LFIs is that they are endowed with linguistic resources that permit to express the notion of consistency of sentences inside the object language by using a sentential unary connective referred to as ‘circle’: A meaning A is consistent Explosion in the presence of contradictions does not hold in LFIs, as much as in any other paraconsistent logic But LFIs are so designed that some contradictions will cause deductive explosion: consistent contradictions lead to triviality–intuitively, one can understand the notion of a ‘consistent contradiction’ as a contradiction involving well-established facts, or involving propositions that have conclusive favorable evidence In this sense, LFIs are logics that permit one to separate the sentences for which explosion hold, from those for which explosion does not hold It is not difficult to see that, in this way, reasoning under LFIs extend and expand the reasoning under classical logic: although LFIs are technically subsystems of classical logic, classical logic can be identified with that portion of LFIs that deals with ‘consistent contradictions’ Therefore LFIs subsume classical reasoning This point will be explained in more detail in Sect 1.2 We may say that a first step in paraconsistency is the distinction between triviality and contradictoriness But there is a second step, namely, the distinction between consistency and non-contradictoriness In LFIs the consistency connective is not only primitive, but it is also not necessarily equivalent to non-contradiction This is the most distinguishing feature of the logics of formal inconsistency Once we break up the equivalence between A and :ðA ^ :AÞ, some quite interesting developments become available Indeed, A may express notions of consistency independent from freedom from contradiction The most important conceptual distinction between LFIs and traditional logic is that LFIs start from the principle that assertions about the world can be divided into two categories: consistent sentences and non-consistent sentences Consistent propositions are subjected to classical logic, and consequently a theory T that contains a pair of contradictory sentences A; :A explodes only if A is taken to be a consistent sentence, linguistically marked as A (or :A) This is the only distinction between LFIs and classical logic, albeit with far-reaching consequences: x Preface classical logic in this form is augmented, in such a way that in most cases an LFI encodes classical logic The concept of LFIs generalizes and extends the famous hierarchy of C-systems introduced in [4] and popularized by hundreds of papers At the same time, LFIs expand the classical logical stance, and consequently the majority of the traditional concepts and methods of classical logic, propositional or quantified (and even higher-order), can be adapted, with careful attention to detail Since, as much as intuitionistic logic, LFIs are more of an epistemic nature, rather than of an ontological, there is no point in advocating the replacement of classical logic with paraconsistent logic Because LFIs extend the classical stance, the analogy with transfinite ordinal numbers and with complex numbers is compelling: in such cases, there is no rejection of what has come before, but a refinement of it It is not infrequent that an argument as of the skeptics, such as that given by Sextus Empiricus1 against the sophists, is trumpeted against the need of paraconsistent logic, in science or reasoning in general: [If an argument] leads to what is inadmissible, it is not we that ought to yield hasty assent to the absurdity because of its plausibility, but it is they that ought to abstain from the argument which constrains them to assent to absurdities, if they really choose to seek truth, as they profess, rather than drivel like children Thus, suppose there were a road leading up to a chasm, we not push ourselves into the chasm just because there is a road leading to it but we avoid the road because of the chasm; so, in the same way, if there should be an argument which leads us to a confessedly absurd conclusion, we shall not assent to the absurdity just because of the argument but avoid the argument because of the absurdity So whenever such an argument is