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Walter Carnielli [64]Walter A. Carnielli [27]Walter Alexandre Carnielli [7]
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Walter Carnielli
University of Campinas
  1.  58
    An Epistemic Approach to Paraconsistency: A Logic of Evidence and Truth.Walter Carnielli & Abilio Rodrigues - 2019 - Synthese 196 (9):3789-3813.
    The purpose of this paper is to present a paraconsistent formal system and a corresponding intended interpretation according to which true contradictions are not tolerated. Contradictions are, instead, epistemically understood as conflicting evidence, where evidence for a proposition A is understood as reasons for believing that A is true. The paper defines a paraconsistent and paracomplete natural deduction system, called the Basic Logic of Evidence, and extends it to the Logic of Evidence and Truth. The latter is a logic of (...)
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  2.  19
    Paraconsistent Logic: Consistency, Contradiction and Negation.Walter Carnielli & Marcelo Esteban Coniglio - 2016 - Basel, Switzerland: Springer International Publishing.
  3. Formal Inconsistency and Evolutionary Databases.Walter A. Carnielli, João Marcos & Sandra De Amo - 2000 - Logic and Logical Philosophy 8 (2):115-152.
    This paper introduces new logical systems which axiomatize a formal representation of inconsistency (here taken to be equivalent to contradictoriness) in classical logic. We start from an intuitive semantical account of inconsistent data, fixing some basic requirements, and provide two distinct sound and complete axiomatics for such semantics, LFI1 and LFI2, as well as their first-order extensions, LFI1* and LFI2*, depending on which additional requirements are considered. These formal systems are examples of what we dub Logics of Formal Inconsistency (LFI) (...)
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  4. Towards a Philosophical Understanding of the Logics of Formal Inconsistency.Walter Carnielli & Abílio Rodrigues - 2015 - Manuscrito 38 (2):155-184.
    In this paper we present a philosophical motivation for the logics of formal inconsistency, a family of paraconsistent logics whose distinctive feature is that of having resources for expressing the notion of consistency within the object language in such a way that consistency may be logically independent of non-contradiction. We defend the view according to which logics of formal inconsistency may be interpreted as theories of logical consequence of an epistemological character. We also argue that in order to philosophically justify (...)
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  5. Twist-Valued Models for Three-Valued Paraconsistent Set Theory.Walter Carnielli & Marcelo E. Coniglio - manuscript
    Boolean-valued models of set theory were independently introduced by Scott, Solovay and Vopěnka in 1965, offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, Löwe and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific 3-valued model called PS3 which satisfies all the axioms of (...)
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  6.  69
    Systematization of Finite Many-Valued Logics Through the Method of Tableaux.Walter A. Carnielli - 1987 - Journal of Symbolic Logic 52 (2):473-493.
    his paper presents a unified treatment of the propositional and first-order many-valued logics through the method of tableaux. It is shown that several important results on the proof theory and model theory of those logics can be obtained in a general way. We obtain, in this direction, abstract versions of the completeness theorem, model existence theorem (using a generalization of the classical analytic consistency properties), compactness theorem and Lowenheim-Skolem theorem. The paper is completely self-contained and includes examples of application to (...)
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  7.  7
    Measuring Evidence: A Probabilistic Approach to an Extension of Belnap–Dunn Logic.Abilio Rodrigues, Juliana Bueno-Soler & Walter Carnielli - forthcoming - Synthese:1-30.
    This paper introduces the logic of evidence and truth \ as an extension of the Belnap–Dunn four-valued logic \. \ is a slightly modified version of the logic \, presented in Carnielli and Rodrigues. While \ is equipped only with a classicality operator \, \ is equipped with a non-classicality operator \ as well, dual to \. Both \ and \ are logics of formal inconsistency and undeterminedness in which the operator \ recovers classical logic for propositions in its scope. (...)
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  8. Two's Company: The Humbug of Many Logical Values.Carlos Caleiro, Walter Carnielli, Marcelo Coniglio & João Marcos - 2005 - In J. Y. Beziau (ed.), Logica Universalis. Birkhäuser Verlag. pp. 169-189.
    The Polish logician Roman Suszko has extensively pleaded in the 1970s for a restatement of the notion of many-valuedness. According to him, as he would often repeat, “there are but two logical values, true and false.” As a matter of fact, a result by W´ojcicki-Lindenbaum shows that any tarskian logic has a many-valued semantics, and results by Suszko-da Costa-Scott show that any many-valued semantics can be reduced to a two-valued one. So, why should one even consider using logics with more (...)
