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Paraconsistent constructive logic with strong negation as a contraction-free relevant logic

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Don Pigozzi on Abstract Algebraic Logic, Universal Algebra, and Computer Science

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 16))

Abstract

Summary Logics with strong negation are a class of sentential calculi that originally arose from concerns about the non-constructive nature of negation in intuitionistic logic. Nelson’s paraconsistent constructive logic with strong negation N4 (Almukdad and Nelson, 1984; Odintsov, 2003, 2004, 2008), the most important member of this class, is an axiomatic expansion of the negation-free fragment of the intuitionistic propositional calculus (Rasiowa, 1974, Chapter X) by a unary logical connective ~ of strong negation. It is well known that strong negation plays an important role as ‘explicit negation’ in logic programming (Akama, 1997; Eiter et al., 1999; Gelfond, 2002; Kamide and Wansing, 2012; Pearce, 1999; Wansing, 1993).

Dedicated to Don Pigozzi on the occasion of his eightieth birthday.

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References

  • Abramsky, S. (1993). Computational intepretations of linear logic, Theoretical Computer Science 111, 3–57.

    Google Scholar 

  • Aglianò, P. (2001). Congruence quasi-orderability in subtractive varieties, Journal of the Australian Mathematical Society. Series A — Pure Mathematics and Statistics 71, 421–445.

    Google Scholar 

  • Akama, S. (1997). Tableaux for logic programming with strong negation, In D. Galmiche (Ed.), Automated Reasoning with Analytical Tableaux and Related Methods. International Conference, TABLEAUX’97. Pont-á-Mousson, France, May, 1997. Proceedings, Volume 1227 of Lecture notes in Artificial Intelligence. Subseries of Lecture Notes in Computer Science, Berlin, pp. 31–42. Springer.

    Google Scholar 

  • Almukdad, A. and Nelson, D. (1984). Constructible falsity and inexact predicates, Journal of Symbolic Logic 49, 231–233.

    Google Scholar 

  • van Alten, C. J. (1998). An Algebraic Study of Residuated Ordered Monoids and Logics without Exchange and Contraction, Ph. D. thesis, University of Natal, Durban, http://hdl.handle.net/10413/3961, accessed 12 March 2016.

  • van Alten, C. J. and Raftery, J. G. (2004). Rule separation and embedding theorems for logics without weakening, Studia Logica 76, 241–274.

    Google Scholar 

  • de Amo, S. and Carnielli, W. A. and Marcos, J. (2002). A logical framework for integrating inconsistent information in multiple databases, In T. Eiter and K.-D. Schewe (Eds.), Foundations of Information and Knowledge Systems. Second international Symposium (FoIKS 2002). Salzau Castle, Germany, February 2002. Proceedings, Volume 2284 of Lecture Notes in Computer Science, Berlin, pp. 67–84. Springer.

    Google Scholar 

  • Anderson, A. R. and Belnap, N. D. (1975). Entailment. The Logic of Relevance and Necessity, Volume 1, Princeton: Princeton University Press.

    Google Scholar 

  • Arieli, O. (1999). Multiple-valued Logics for Reasoning with Uncertainty, Ph. D. thesis, Tel-Aviv University, http://www2.mta.ac.il/˜oarieli/Papers/phd.pdf, accessed 12 March 2016.

  • Arieli, O. and Avron, A. (1994). Logical bilattices and inconsistent data, In Ninth Annual IEEE Symposium on Logic in Computer Science (LICS’94). Proceedings, Los Alamitos, pp. 468–476. IEEE Computer Society Press.

    Google Scholar 

  • Arieli, O. and Avron, A. (1996). Reasoning with logical bilattices, Journal of Logic, Language and Information 5, 25–63.

    Google Scholar 

  • Arieli, O. and Avron, A. (2015). Three-valued paraconsistent propositional logics, In J.-Y. Beziau, M. Chakraborty, and S. Dutta (Eds.), New Directions in Paraconsistent Logic. 5th WCP, Kolkata, India, February 2014, Volume 152 of Springer Proceedings in Mathematics & Statistics, New Delhi, pp. 91–129. Springer.