propounded to us we shall suspend judgement regarding each premiss, and when finally the whole argument is propounded we shall draw what conclusions we approve This argument, however, if it is not against the use of any logic, is indeed favorable to the kind of paraconsistency represented by LFIs The notion of consistency—symbolized as when applied to propositions—actually increases our wisdom: it does not stop one to jump into the chasm, but rather marks out the dangerous roads and, precisely, helps avoid such roads because of the chasm! The idea that consistency can be taken as a primitive, independent notion, and be axiomatized for the good profit of logic is a new idea, which permits one to separate not only the notion of contradiction from the notion of deductive triviality, which is true of all paraconsistent logics, but also the notion of inconsistency from the notion of contradiction—as well as consistency from non-contradiction This refined idea of consistency has great potential, as we shall see in detail in this book, as unanticipated as the possibilities that imaginary numbers, completed infinite, and Frege’s idealization of a ‘concept’ as a function mapping arguments to one of the truth-values represented in mathematics, logic and philosophy The rest of the book will speak for itself Sextus Empiricus, Outlines of Pyrrhonism, LCL 273: 318–319 http://www.loebclassics.com/ view/sextus_empiricus-outlines_pyrrhonism/1933/pb_LCL273.3.xml 9.3 Quasi-truth and the Reconciliation of Science and Rationality 383 true If this view seems appropriate for an anti-realist view of natural sciences, it is debatable whether it is also good for mathematics, logic, statistics, and other formal sciences, such as theoretical computer science, information theory, game theory, theoretical linguistics and cognitive science Nothing, however, in principle impeaches a partial-truth account of such disciplines One of the main problems concerning scientific rationality is to try to understand how science develops, and how theories are selected and substituted in the long term, as put in [32]: How to entertain with the best rational attitude the periods when contradictions and apparently irreconcilable opposition between theories coexist? This problem involves two main interconnected issues according to [32]: (a) How should we rationally understand theory change in science? In particular, how should we accommodate, in rational terms, the apparent lack of a cumulative development in scientific development? What is at stake here are the criteria of theory selection, and the difficulties of rationally accommodating scientific development given the presence of radical theory changes in science This immediately raises the second issue: (b) How should we make sense of episodes that apparently challenge the rationality of scientific theorizing? For example, how should we understand the various situations in which scientists, or even mathematicians, entertain inconsistent theories? Are these simply cases of scientific irrationality? Noting the blatant diversity of cases in which dramatic theoretical changes are involved, leaving scientists and mathematicians facing the trouble of (even if temporarily) entertaining contradictory theories, a model of scientific rationality is outlined in [32] The intention is to accommodate these two issues, making explicit the role of such contradictory theories and yielding ‘an account of scientific rationality that is able to make better sense of scientific and mathematical activity’ Now, if scientific theories can rationally be taken to be quasi-true, the underlying logic is necessarily paraconsistent, on pain of trivialism, a disaster to be avoided at all costs A paraconsistent logic—in fact, a first-order three-valued LFI, called LPT1– providing support for quasi-truth, was proposed in [33], and proven to be sound and complete with respect to a certain semantics of triples A revised version of this proposal was given in Sect 7.9 of Chap The strategy in [33] avoids constructing total structures, with a minimal detour from the Tarskian notion of satisfaction Then it is proved that LPT1 coincides (up to language) with LFI1∗, the quantified version of LFI1 proposed by