     
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  9.  46
    Limits for Paraconsistent Calculi.Walter A. Carnielli & João Marcos - 1999 - Notre Dame Journal of Formal Logic 40 (3):375-390.
    This paper discusses how to define logics as deductive limits of sequences of other logics. The case of da Costa's hierarchy of increasingly weaker paraconsistent calculi, known as $ \mathcal {C}$n, 1 $ \leq$ n $ \leq$ $ \omega$, is carefully studied. The calculus $ \mathcal {C}$$\scriptstyle \omega$, in particular, constitutes no more than a lower deductive bound to this hierarchy and differs considerably from its companions. A long standing problem in the literature (open for more than 35 years) is (...)
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  10.  37
    Maximal Weakly-Intuitionistic Logics.A. M. Sette & Walter A. Carnielli - 1995 - Studia Logica 55 (1):181 - 203.
    This article introduces the three-valuedweakly-intuitionistic logicI 1 as a counterpart of theparaconsistent calculusP 1 studied in [11].I 1 is shown to be complete with respect to certainthree-valued matrices. We also show that in the sense that any proper extension ofI 1 collapses to classical logic.The second part shows thatI 1 is algebraizable in the sense of Block and Pigozzi (cf. [2]) in a way very similar to the algebraization ofP 1 given in [8].
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  11.  93
    New Dimensions on Translations Between Logics.Walter A. Carnielli, Marcelo E. Coniglio & Itala M. L. D’Ottaviano - 2009 - Logica Universalis 3 (1):1-18.
    After a brief promenade on the several notions of translations that appear in the literature, we concentrate on three paradigms of translations between logics: ( conservative ) translations , transfers and contextual translations . Though independent, such approaches are here compared and assessed against questions about the meaning of a translation and about comparative strength and extensibility of a logic with respect to another.
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  12.  49
    Computability Computable Functions, Logic, and the Foundations of Mathematics.Richard L. Epstein & Walter A. Carnielli - 1989
    This book is dedicated to a classic presentation of the theory of computable functions in the context of the foundations of mathematics. Part I motivates the study of computability with discussions and readings about the crisis in the foundations of mathematics in the early 20th century, while presenting the basic ideas of whole number, function, proof, and real number. Part II starts with readings from Turing and Post leading to the formal theory of recursive functions. Part III presents sufficient formal (...)
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  13.  66
    The Dialogical Approach to Paraconsistency.Sahid Rahman & Walter A. Carnielli - 2000 - Synthese 125 (1-2):201-232.
    Being a pragmatic and not a referential approach tosemantics, the dialogical formulation ofparaconsistency allows the following semantic idea tobe expressed within a semi-formal system: In anargumentation it sometimes makes sense to distinguishbetween the contradiction of one of the argumentationpartners with himself (internal contradiction) and thecontradiction between the partners (externalcontradiction). The idea is that externalcontradiction may involve different semantic contextsin which, say A and ¬A have been asserted.The dialogical approach suggests a way of studying thedynamic process of contradictions through which thetwo (...)
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  14.  19
    On the Way to a Wider Model Theory: Completeness Theorems for First-Order Logics of Formal Inconsistency.Walter Carnielli, Marcelo E. Coniglio, Rodrigo Podiacki & Tarcísio Rodrigues - 2014 - Review of Symbolic Logic 7 (3):548-578.
    This paper investigates the question of characterizing first-order LFIs (logics of formal inconsistency) by means of two-valued semantics. LFIs are powerful paraconsistent logics that encode classical logic and permit a finer distinction between contradictions and inconsistencies, with a deep involvement in philosophical and foundational questions. Although focused on just one particular case, namely, the quantified logic QmbC, the method proposed here is completely general for this kind of logics, and can be easily extended to a large family of quantified paraconsistent (...)
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  15. On Formal Aspects of the Epistemic Approach to Paraconsistency.Walter Carnielli, Marcelo E. Coniglio & Abilio Rodrigues - 2018 - In Max Freund, Max Fernandez de Castro & Marco Ruffino (eds.), Logic and Philosophy of Logic: Recent Trends in Latin America and Spain. London: College Publications. pp. 48-74.