    Google Scholar 

  • Arieli, O. and Avron, A. and Zamansky, A. (2010). Maximally paraconsistent three-valued logics, In F. Lin, U. Sattler, and M. Truszczyński (Eds.), Twelfth International Conference on the Principles of Knowledge Representation and Reasoning (KR 2010). Toronto, Canada, May 2010. Proceedings, Massachusetts, pp. 310–318. AAAI Press.

    Google Scholar 

  • Avron, A. (1986). On an implication connective of RM, Notre Dame Journal of Formal Logic 27, 201–209.

    Google Scholar 

  • Avron, A. (1988). The semantics and proof theory of linear logic, Theoretical Computer Science 57, 161–184.

    Google Scholar 

  • Avron, A. (1991). Natural 3-valued logics — characterization and proof theory, Journal of Symbolic Logic 56, 276–294.

    Google Scholar 

  • Avron, A. (1999). On the expressive power of three-valued and four-valued languages, Journal of Logic and Computation 9, 977–994.

    Google Scholar 

  • Avron, A. (2016). RM and its nice properties, In K. Bimbó (Ed.), J. Michael Dunn on Information Based Logics, Volume 8 of Outstanding Contributions to Logic, pp. 15–43. Switzerland: Springer International Publishing.

    Google Scholar 

  • Baldwin, J. T. and Berman, J. (1975). The number of subdirectly irreducible algebras in a variety, Algebra Universalis 5, 379–389.

    Google Scholar 

  • Barbour, G. D. and Raftery, J. G. (2003). Quasivarieties of logic, regularity conditions and parameterized algebraization, Studia Logica 74, 99–152.

    Google Scholar 

  • Batens, D. (1980). Paraconsistent extensional propositional logics, Logique et Analyse. Nouvelle Série 90–91, 195–234.

    Google Scholar 

  • Belnap, N. D. (1976). How a computer should think, In G. Ryle (Ed.), Contemporary Aspects of Philosophy. Papers of the Oxford international Symposium of 29 September–4 October 1975, held in Christ Church College, pp. 30–55. Stocksfield: Oriel Press.

    Google Scholar 

  • Belnap, N. D. (1977). A useful four-valued logic, In J. M. Dunn and G. Epstein (Eds.), Modern Uses of Multiple-Valued Logics. Invited papers from the fifth International Symposium on Multiple-Valued Logic held at Indiana University, Bloomington, Indiana, May 13–16, 1975, with a bibliography of many-valued logic by Robert. G. Wolf, Volume 2 of Episteme. A Series in the Foundational, Methodological, Philosophical, Psychological, Sociological and Political Aspects of the Sciences, Pure and Applied, pp. 8–37. Dordrecht: D. Riedel Publishing Company.

    Google Scholar 

  • Białynicki-Birula, A. and Rasiowa, H. (1958). On constructible falsity in the constructive logic with strong negation, Colloquium Mathematicum 6, 287–310.

    Google Scholar 

  • Bianchi, M. (2011). On some Axiomatic Extensions of the Monoidal T-norm Based Logic MTL: an Analysis in the Propositional and in the First-order Case, Milan: Ledizioni LediPublishing.

    Google Scholar 

  • Blok, W. J. and Dziobiak, W. (1986). On the lattice of quasivarieties of Sugihara algebras, Studia Logica 45, 275–280.

    Google Scholar 

  • Blok, W. J. and Köhler, P. and Pigozzi, D. (1984). On the structure of varieties with equationally definable principal congruences II, Algebra Universalis 18, 334–379.

    Google Scholar 

  • Blok, W. J. and Pigozzi, D. (1982). On the structure of varieties with equationally definable principal congruences I, Algebra Universalis 15, 195–227.

    Google Scholar 

  • Blok, W. J. and Pigozzi, D. (1989). Algebraizable logics, Memoirs of the American Mathematical Society 77 (396).

    Google Scholar 

  • Blok, W. J. and Pigozzi, D. (1994a). On the structure of varieties with equationally definable principal congruences III, Algebra Universalis 32, 545–608.

    Google Scholar 

  • Blok, W. J. and Pigozzi, D. (1994b). On the structure of varieties with equationally definable principal congruences IV, Algebra Universalis 31, 1–35.

    Google Scholar 

  • Blok, W. J. and Pigozzi, D. (2001). Abstract algebraic logic and the deduction theorem, Manuscript, http://orion.math.iastate.edu/dpigozzi/papers/aaldedth.pdf, accessed 12 March 2016.