Carnielli et al in [34], and also with the quantified version of J3 studied several decades ago by I.M.L D’Ottaviano (see [35–38]) After this, LPT1 is compared with the logic LP, introduced by G Priest in [39] from F Asenjo’s proposal in [40], as a formal framework for studying antinomies The ‘logic of paradox’ LP, studied in more detail in Sect 4.4.5, is one of the main 3-valued paraconsistent logics introduced in the literature, and has a certain kinship with Łukasiewicz and Kleene’s 3-valued logics It was proved in [33] that LPQ, the first-order version of LP, is a fragment of LPT1, and so the latter is a conservative extension of the former The three-valued propositional logic underlying LPT1, 384 Paraconsistency and Philosophy of Science: Foundations … called MPT, was described in Sect 4.4.7 of Chap 4, while J3 and LP were briefly described in Sects 4.4.3 and 4.4.5 of that chapter, respectively The notion of quasi-truth, or partial truth, is therefore, closely connected to the paraconsistent LFI paradigm and constitutes a non-dogmatic overture to the dynamics of theory change in science, tolerant to the flounderings of scientific practice In [41] and [42] this question is studied under the perspective of the AGM theory of Belief Change based on LFIs (a good general reference for Belief Change is S Hansson’s book [43]) Paraconsistent Belief Revision systems apply their tools to elicit the very notion of rationality within a paraconsistent setting It is possible to explain, in particular, the role the consistency operator, as introduced by LFIs, has to play within a dynamic context By considering the existence of contradictions as a natural consequence of the dynamics of rational thinking, the strictures set by the Belief Revision systems’ operations within a paraconsistent approach are to be reinterpreted Thus, such approach could also be taken as a paradigm for scientific reasoning One innovation of [41, 42] is to understand consistency as an epistemic attitude, thus clearing the way for further inquiries about the epistemological features of paraconsistency, as the ones discussed in Chap and in the present chapter 9.4 An Evidence-Based Approach to Paraconsistency A pair of contradictory sentences may be understood as conveying conflicting information, or conflicting scenarios, or imposing a rational deadlock, and when one designs a paraconsistent logic aiming at giving rational tools to deal with them, the question of acceptance of a contradictory pair of sentences without falling into trivialism requires some philosophically palatable position; the notion of evidence seems particularly promising as a vantage point from which to understand paraconsistent logics Evidence may be understood, as it is usual in epistemology, as the relevant ingredient for justified belief [1] taking into account that justified belief is, in traditional terms, a necessary condition for knowledge From this point of view, the idea of preservation of evidence suggests itself as a topic to be further developed in LFIs, in an anlogous way to the idea of preservation of certain constructions, as in the well-known BHK interpretation of intuitionistic logic On the other hand, however, a paraconsistentist approach to evidence may also be characterized from a probabilistic point of view The aim of evidence theory is to give an account of reasoning under epistemic uncertainty, that may happen due to a lack of data and/or a defective understanding of the available data related to some scenarios It is distinguished from randomic uncertainty, that is due to the inherently aleatory character of some scenarios One of the key points of evidence theory is that in imprecise events, uncertainty about an event can be quantified by the maximum and minimum probabilities of that 9.4 An Evidence-Based Approach to Paraconsistency 385 event This directly connects evidence theory to probability theory, and therefore to Bayesian conditionalization In [44] the authors investigate a paraconsistent approach to theory of probability based on the logics of formal inconsistency The paper shows that LFIs naturally encode an extension of the notion of probability able to express the notion of probabilistic reasoning under contradictions