    This paper reviews the central points and presents some recent developments of the epistemic approach to paraconsistency in terms of the preservation of evidence. Two formal systems are surveyed, the basic logic of evidence (BLE) and the logic of evidence and truth (LET J ), designed to deal, respectively, with evidence and with evidence and truth. While BLE is equivalent to Nelson’s logic N4, it has been conceived for a different purpose. Adequate valuation semantics that provide decidability are given for (...)
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  16.  8
    Anti-Intuitionism and Paraconsistency.Andreas B. M. Brunner & Walter A. Carnielli - 2005 - Journal of Applied Logic 3 (1):161-184.
  17.  70
    Modulated Fibring and the Collapsing Problem.Cristina Sernadas, João Rasga & Walter A. Carnielli - 2002 - Journal of Symbolic Logic 67 (4):1541-1569.
    Fibring is recognized as one of the main mechanisms in combining logics, with great signicance in the theory and applications of mathematical logic. However, an open challenge to bring is posed by the collapsing problem: even when no symbols are shared, certain combinations of logics simply collapse to one of them, indicating that bring imposes unwanted interconnections between the given logics. Modulated bring allows a ner control of the combination, solving the collapsing problem both at the semantic and deductive levels. (...)
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  18.  10
    On Paraconsistent Deontic Logic.Newton C. A. Da Costa & Walter A. Carnielli - 1986 - Philosophia 16 (3-4):293-305.
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  19.  8
    On Epistemic and Ontological Interpretations of Intuitionistic and Paraconsistent Paradigms.Walter Carnielli & Abilio Rodrigues - forthcoming - Logic Journal of the IGPL.
    From the technical point of view, philosophically neutral, the duality between a paraconsistent and a paracomplete logic lies in the fact that explosion does not hold in the former and excluded middle does not hold in the latter. From the point of view of the motivations for rejecting explosion and excluded middle, this duality can be interpreted either ontologically or epistemically. An ontological interpretation of intuitionistic logic is Brouwer’s idealism; of paraconsistency is dialetheism. The epistemic interpretation of intuitionistic logic is (...)
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  20.  11
    Paraconsistency:The Logical Way to the Inconsistent.Walter Carnielli (ed.) - 2002 - CRC Press.
    The Logical Way to the Inconsistent Walter Alexandr Carnielli, Marcelo Coniglio, Itala Maria Lof D'ottaviano. Beyond Truth(-Preservation) R.E. JENNINGS Laboratory for Logic and Experimental Philosophy, Simon Fraser University, Burnaby, ...
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  21.  96
    Surviving Abduction.Walter Carnielli - 2006 - Logic Journal of the IGPL 14 (2):237-256.
    Abduction or retroduction, as introduced by C.S. Peirce in the double sense of searching for explanatory instances and providing an explanation is a kind of complement for usual argumentation. There is, however, an inferential step from the explanandum to the abductive explanans . Whether this inferential step can be captured by logical machinery depends upon a number of assumptions, but in any case it suffers in principle from the triviality objection: any time a singular contradictory explanans occurs, the system collapses (...)
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  22. Translations Between Logical Systems: A Manifesto.Walter A. Carnielli & Itala Ml D'Ottaviano - 1997 - Logique Et Analyse 157:67-81.
    The main objective o f this descriptive paper is to present the general notion of translation between logical systems as studied by the GTAL research group, as well as its main results, questions, problems and indagations. Logical systems here are defined in the most general sense, as sets endowed with consequence relations; translations between logical systems are characterized as maps which preserve consequence relations (that is, as continuous functions between those sets). In this sense, logics together with translations form a (...)
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  23.  12
    Computability. Computable Functions, Logic, and the Foundations of Mathematics.Richard L. Epstein & Walter A. Carnielli - 2002 - Bulletin of Symbolic Logic 8 (1):101-104.
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  24. On Paraconsistent Deontic Logic.Newton C. A. Costa & Walter A. Carnielli - 1986 - Philosophia 16 (3-4):293-305.
    This paper develops the first deontic logic in the context of paraconsistent logics.
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  25.  4
    Recovery Operators, Paraconsistency and Duality.Walter Carnielli, Marcelo E. Coniglio & Abilio Rodrigues - 2020 - Logic Journal of the IGPL 28 (5):624-656.