  • Blok, W. J. and Raftery, J. G. (2004). Fragments of R-Mingle, Studia Logica 78, 59–106.

    Google Scholar 

  • Blok, W. J. and Raftery, J. G. (2008). Assertionally equivalent quasivarieties, International Journal of Algebra and Computation 18, 589–681.

    Google Scholar 

  • Bloom, S. L. and Suszko, R. (1972). Investigations into the sentential calculus with identity, Notre Dame Journal of Formal Logic 13, 289–308.

    Google Scholar 

  • Bou, F. and Rivieccio, U. (2011). The logic of distributive bilattices, Logic Journal of the IGPL. Interest Group in Pure and Applied Logic 19, 183–216.

    Google Scholar 

  • Bou, F. and Rivieccio, U. (2013). Bilattices with implications, Studia Logica 101, 651–675.

    Google Scholar 

  • Brady, R. T. (1982). Completeness proofs for the systems RM3 and BN4, Logique et Analyse. Nouvelle Série 25, 9–32.

    Google Scholar 

  • Brady, R. T. (1990). The Gentzenization and decidability of RW, Journal of Philosophical Logic 19, 35–73.

    Google Scholar 

  • Brady, R. T. (1991). Gentzenization and decidability of some contraction-less relevant logics, Journal of Philosophical Logic 20, 97–117.

    Google Scholar 

  • Brady, R. T. (1996a). Gentzenization of relevant logics without distribution. I, Journal of Symbolic Logic 61, 353–378.

    Google Scholar 

  • Brady, R. T. (1996b). Gentzenization of relevant logics without distribution. II, Journal of Symbolic Logic 61, 379–401.

    Google Scholar 

  • Brignole, D. (1969). Equational characterisation of Nelson algebra, Notre Dame Journal of Formal Logic 10, 285–297.

    Google Scholar 

  • Burris, S. (1992). Discriminator varieties and symbolic computation, Journal of Symbolic Computation 13, 175–207.

    Google Scholar 

  • Burris, S. and Sankappanavar, H. P. (1981). A Course in Universal Algebra, Volume 78 of Graduate Texts in Mathematics, New York: Springer-Verlag.

    Google Scholar 

  • Burris, S. N. (1998). Logic for Mathematics and Computer Science, New Jersey: Prentice-Hall.

    Google Scholar 

  • Busaniche, M. and Cignoli, R. (2010). Constructive logic with strong negation as a substructural logic, Journal of Logic and Computation 20, 761–793.

    Google Scholar 

  • Caleiro, C. and Gonçalves, R. (2005). Equipollent logical systems, In J.-Y. Beziau (Ed.), Logica Universalis: Towards a General Theory of Logic, pp. 99–111. Basel: Birkhäuser Verlag.

    Google Scholar 

  • Caret, C. and Weber, Z. (2015). A note on contraction-free logic for validity, Topoi. An international review of philosophy 34, 63–74.

    Google Scholar 

  • Carnielli, W. and Coniglio, M. E. and Marcos, J. (2007). Logics of formal inconsistency, In D. M. Gabbay and F. Guenther (Eds.), The Handbook of Philosophical Logic (2nd ed.), Volume 14, pp. 1–93. Dordrecht: Springer.

    Google Scholar 

  • Carnielli, W. and Marcos, J. (2002). A taxonomy of C-systems, In W. Carnielli, M. E. Coniglio, and I. M. L. D’Ottaviano (Eds.), Paraconsistency. The Logical Way to the Inconsistent. Second World Congress on Paraconsistency (WCP 2000). São Paulo, Brazil, May 2000. Proceedings, Volume 228 of Lecture Notes in Pure and Applied Mathematics, New York, pp. 1–94. Marcel Dekker, Inc.

    Google Scholar 

  • Carnielli, W. A. (2002). How to build your own paraconsistent logic: An introduction to the logics of formal (in)consistency, In J. Marcos, D. Batens, and W. A. Carnielli (Eds.), Workshop on Paraconsistent Logic (WoPaLo). Held in Trento, Italy, as part of the 14th European Summer School on Logic, Language, and Information (ESSLLI’02), 5–9 August 2002. Proceedings, CLE e-Prints, vol. 2, pp. 58–72.