by means of appropriate notions of conditional probability and paraconsistent updating, via a version of Bayes’ Theorem for conditionalization The paper argues that the dissimilarity between the notions of inconsistency and contradiction plays a central role in an extended notion of probability that supports contradictory reasoning Actually, evidence theory and Bayesian theory of subjective probability are simultaneously generalized by Dempster-Shafer theory, which concerns belief and plausibility Beliefs from different sources can be combined (by means of the so-called Dempster rule of combination) with various operators used to model specific situations of belief diffusion An analogous treatment can be given using paraconsistent probability Another approach to an evidence-based interpretation to paraconsistency, related (though not identical) to paraconsistent probability, is the game-theoretical view of paraconsistency, still to be developed in full A game-theoretical account (by means of dialogical logic) of paraconsistency is defended in [45] Dialogical logic makes it possible to accommodate the occurrence of contradictions in two (or more) persons’ reasoning, and contributes positively to debates concerning the ontological versus epistemological nature of contradictions Rahman and Carnielli’s paper has attracted some attention (see e.g [46]) as an relevant first step into reformulating paraconsistent logic in a dialogue format Dialogical logic is not the only approach to paraconsistency from the point of view of game theory For instance, in a paraconsistent game-theoretical scenario the truth of a sentence can be defined in terms of the lack of winning strategies for the Opponent, instead of in terms of existence of winning strategies of the Proponent; a similar view is defended in [47], but with different assumptions (namely, the existence of true contradictions or dialetheias) A related possibility is granted by the definition of so-called ‘team semantics’ in [48], although the idea of a society producing semantics had been introduced (years before) in [49] 9.5 Summing Up A logic has epistemic rather than ontological character when its subject matter refers not only to truth, but also to some other concept strictly related to reason This is the case of intuitionistic logic, which is concerned with truth attained in a specific way, by means of a constructive proof And we claim that this is also a way of understanding paraconsistency in general, and particularly the logics of formal inconsistency The latter, we may say, is concerned with truth, since classical logic can be recovered, but it is also concerned with a notion weaker than truth, and it is precisely this 386 Paraconsistency and Philosophy of Science: Foundations … notion weaker than truth that allows an intuitive and plausible understanding of the acceptance of contradictions in some contexts of reasoning Intuitionistic logic is a special sort of paracomplete logics, that is, logics in which there is a model M and a sentence A such that both A and its negation not hold in M Mathematicians deal with lack of information in the sense that there are many unsolved mathematical problems This is one of the reasons for the rejection of the law of excluded middle by intuitionists In empirical sciences, on the other hand, although obviously many things are not known, the researcher sometimes deals with conflicting information, and very often with contradictory information Thus (s)he sometimes may have to provisionally consider two contradictory claims, one of which in due time will be rejected Both intuitionistic and paraconsistent logics may be conceived as normative theories of logical consequence with an epistemic character Notwithstanding, both are also descriptive: the first, according to Brouwer and Heyting, is intended to represent how the mind works in constructing correct mathematical proofs, while the latter, we argue, represents how we draw inferences correctly when faced with contradictions In fact, it is not surprising that we find a kind of duality in the motivations for intuitionistic and paraconsistent logics Dual-intuitionistic logics have been shown in [50] to be essentially paraconsistent, and