    There are two foundational, but not fully developed, ideas in paraconsistency, namely, the duality between paraconsistent and intuitionistic paradigms, and the introduction of logical operators that express metalogical notions in the object language. The aim of this paper is to show how these two ideas can be adequately accomplished by the logics of formal inconsistency and by the logics of formal undeterminedness. LFIs recover the validity of the principle of explosion in a paraconsistent scenario, while LFUs recover the validity of (...)
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  26.  24
    The Single-Minded Pursuit of Consistency and its Weakness.Walter Carnielli - 2011 - Studia Logica 97 (1):81 - 100.
    I argue that a compulsive seeking for just one sense of consistency is hazardous to rationality, and that observing the subtle distinctions of reasonableness between individual and groups may suggest wider, structuralistic notions of consistency, even relevant to re-assessing Gödei's Second Incompleteness Theorem and to science as a whole.
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  27.  26
    Polynomizing: Logic Inference in Polynomial Format and the Legacy of Boole.Walter Carnielli - 2007 - In L. Magnani & P. Li (eds.), Model-Based Reasoning in Science, Technology, and Medicine. Springer. pp. 349--364.
    Polynomizing is a term that intends to describe the uses of polynomial-like representations as a reasoning strategy and as a tool for scientific heuristics. I show how proof-theory and semantics for classical and several non-classical logics can be approached from this perspective, and discuss the assessment of this prospect, in particular to recover certain ideas of George Boole in unifying logic, algebra and the differential calculus.
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  28.  43
    MODEL-BASED REASONING IN SCIENCE AND TECHNOLOGY.Lorenzo Magnani, Walter Carnielli & Claudio Pizzi (eds.) - 2010 - Springer.
    This volume is based on the papers presented at the international conference Model-Based Reasoning in Science and Technology (MBR09_BRAZIL), held at the University of Campinas (UNICAMP), Campinas, Brazil, December 2009. The presentations given at the conference explored how scientific cognition, but several other kinds as well, use models, abduction, and explanatory reasoning to produce important or creative changes in theories and concepts. Some speakers addressed the problem of model-based reasoning in technology, and stressed the issue of science and technological innovation. (...)
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  29. A Categorial Approach to the Combination of Logics.Walter A. Carnielli & Marcelo E. Coniglio - 1999 - Manuscrito 22 (2):69-94.
    In this paper we propose a very general de nition of combination of logics by means of the concept of sheaves of logics. We first discuss some properties of this general definition and list some problems, as well as connections to related work. As applications of our abstract setting, we show that the notion of possible-translations semantics, introduced in previous papers by the first author, can be described in categorial terms. Possible-translations semantics constitute illustrative cases, since they provide a new (...)
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  30.  28
    Special Issue: Formal Representations in Model-Based Reasoning and Abduction.Lorenzo Magnani, Walter Carnielli & Claudio Pizzi - 2012 - Logic Journal of the IGPL 20 (2):367-369.
    This is the preface of the special Issue: Formal Representations in Model-based Reasoning and Abduction, published at the Logic Jnl IGPL (2012) 20 (2): 367-369. doi: 10.1093/jigpal/jzq055 First published online: December 20, 2010.
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  31.  20
    On Discourses Addressed by Infidel Logicians.Walter Carnielli & Marcelo E. Coniglio - 2013 - In Francesco Berto, Edwin Mares, Koji Tanaka & Francesco Paoli (eds.), Paraconsistency: Logic and Applications. Springer. pp. 27--41.
    We here attempt to address certain criticisms of the philosophical import of the so-called Brazilian approach to paraconsistency by providing some epistemic elucidations of the whole enterprise of the logics of formal inconsistency. In the course of this discussion, we substantiate the view that difficulties in reasoning under contradictions in both the Buddhist and the Aristotelian traditions can be accommodated within the precepts of the Brazilian school of paraconsistency.
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  32.  35
    Combining Logics.Walter Carnielli & Marcelo E. Coniglio - 2008 - Stanford Encyclopedia of Philosophy.
    Although a very recent topic in contemporary logic, the subject of combinations of logics has already shown its deep possibilities. Besides the pure philosophical interest offered by the possibility of defining mixed logic systems in which distinct operators obey logics of different nature, there are also several pragmatical and methodological reasons for considering combined logics. We survey methods for combining logics (integration of several logic systems into a homogeneous environment) as well as methods for decomposing logics, showing their interesting properties (...)