    Google Scholar 

  • Carnielli, W. A. and Marcos, J. and de Amo, S. (2000). Formal inconsistency and evolutionary databases, Logic and Logical Philosophy 8, 115–152.

    Google Scholar 

  • Cheng, G. and Wang, G. J. (1999). R0-algebras and their basic structure, Acta Mathematica Scientia. Series A. Shuxue Wuli Xuebao. Chinese Edition 19, 584–588, (Chinese).

    Google Scholar 

  • da Costa, N. C. A. (1974). Theory of inconsistent formal systems, Notre Dame Journal of Formal Logic 15, 497–510.

    Google Scholar 

  • Czelakowski, J. (2001). Protoalgebraic Logics, Volume 10 of Trends in Logic. Studia Logica Library, Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Czelakowski, J. and Pigozzi, D. (2004). Fregean logics, Annals of Pure and Applied Logic 127, 17–76.

    Google Scholar 

  • De, M. and Omori, H. (2015). Classical negation and expansions of Belnap- Dunn logic, Studia Logica 103, 825–851.

    Google Scholar 

  • D’Ottaviano, I. M. L. and da Costa, N. C. A. (1970). Sur un problème de Jaśkowski, Comptes Rendus de l’Académie des Sciences Paris. Seríe A (Sciences mathématiques) 270A, 1349–1353.

    Google Scholar 

  • Došen, K. (1993). A historical introduction to substructural logics, In K. Došen and P. Schroeder-Heister (Eds.), Substructural Logics, Volume 2 of Studies in Logic and Computation, pp. 1–30. Oxford: Clarendon Press.

    Google Scholar 

  • Dunn, J. M. (1970). Algebraic completeness results for R-mingle and its extensions, Journal of Symbolic Logic 35, 1–13.

    Google Scholar 

  • Dunn, J. M. (1976). Intuitive semantics for first-degree entailments and ‘coupled trees’, Philosophical Studies. An international journal of philosophy in the analytic tradition. 29, 149–168.

    Google Scholar 

  • Eiter, T. and Leone, N. and Pearce, D. (1999). Assumption sets for extended logic programs, In J. Gerbrandy, M. Marx, M. de Rijke, and Y. Venema (Eds.), JFAK. Essays dedicated to Johan van Benthem on the occasion of his 50th birthday, pp. 167–186. Amsterdam: Vossiuspers, Amsterdam University Press, http://www.illc.uva.nl/j50/contribs/pearce/pearce.pdf, accessed 12 March 2016.

  • Epstein, R. L. (1995). The Semantic Foundations of Logic. Volume I. Propositional Logics, Volume 35 of Nijhoff International Philosophy Series, Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Esteva, F. and Godo, L. (2001). Monoidal t-norm based logic: towards a logic for left continuous t-norms, Fuzzy Sets and Systems. An international journal in Information Science and Engineering 124, 271–288.

    Google Scholar 

  • Fitting, M. (1989). Bilattices and the theory of truth, Journal of Philosophical Logic 18, 225–256.

    Google Scholar 

  • Fitting, M. (1991). Bilattices and the semantics of logic programming, The Journal of Logic Programming 11, 91–116.

    Google Scholar 

  • Fitting, M. (1994). Kleene’s three-valued logic and their children, Fundamenta Informaticae 20, 113–131.

    Google Scholar 

  • Fodor, J. C. (1995). Contrapositive symmetry of fuzzy implications, Fuzzy Sets and Systems. An international journal in Information Science and Engineering 69, 142–156.

    Google Scholar 

  • Font, J. M. (1993). On the Leibniz congruences, In C. Rauszer (Ed.), Algebraic Methods in Logic and Computer Science, Volume 28 of Banach Center Publications, pp. 17–36. Warszawa: Institute of Mathematics, Polish Academy of Sciences.

    Google Scholar 

  • Font, J. M. and Jansana, R. (2009). A General Algebraic Semantics for Sentential Logics (2nd ed.), Volume 7 of Lecture Notes in Logic, New York: Association for Symbolic Logic.

    Google Scholar 

  • Font, J. M. and Jansana, R. and Pigozzi, D. (2003). A survey of abstract algebraic logic, Studia Logica 74, 13–97.

    Google Scholar 

  • Font, J. M. and Pérez, G. R. (1992). A note on Sugihara algebras, Publicacions Matemàtiques 36, 591–599.