similar results in the other direction have been investigated Nevertheless, the dual of Heyting’s well-known intuitionistic logic gives rise to a new paraconsistent logic, that is, one that is not a familiar paraconsistent logic Another dualities between paraconsistency and intuitionism, even from a topological perspective, were also investigated in [51–53] These results, in any case, show an intrinsic relationship between both paradigms, but there is of course much to be investigated in this regard Conceptions of consistency as coherence among a collection of statements, or as continuity or persistence of a collection of statements in time, could also be studied from a paraconsistent point of view It is worth noting that the ideas presented here indicate that there is much more to be explored in Hegel than just the (maybe misleading) idea of true contradictions Although classical first-order logic is a powerful tool for modeling reasoning, the fact that it does not handle contradictions in a sensible way, due to the insistence on the Principle of Explosion, is an intrinsic drawback In essence, classical logic is too brutal to sense contradictions and to appraise their meaning In theoretical computer science, for example, facts and rules of knowledge bases, as well as integrity constraints, can produce contradictions when combined, even if they are sound when separate, and this is also the case for description logics For this reason, the development of paraconsistent tools has turned out to be an important issue in working with description logics and expandable knowledge bases, as well as with large or combined databases (see [34, 54]) The new approach to consistency delivered by the logics of formal inconsistency has also generated a good deal of interest in the field of inferential probability and confirmation theory Because of the well-known debate in the philosophical literature on the long-standing confusion about probability when confronted with confirmation (see e.g [55, 56]), notions such as coherence, credence, consistent or coherent 9.5 Summing Up 387 individual profiles versus group profiles, etc (see [57]), can be fruitfully approached from the point of view of the logics of formal inconsistency, although there is still 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2012 Dialetheism, game theoretic semantics, and paraconsistent team semantics ArXiv e-prints http://arxiv.org/abs/1206.6573 48 Väänänen, Jouko 2007 Dependence Logic: A New Approach to Independence Friendly Logic Cambridge: Cambridge University Press 49 Carnielli, Walter A., and Mamede Lima-Marques 1999 Society semantics and multiple-valued logics In Advances in Contemporary Logic and Computer Science Proceedings of the XI Brazilian Conference on Mathematical Logic, May 1996, Salvador, Bahia, Brazil Contemporary Mathematics, vol 235, ed by Walter A Carnielli and Itala M.L D’Ottaviano, 33–52 American Mathematical Society 50 Brunner, Andreas, and Walter A Carnielli 2005 Anti-intuitionism and paraconsistency Journal of Applied Logic 3(1): 161–184 51 Goodman, Nicolas D 1981 The logic of contradiction Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 27(8/10): 119–126 52 Priest, Graham 2009 Dualising intuitionistic negation Principia 13(3): 165–184 53 Ba¸skent, Can 2013 Some topological properties of paraconsistent models Synthese 190(18): 4023–4040 doi:10.1007/s11229-013-0246-8 54 Grant, John, and V.S Subrahmanian 2000 Applications of paraconsistency in data and knowledge bases Synthese 125(1–2): 121–132 55 Fitelson, Branden 2012 Accuracy, language dependence, and Joyce’s argument for probabilism Philosophy of Science 79(1): 167–174 56 Fitelson, Branden 2006 Logical foundations of evidential support Philosophy of Science 73(5): 500–512 57 Fitelson, Branden 2007 Likelihoodism, Bayesiansim, and relational confimation Synthese 156(3): 473–489 Index A Ackermann, W., 2, 361 Aether theory, 373 AGM theory of Belief Change, 384 Algebra, 121 BL, 193 MTL, 193 De Morgan, 213 Lindenbaum of TM4, 228 Ockham, 279 t-norm continuous, 192 left-continuous, 192 tetravalent modal, 213 Almukdad, A., 179 Alves, E.H., 35, 40, 293 Anellis, I., 121 Aristotle, vii, xi, 11, 12, 16, 17 Arruda, A., 347, 354 Asenjo, F., 11, 149, 383 Avron, A., 39, 40, 251–253, 256, 260, 272, 279, 293 Axiom of separation (Aussonderung), 347 B Baaz, M., 112 Bar-Hillel, Y., Bar-Hillel-Carnap’s paradox, Batens, D., 77, 141, 232, 347 Beall, J.C., 24 Belnap, N., xiv, 171, 176, 211–214, 216 Bernoulli, J., 380 Berry, G.G., 365 Béziau, J.