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  33. Book Reviews: Anita Burdman Feferman and Solomon Feferman, "Alfred Tarski: Life and Logic", Cambridge University Press, Cambridge.Walter Carnielli - 2006 - Logic and Logical Philosophy 15 (1):91-96.
    Anita Burdman Feferman and Solomon Feferman, "Alfred Tarski: Life and Logic", Cambridge University Press, Cambridge, UK, 2004, pp. 432.
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  34.  29
    Paraconsistent Algebras.Walter Alexandre Carnielli & Luiz Paulo Alcantara - 1984 - Studia Logica 43 (1-2):79 - 88.
    The prepositional calculiC n , 1 n introduced by N.C.A. da Costa constitute special kinds of paraconsistent logics. A question which remained open for some time concerned whether it was possible to obtain a Lindenbaum''s algebra forC n . C. Mortensen settled the problem, proving that no equivalence relation forC n . determines a non-trivial quotient algebra.The concept of da Costa algebra, which reflects most of the logical properties ofC n , as well as the concept of paraconsistent closure system, (...)
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  35.  21
    Interpolation Via Translations.João Rasga, Walter Carnielli & Cristina Sernadas - 2009 - Mathematical Logic Quarterly 55 (5):515-534.
    A new technique is presented for proving that a consequence system enjoys Craig interpolation or Maehara interpolation based on the fact that these properties hold in another consequence system. This technique is based on the existence of a back and forth translation satisfying some properties between the consequence systems. Some examples of translations satisfying those properties are described. Namely a translation between the global/local consequence systems induced by fragments of linear logic, a Kolmogorov-Gentzen-Gödel style translation, and a new translation between (...)
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  36. Erratum to “The Ricean Objection: An Analogue of Rice's Theorem for First-Order Theories” Logic Journal of the IGPL, 16: 585–590. [REVIEW]Igor Oliveira & Walter Carnielli - 2009 - Logic Journal of the IGPL 17 (6):803-804.
    This note clarifies an error in the proof of the main theorem of “The Ricean Objection: An Analogue of Rice’s Theorem for First-Order Theories”, Logic Journal of the IGPL, 16(6): 585–590(2008).
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  37. Meeting Hintikka's Challenge to Paraconsistentism.Walter Carnielli - 2009 - Principia: An International Journal of Epistemology 13 (3):283-297.
    Jaakko Hintikka, in a series of talks in Brazil in 2008, defended that IF logic and paraconsistent logic are, in a sense, very similar. Having sketched the proposal of a new paraconsistent system, he maintains that several achievements of IF logic could be reproducible in paraconsistent logic. One of the major difficulties, left as a challenge, would be to formulate some truth conditions for this new paraconsistent first-order language in order to make IF logic and paraconsistent logic more inter-related. My (...)
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  38.  15
    Some Results on Polarized Partion Relations of Higher Dimension.Walter Alexandre Carnielli & Carlos Augusto Di Prisco - 1993 - Mathematical Logic Quarterly 39 (1):461-474.
    Several types of polarized partition relations are considered. In particular we deal with partitions defined on cartesian products of more than two factors. MSC: 03E05.
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  39.  16
    Non-Standard Numbers: A Semantic Obstacle for Modelling Arithmetical Reasoning.Anderson De Araújo & Walter Carnielli - 2012 - Logic Journal of the IGPL 20 (2):477-485.
    The existence of non-standard numbers in first-order arithmetics is a semantic obstacle for modelling our arithmetical skills. This article argues that so far there is no adequate approach to overcome such a semantic obstacle, because we can also find out, and deal with, non-standard elements in Turing machines.
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  40.  33
    An Event on Brazilian Logic: Proceedings of the XIII Brazilian Logic Conference.Walter Carnielli, Marcelo Coniglio & Itala D'ottaviano - 2005 - Logic Journal of the IGPL 13 (1):1-3.
    This volume corresponds to the Proceedings of the XIII Brazilian Logic Conference held at the CLE - Centre for Logic, Epistemology and the History of Science in Campinas, SP, Brazil from May 26-30, 2003 under the auspices of the SBL - Brazilian Logic Society and the ASL - Association for Symbolic Logic.
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  41.  9
    The Ricean Objection: An Analogue of Rice's Theorem for First-Order Theories.Igor Oliveira & Walter Carnielli - 2008 - Logic Journal of the IGPL 16 (6):585-590.