    Google Scholar 

  • Font, J. M. and Rodríguez, G. (1990). Note on algebraic models for relevance logic, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 36, 535–540.

    Google Scholar 

  • Fried, E. and Grätzer, G. and Quackenbush, R. (1980). Uniform congruence schemes, Algebra Universalis 10, 176–188.

    Google Scholar 

  • Fried, E. and Kiss, E. W. (1983). Connections between congruence-lattices and polynomial properties, Algebra Universalis 17, 227–262.

    Google Scholar 

  • Galatos, N. and Jipsen, P. and Kowalski, T. and Ono, H. (2007). Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Volume 151 of Studies in Logic and the Foundations of Mathematics, Amsterdam: Elsevier.

    Google Scholar 

  • Gelfond, M. (2002). Representing knowledge in A-Prolog, In A. C. Kakas and F. Sadri (Eds.), Computational Logic: Logic Programming and Beyond. Essays in Honour of Robert A. Kowalski. Part II, Volume 2408 of Lecture notes in Artificial Intelligence. Subseries of Lecture Notes in Computer Science, Berlin, pp. 413–451.

    Google Scholar 

  • Ginsberg, M. (1990). Bilattices and modal operators, Journal of Logic and Computation 1, 41–69.

    Google Scholar 

  • Girard, J.-Y. (1987). Linear logic, Theoretical Computer Science 50, 1–102.

    Google Scholar 

  • Girard, J.-Y. (1995). Linear logic: its syntax and semantics, In J.-Y. Girard, Y. Lafont, and L. Regnier (Eds.), Advances in Linear Logic, Volume 222 of London Mathematical Society Lecture Note Series, pp. 1–42. Cambridge: Cambridge University Press.

    Google Scholar 

  • Gispert, J. (2003). Axiomatic extensions of the nilpotent minimum logic, Reports on Mathematical Logic 37, 113–123.

    Google Scholar 

  • Grätzer, G. (2008). Universal Algebra (2nd ed.), New York: Springer.

    Google Scholar 

  • Gyuris, V. (1999). Variations of Algebraizability, Ph. D. thesis, The University of Illinois at Chicago.

    Google Scholar 

  • Hart, J. B. and Rafter, L. and Tsinakis, C. (2002). The structure of commutative residuated lattices, International Journal of Algebra and Computation 12, 509–524.

    Google Scholar 

  • Hsieh, A. (2008). Embedding Theorems and Finiteness Properties for Residuated Structures and Substructural Logics, Ph. D. thesis, University of KwaZulu-Natal, Durban, South Africa, http://hdl.handle.net/10413/446, accessed 12 March 2016.

  • Hsieh, A. and Raftery, J. G. (2006). A finite model property for RMI min, Mathematical Logic Quarterly 52, 602–612.

    Google Scholar 

  • Hsieh, A. and Raftery, J. G. (2007). Conserving involution in residuated structures, Mathematical Logic Quarterly 53, 583–609.

    Google Scholar 

  • Humberstone, L. (2005). Logical discrimination, In J.-Y. Béziau (Ed.), Logica Universalis : Towards a General Theory of Logic, pp. 207–228. Basel: Birkhäuser Verlag.

    Google Scholar 

  • Humberstone, L. (2011). The Connectives, Cambridge (Massachusetts): The MIT Press.

    Google Scholar 

  • Humberstone, L. (2015). Béziau on And and Or, In A. Koslow and A. Buchsbaum (Eds.), The Road to Universal Logic : Festschrift for 50th Birthday of Jean-Yves Béziau. Volume I, Studies in Universal Logic, pp. 283–307. Basel: Birkhäuser.

    Google Scholar 

  • Idziak, P. M. and Słomczyńska, K. and Wroński, A. (2009). Fregean varieties, International Journal of Algebra and Computation 19, 595–645.

    Google Scholar 

  • Ishii, T. (2000). Nonclassical Logics with Identity Connective and their Algebraic Characterization, Ph. D. thesis, Japan Advanced Institute of Science and Technology, http://hdl.handle.net/10119/898, accessed 12 March 2016.

  • Jansana, R. and Rivieccio, U. (2013). Priestley duality for N4-lattices, In Eighth Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2013). Proceedings, Volume 32 of Advances in Intelligent Systems Research (AISR), Amsterdam, pp. 223–229. Atlantis Press.