-Y., 115, 116 BHK interpretation, 384 Blok, W., xiii, xiv, 43, 119, 121, 129, 130, 135, 136, 139, 140, 144, 145, 151, 155, 171, 180, 193, 199, 207, 209, 210, 237 Bôcher, M., 381 Bochvar, D.A., 10, 138, 143 Bohr, N., 324, 371, 374 Boole, G., 2, 129 Brady, R., 348 Brouwer, L.E.J., 13, 14 Bueno, O., 324 Bueno-Soler, J., 230 Burali-Forti’s paradox, 360 Burali-Forti, C., 360, 361, 365 Byers, W., 364 C Cantor, G., vii, xvi, 3, 345–347, 349, 360– 362 Cantor’s Theorem, 346 Cardano, G., vii Cardinal of a first-order signature, 316 Cardinal of a first-order structure, 316 Carnap, R., Carnielli, W.A., 141, 158, 280, 383 C-system, 73 Chisholm, R., 233 Chomsky, N., 382 Chuaqui, R., 142, 158, 324 Church, A., 52 Closed theory, 36 Cohen, K.J., 10 Cohen, P., Columbus, C., 11 © Springer International Publishing Switzerland 2016 W Carnielli and M.E Coniglio, Paraconsistent Logic: Consistency, Contradiction and Negation, Logic, Epistemology, and the Unity of Science 40, DOI 10.1007/978-3-319-33205-5 391 392 Commutative monoid, 192 Complex number, 380 logarithm of a, 380 Complexity of a formula, 33, 50, 78–80, 294 of a term, 294 Concepts, 2, 52 Congruence, 130 compatible with a theory, 131 logical, 131 trivial, 131 Coniglio, M.E., 142, 158, 280 Consistency for sets, 349 retropropagation, 359 operator, 32 in a core fuzzy logic, 198 propagation, 104 w.r.t a subsignature, 205 strong, 115 retropropagation strong, 132 Context for a formula, 303 for a term, 303 Cornwall, J., Curry’s paradox, 354, 379 D Da Costa, N.C.A., xiii, 3, 9, 11, 25, 35, 40, 48, 71, 104, 111, 121, 141, 142, 144, 145, 158, 205, 279, 282, 293, 324, 347, 348, 363 Dauben, J.W., 347, 360 dC-system, 73 de Amo, S., 141, 158, 383 De Clercq, K., 141 Dedekind, R., xiv, 345, 360, 361, 379 Deduction Meta-Theorem (DMT), 34, 298, 300 De Morgan, A., xiv, 129, 179 Dempster-Shafer theory, 385 Derivability Adjustment Theorem (DAT), 46, 190, 207, 220, 352 Derivation in mbC, 34 in TM4, 227 Descartes, R., viii Designated truth-values, 122, 252 Diagram language, 303 Dialetheia, 17, 362, 376, 385 Dialetheism, 3, 17, 25, 151, 369, 382 Index Dialogical logic, 385 Dirac, P., 365 Distinguished truth-values, 122 D’Ottaviano, I.M.L., xiii, 141, 142, 282, 293, 383 Dugundji, J., 63, 251 Dugundji-like theorem, 126, 129 w.r.t Nmatrices, 253, 272 Dugundji’s Theorem, 121, 122 Dummet, M., xiv, 191 Dunn, J.M., xiii, 141, 176, 211–214, 216 E Ehrenfest, P., 371 Einstein, A., 373–375, 377–379 Empty set strong, 350 weak, 363 Equality relation, 318 Esteva, F., 191, 193 Euclid, vii Euler, L., vii, 380 Evidence, 370, 384 Ex Contradictione Sequitur Quodlibet (ECSQ), Explosion law, 31 F Fidel structure for bC, 248 for Ci, 248 for Cil, 249 for Cila, 250 for mbCci, 244 for mbCciw, 242 for mbCcl, 245 for mbC, 238 for N4, 181 Lindenbaum for mbCciw, 243 for mbCci, 245 for mbCcl, 246 for mbC, 241 for N4, 183 Fidel, M., xiv, 40, 180, 181, 184, 238, 256, 279 Figallo, A., 224 Figallo, M., 217, 218 Finsler, P., 361 First-order language, 294 First-order signature, 294 First-order structure, 302 Index 393 extended, 303 Fodor, J., 192 Font, J.M., 135, 214, 217 Frege, G., viii, x, xvi, 2, 5, 12, 13, 345, 346, 364, 380 Fresnel, A.J., 373 for QCi, 323 for QmbC , 304 canonical, 313 for QmbC≈ , 320 for QP1, 336 Ivlev, Y.V., 40, 251 G Galle, J., 375 Game-theoretical view of paraconsistency, 385 Gauss, C.F., vii General relativity, 374 Generalized functions, 365 Gödel, K., xiv, 3, 191, 200, 379 Gödel’s Incompleteness Theorems, 346 Godo, L., 191, 193 J Jansana, R., 135 Ja´skowski, S., 7, 10, 136, 140, 141, 229 Jevons, W.S., 129 Johansson, I., xiii, 171, 175, 179, 187, 221 Jourdain, P., 347 H Hadamard, J., 347 Hájek, P., xiv, 191, 193 Halldén, S., xiii, 10, 136, 138, 159 Hansson, S.O., 384 Harman, G., Hawking, S., 372 Hegel, G.W.F., 17, 19, 25 Henkin theory, 311 Henkin, L., 231 Heyting, A., 14 Hilbert, D., 2, 360, 380 Hintikka, J., 381 Humberstone, L., 176 Husserl, E., 380 Hyperalgebra, 251 Hyperoperation, 251 