    We propose here an extension of Rice's Theorem to first-order logic, proven by totally elementary means. If P is any property defined over the collection of all first-order theories and P is non-trivial over the set of finitely axiomatizable theories , then P is undecidable. This not only means that the problem of deciding properties of first-order theories is as hard as the problem of deciding properties about languages accepted by Turing machines, but also offers a general setting for proving (...)
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  42.  57
    Advances in Contemporary Logic and Computer Science Proceedings of the Eleventh Brazilian Conference on Mathematical Logic, May 6-10, 1996, Salvador, Bahia, Brazil. [REVIEW]Walter A. Carnielli, Itala M. L. D'ottaviano & Brazilian Conference on Mathematical Logic - 1999
    This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society (co-sponsored by the Centre for Logic, Epistemology and the History of Science, State University of Campinas, Sao Paulo) in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their (...)
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  43.  8
    The Problem Of Quantificational Completeness and the Characterization of All Perfect Quantifiers in 3‐Valued Logics.Walter A. Carnielli - 1987 - Mathematical Logic Quarterly 33 (1):19-29.
    This paper introduces the notions of perfect quantifiers in general many-valued logics and investigates the problem of quantificational completeness for such logics as well as the problem of characterizing all perfect quantifiers in 3-valued logics using techniques of combinatorial group theory.
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  44.  4
    Volume II: New Advances in Logics of Formal Inconsistency.Eduardo Alejandro Barrio & Walter Carnielli - 2020 - Logic Journal of the IGPL 28 (5):845-850.
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  45.  13
    Kripke-Style Models for Logics of Evidence and Truth.Henrique Antunes, Walter Carnielli, Andreas Kapsner & Abilio Rodrigues - 2020 - Axioms 9 (3).
    In this paper, we propose Kripke-style models for the logics of evidence and truth LETJ and LETF. These logics extend, respectively, Nelson’s logic N4 and the logic of first-degree entailment with a classicality operator ∘ that recovers classical logic for formulas in its scope. According to the intended interpretation here proposed, these models represent a database that receives information as time passes, and such information can be positive, negative, non-reliable, or reliable, while a formula ∘A means that the information about (...)
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  46. Tableaux sin refutación.Tomás Barrero & Walter Carnielli - 2005 - Matemáticas: Enseñanza Universitaria 13 (2):81-99.
    Motivated by H. Curry’s well-known objection and by a proposal of L. Henkin, this article introduces the positive tableaux, a form of tableau calculus without refutation based upon the idea of implicational triviality. The completeness of the method is proven, which establishes a new decision procedure for the (classical) positive propositional logic. We also introduce the concept of paratriviality in order to contribute to the question of paradoxes and limitations imposed by the behavior of classical implication.
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  47.  7
    Volume I: Recovery Operators in Logics of Formal Inconsistency.Eduardo Alejandro Barrio & Walter Carnielli - 2020 - Logic Journal of the IGPL 28 (5):615-623.
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  48. Advances in Contemporary Logic and Computer Science: Proceedings of the Eleventh Brazilian Conference on Mathematical Logic, May 6-10, 1996, Salvador, Bahia, Brazil. [REVIEW]Walter A. Carnielli & Itala M. L. D'ottaviano - 2001 - Studia Logica 69 (3):439-439.
     
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  49. Aristóteles, paraconsistentismo e a tradição budista.Walter Carnielli & Marcelo Coniglio - 2008 - O Que Nos Faz Pensar 23:163-175.
    This paper (in Portuguese) defends that the both the Buddhist tradition and the Aristotelian allow us to think of the distinction between to reason with contradictions and to accept them, understanding ' accept a contradiction ' by taking it as consistent. From this viewpoint, none of two would disagree with most contemporary paraconsistent views. -/- The conclusions are, thus, that, firstly, there is no compelling reason to endorse any kind of metaphysical dialetheism, and, second, that a coherent form of reasoning (...)
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  50. Book Reviews: Claude P. Bruter (Editor), Mathematics in Art: Mathematical Visualization in Art and Education.Walter Carnielli - 2004 - Logic and Logical Philosophy 13:163-166.
    Claude P. Bruter (editor), Mathematics in Art: Mathematical Visualization in Art and Education, Springer-Verlag, New York, 2002, pp. X + 337, ISBN 3-540-43422-4.
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