    Google Scholar 

  • Järvinen, J. and Pagliani, P. and Radeleczki, S. (2013). Information completeness in Nelson algebras of rough sets induced by quasiorders, Studia Logica 101, 1073–1092.

    Google Scholar 

  • Järvinen, J. and Radeleczki, S. (2011a). Erratum to: Representation of Nelson algebras by rough sets determined by quasiorders, Algebra Universalis 66, 181.

    Google Scholar 

  • Järvinen, J. and Radeleczki, S. (2011b). Representation of Nelson algebras by rough sets determined by quasiorders, Algebra Universalis 66, 163–179.

    Google Scholar 

  • Järvinen, J. and Radeleczki, S. (2014). Monteiro spaces and rough sets determined by quasiorder relations: Models for Nelson algebras, Fundamenta Informaticae 131, 205–215.

    Google Scholar 

  • Jipsen, P. and Tsinakis, C. (2002). A survey of residuated lattices, In J. Martínez (Ed.), Ordered Algebraic Structures. Proceedings of the Gainesville Conference. Sponsored by the University of Florida. 28th February–3rd March, 2001, Volume 7 of Developments in Mathematics, Dordrecht, pp. 19–56. Kluwer Academic Publishers.

    Google Scholar 

  • Johansson, I. (1936). Der minimalkalkül, ein reduzierter intuitionistischer formalismus, Compositio Mathematica 4, 119–136.

    Google Scholar 

  • Jónsson, B. (1995). Congruence distributive varieties, Mathematica Japonica 42, 353–401.

    Google Scholar 

  • Jung, A. and Rivieccio, U. (2013). Kripke semantics for modal bilattice logics (extended abstract), In Twenty-eighth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS’13). Proceedings, Los Alamitos, pp. 438–447. IEEE Computer Society Press.

    Google Scholar 

  • Kachi, D. (2002). Validity in simple partial logic, Annals of the Japan Association for Philosophy of Science 10, 139–153.

    Google Scholar 

  • Kachi, D. (2007). Partial logic as a logic of extensional alethic modality, Journal of the Japan Association for Philosophy of Science 34, 13–22, (Japanese).

    Google Scholar 

  • Kamide, N. (2016). A decidable paraconsistent relevant logic: Gentzen system and Routley-Meyer semantics, Mathematical Logic Quarterly 62, 177–189.

    Google Scholar 

  • Kamide, N. and Wansing, H. (2012). Proof theory of Nelson’s paraconsistent logic: A uniform perspective, Theoretical Computer Science 415, 1–38.

    Google Scholar 

  • Kapsner, A. (2014). Logic and Falsifications. A New Perspective on Constructivist Semantics, Volume 40 of Trends in Logic. Studia Logica Library, Basel: Springer.

    Google Scholar 

  • Köhler, P. and Pigozzi, D. (1980). Varieties with equationally definable principal congruences, Algebra Universalis 11, 213–219.

    Google Scholar 

  • Kracht, M. (1998). On extensions of intermediate logics by strong negation, Journal of Philosophical Logic 27, 49–73.

    Google Scholar 

  • Łukasiewicz, J. (1970a). On three-valued logic, In L. Borkowski (Ed.), Selected Works of Jan Łukasiewicz, Studies in Logic and the Foundations of Mathematics, pp. 87–88. Warsaw: North-Holland Publishing Company.

    Google Scholar 

  • Łukasiewicz, J. (1970b). Philosophical remarks on many-valued systems of propositional logic, In L. Borkowski (Ed.), Selected Works of Jan Łukasiewicz, Studies in Logic and the Foundation of Mathematics, pp. 153–178. Warsaw: North-Holland Publishing Company.

    Google Scholar 

  • Malinowski, G. (1993). Many-Valued Logics, Volume 25 of Oxford Logic Guides, Oxford: Clarendon Press.

    Google Scholar 

  • Marcos, J. (2005). On a problem of da Costa, In G. Sica (Ed.), Essays on the Foundations of Mathematics and Logic, Volume 2, pp. 53–69. Monza: Polimetrica.

    Google Scholar 

  • McCune, W., Prover 9, http://www.cs.unm.edu/˜mccune/prover9/, accessed12 March 2016.