Hyperstructure, 251 I Identity relation, 318 Imaginary number, 380 logarithm of a, 380 Implication contrapositive, 224 deductive, 31 Inconsistency operator, 95 for sets, 355 in a core fuzzy logic, 208 Inconsistent mathematics, 365 Inconsistent set, 347 Interpretation for QLFI1◦ , 333 K Kanamori, A., 361 Kant, I., 19, 25, 371, 372 Kapsner, A., 10 Kearns, J.T., 251 Kleene, S., 11, 149 Klein, F., 380 Kolmogorov, A., 175 König, J., 361 Kripke model for IPL+ , 172 for imbC, 188 for LFIs, 230 for Min, 176 Kripke, S., xiv, 173 Kronecker, L., 361, 365 L Landini, P., 224 Lattice bounded comutative, 193 distributive, 216, 222 De Morgan, 213 implicative, 181 residuated comutative, 192 divisible, 192 prelinear, 193 Lavoisier, A.-L., 376, 377 Leibniz, G., 380 Leibniz, G.W., 371, 380 Lev, I., 39, 40, 251 Level valuations, 251 Le Verrier, U.J.J., 374, 375 Lewin, R.A., 145 Liar paradox, 379 394 Libert, T., 364 Lindenbaum, A., xiii, 37, 130, 131, 227 Logic adequate for an Nmatrix, 252 (Blok and Pigozzi) algebraizable, 130 classical propositional, 44 discussive, 141 finitary, 30 fuzzy basic BL, 193 core, 195 gently explosive, 9, 32 Johansson’s minimal, 175 maximal, 155 strong, 155 weak, 155 modal anodic, 230 cathodic, 230 tetravalent, 214 Monoidal t-norm based (MTL), 193 monotonic, 30 Nelson’s paraconsistent (N4), 180 of deontic inconsistency (LDI), 233 of formal inconsistency (LFI), 31 strong, 32 weak, 32 paracomplete, paraconsistent, boldly, 176, 220 paranormal, 39 partially explosive, 176 Positive classical, 34 Positive intuitionistic, 111 preserving degrees of truth, 196, 214, 222, 223 reducible to a signature, 72 standard, 30 Strict monoidal t-norm based (SMTL), 197 structural, 30 Tarskian, 30 paraconsistent, 31 Logical matrix, 122 Lopari´c, A., 112 Lorentz, H.A., 373, 378 Łós, J., 37 Loureiro, I., 213, 224 Lưwe, B., 142 Lowenhëim, L., xv Lowenhëim-Skolem Theorem for QmbC downward, 318 upward, 318 Index Łukasiewicz, J., xiv, 10, 16, 121, 141, 142, 149, 159, 200, 229 M Maddy, P., 361 Marcos, J., 141, 155, 158, 280, 282, 383 Marden’s Theorem, 381 Marden, M., 381 Marty, F., 251 Mathematical Fuzzy Logic, 191 Matrix semantics, 122 Maximal non-trivial set w.r.t a formula, 36 Maxwell, J.C., 377, 378 Mercury, 374 Mikenberg, I., 142, 145, 158, 324 Monteiro, A., xiv, 39, 213 Monteiro, L., 213 Moore, G.H., 381 Mortensen, C., 237, 365 Multialgebra, 251 Multifunction, 251, 380 Multivalued function, 251, 380 Multivalued operation, 251 N Negation classical, paracomplete, paraconsistent, strong, strong (Nelson), 10 Nelson, D., xiii, 10, 106, 108, 160, 171, 179, 180, 187, 212, 233, 279 Newton, I., 371, 374, 375, 378, 379 Nmatrix, 252 associated to a swap structure for bC, 267 for Ci, 271 for mbC, 255 for mbCci, 264 for mbCciw, 261 Non-deterministic algebra, 251 Non-deterministic matrix, 252 Norm continuous, 191 O Odintsov, S., 180, 181, 184, 238, 253 Ono, H., 180 Index P Paraconsistent Belief Revision, 384 Paraconsistent probability, 385 Paraconsistent set theory, 347 Paraconsistent updating, 385 Paraconsistent version of Bayes’ Theorem, 385 Partial relation, 325 Partial structure, 325 associated to an interpretation for QLFI1◦ , 333 associated to an interpretation for QP1, 342 extended, 330 for LFI1’, 325, 326 for QP1, 338 Partial truth, 324, 382 Pauli, W., 371 Peirce, C.S., 121, 129 Phlogiston, 376 Phlogiston hypothesis, 376 Pigozzi, D., xiii, 43, 119, 121, 129, 130, 135, 136, 139, 140, 144, 145, 151, 155, 171, 180, 193, 199, 207, 209, 210, 237 Polarities, 211 Popper, K., 10 Possible translations algebraizability, 119 Possible translations semantics, 280 for bC, 281 for Cila, 283 for Ci, 281 for mbC, 281 for mCi, 281 Pragmatic truth, 324, 382 Priest, G., xiii, 2, 11, 25, 150, 151, 383 Priestley, J., 376 Principle of (unrestricted) Abstraction, 346 Principle of Comprehension, 346 Principle of Explosion (PE), Proper class, 361 Proposition surrogates, 211 Propositional language, 30 Propositional signature, 29 Propositional variable, 30 Q Quantum Theory, 374 Quasi-matrices, 40 Quasi-truth, 324, 382 Quesada, F.M., 11 Quine, W.V.O., 5, 16, 24, 347 395 R Reduct of a first-order structure, 316 Rescher, N., 251 Restall, G., 24 Rius, M., 214, 217 Rivieccio, U., 180 Robinson, A., Routley, R., 348 Russell set, 363 strong, 353 Russell’s paradox, 346, 379 Russell, B., 2, 5, 24, 346, 360, 361, 363–365 S Schröder, E., 129 Schumacher, vii Schütte, K., 141 Schwarz, L., 365 Schwarze, M.G., 145 Segerberg, K., 138, 141, 145 Sette, A.M.A., xiii, 144, 145, 155, 158, 335 Sextus Empiricus, x Siebeck, J., 381 Signature , 30 + , 30 , 30 , 30 , 30 ⊥ , 30 • , 30 c , 30 Silvestrini, L.H., 142, 158 Skolem, T., xv Slater, H., 23 Snapshots, 255 Society semantics, 289 Splicing of logics, 289 Splitting of logics, 289 Stokes, G.G., 373 Subcontraries sentences, 23 Substitution, 30 Lemma for QmbC , 306 for QmbC≈ , 320 for QP1, 337 multiple, 303 Swap structure for bC, 267 for Ci, 270 for mbC, 254 for mbCci, 263 396 for mbCciw, 260 Sylvan, R., 151, 348 T Tarafder, S., 142 Tarski, A., xiii, 5, 121, 130, 131, 227, 324 Theory contradictory, explosive, trivial, Thomason, R., 180 Translation between logics, 43 conservative, 43 Truth-preserving semantics, 194, 214 T-(triangular) norm, 191 left-continuous, 191 Twist structure for N4, 184 full, 184 Lindenbaum for N4, 185 U Universal set, 353 Uranus, 374 V Vakarelov, D., 180, 184, 279 Valuation extended, 305 for LFI1◦ , 160 for QLFI1◦ , 333 for P1 , 146 for Cn , 113 for Cio, 132 for Ciore, 133 for LFI2, 152 for LPT0, 166 for mbC⊥ ciw, 94 for mbCclND, 275 for mbC⊥ , 53 for mCi, 102 for mbC, 35 for mbC• , 97 for mbC• cew, 99 for mbCcew, 99 Index for mbCciciβ , 89 for mbCciwciβ , 70 for mbCcicl, 88 for mbCciw, 64 for mbCcl− , 84 for mbC⊥ ci, 95 for mbCciciβ cl, 89 for mbCci, 66 for mbCcl, 82 for QCi, 323 for QmbC , 303 for QmbC≈ , 320 for QP1, 335 for PC→,∧ , 231 for PC→ , 231 for PI, 232 over an F-structure for bC, 248 over an F-structure for Ci, 249 over an F-structure for Cil, 249 over an F-structure for Cila, 250 over an F-structure for mbCci, 244 over an F-structure for mbCciw, 242 over an F-structure for mbCcl, 246 over an F-structure for mbC, 239 over an F-structure for N4, 182 over an MTL-algebra, 194 over an Nmatrix, 252 over a matrix logic, 122 over a swap structure for mbC, 255 over a twist structure for N4, 184 Variant formulas, 295 Vasiliev, N.A., 10 Von Seeliger, H., 374 Vulcan, 375 Vulcan hypothesis, 375 W Wajsberg, M., 91, 144 Weber, Z., 348 Weir, A., 361 Whitehead, A., 381 Wójcicki, R., 43, 122, 196 Z Zamansky, A., 293 Zermelo, E., 347 Index of Logic Systems Symbols C1+ , 116 C1s , 110 C2∼ , 75 Cω , 112 Cn , 113 Cn∗ , 293 Cn= , 293 CLim , 115 M4m , 214 PS3 , 144 c , 225 M4m P1 , 52 F OU R, 212 W3+ , 282 LBDL , 222 LDMA , 223 LDML , 222 LTMAc , 225 LTMA , 214 Ciaeciβ , 123 Cilaeciβ , 126 Cl− , 128 mbC⊥ ciw, 94 mbC⊥ ci, 95 mbC• cew, 99 mbC• , 96 mbC− , 77 mbCciciβ , 89 mbCciwciβ , 70 CPL+ , 34 mbC⊥ , 51 M B , 215 T ML, 217 BD, 212 mbCciciβ cl, 89 CPL◦ , 45 CPLW , 52 LFI1 P , 330 LFI1• , 159 LFI1◦ , 160 P1 , 146 QLFI1◦ , 332 QP1, 335 QP1 P , 341 QmbC≈ , 319 mbCciw, 64 MTL, 193 mbCcew, 99 CLuN, 77 CPL, 44 Ci, 103 Cil, 103 Cilo, 116 Cilore, 154 Cio, 131 Ciore, 133 H3, 136 J3, 141 K3, 149 LFI1, 159 LP, 150 LPQ, 151 LPT0, 165 MPT, 163 MPT0, 164 N4, 180 P1, 145 PI, 77 QCi, 323 QmbC , 295 S3, 138 © Springer International Publishing Switzerland 2016 W Carnielli and M.E Coniglio, Paraconsistent Logic: Consistency, Contradiction and Negation, Logic, Epistemology, and the Unity of Science 40, DOI 10.1007/978-3-319-33205-5 397 398 TM4, 226 Ł3, 142 bC, 103 imbC, 188 mCi, 102 mbC, 33 mbCcicl, 88 mbCcl, 82 mbCclND, 275 mbcCe, 103 LPT1, 383 ZFCil, 356 ZFCi, 356 ZFmCi, 355 ZFmbC, 349 Index of Logic Systems K→,∧,♦ , 231 K→,∧ , 231 K→ , 231 PC, 231 PC→,∧ , 231 PC→ , 231 PI ,♦ , 232 BL, 191 SMTL, 197 Ci ,♦ , 233 LFI1’, 159 LFI2, 152 bC ,♦ , 233 mbCcl− , 84 mbC ,♦ , 233 ... Switzerland 2016 W Carnielli and M.E Coniglio, Paraconsistent Logic: Consistency, Contradiction and Negation, Logic, Epistemology, and the Unity of Science 40, DOI 10.1007/978-3-319-33205-5_1 Contradiction. .. Marcelo Esteban Coniglio • Paraconsistent Logic: Consistency, Contradiction and Negation 123 Walter Carnielli Department of Philosophy and Centre for Logic, Epistemology and the History of Science... misunderstandings should be noted and avoided Firstly, paraconsistent logic in general, and LFIs in particular, not prove contradictions: these logic systems only support reasoning under hypothetical contradictions