  • McCune, W. and Padmanabhan, R. (1996). Automated Deduction in Equational Logic and Cubic Curves, Volume 1095 of Lecture notes in Artificial Intelligence. Subseries of Lecture Notes in Computer Science, Berlin: Springer-Verlag.

    Google Scholar 

  • McKenzie, R. N. and McNulty, G. F. and Taylor, W. F. (1987). Algebras, Lattices, Varieties. Volume I, The Wadsworth & Brooks/Cole Mathematics Series. Monterey: Wadsworth & Brooks/Cole Advanced Books & Software.

    Google Scholar 

  • McNulty, G. (1992). A field guide to equational logic, Journal of Symbolic Computation 14, 371–397.

    Google Scholar 

  • Méndez, J. M. and Robles, G. (2016). Strengthening Brady’s paraconsistent 4-valued logic BN4 with truth-functional modal operators, Journal of Logic, Language and Information 25, 163–189.

    Google Scholar 

  • Meyer, R. K. (2004). Ternary relations and relevant semantics, Annals of Pure and Applied Logic 127, 195–217.

    Google Scholar 

  • Meyer, R. K. and Giambrone, S. and Brady, R. T. (1984). Where Gamma fails, Studia Logica 43, 247–256.

    Google Scholar 

  • Monteiro, A. (1963). Algèbres de Nelson semi-simples. Résumé d’une communication présentée à la U.M.A. en octobre 1962, Revista de la Unión Mathemática Argentina 21, 145–146.

    Google Scholar 

  • Monteiro, A. (1995). Les algèbres de Nelson semi-simples, Technical Report 50, Universidad Nacional del Sur, BahÍa Blanca, Also available in Unpublished papers I, Notas de Lógica Matemática 40, Instituto de Matemática, Universidad Nacional del Sur, Bahía Blanca (1996).

    Google Scholar 

  • Moore, R. (1985). Semantical considerations on nonmonotonic logic, Artificial Intelligence. An International Journal 25, 75–94.

    Google Scholar 

  • Muskens, R. (1995). Meaning and Partiality, Studies in Logic, Language, and Information. Stanford: CSLI Publications and FoLLI.

    Google Scholar 

  • Nelson, D. (1949). Constructible falsity, Journal of Symbolic Logic 14, 16–26.

    Google Scholar 

  • Nemitz, W. (1965). Implicative semilattices, Transactions of the American Mathematical Society 117, 128–142.

    Google Scholar 

  • Noguera, C. (2007). Algebraic Study of Axiomatic Extensions of Triangular Norm Based Fuzzy Logics, Number 27 in Monografies de l’Institut d’Investigació en Intel·ligència Artificial. Bellaterra: Institut d’Investigació en Intel·ligència Artificial.

    Google Scholar 

  • Noguera, C. and Esteva, F. and Gispert, J. (2008). On triangular norm based axiomatic extensions of the weak nilpotent minimum logic, Mathematical Logic Quarterly 54, 403–425.

    Google Scholar 

  • Odintsov, S. P. (2003). Algebraic semantics for paraconsistent Nelson’s logic, Journal of Logic and Computation 13, 453–468.

    Google Scholar 

  • Odintsov, S. P. (2004). On the representation of N4-lattices, Studia Logica 76, 385–405.

    Google Scholar 

  • Odintsov, S. P. (2005). The class of extensions of Nelson’s paraconsistent logic, Studia Logica 80, 291–320.

    Google Scholar 

  • Odintsov, S. P. (2007). On extensions of Nelson’s logic satisfying Dummett’s axiom, Siberian Mathematical Journal 48, 232–247.

    Google Scholar 

  • Odintsov, S. P. (2008). Constructive Negations and Paraconsistency, Volume 26 of Trends in Logic. Studia Logica Library, Berlin: Springer.

    Google Scholar 

  • Odintsov, S. P. and Speranski, S. O. (2016). The lattice of Belnapian modal logics: Special extensions and counterparts, Logic and Logical Philosophy 25, 3–33.

    Google Scholar 

  • Omori, H. and Sano, K. (2014). da Costa meets Belnap and Nelson, In R. Ciuni, H. Wansing, and C. Willkommen (Eds.), Recent Trends in Philosophical Logic, Volume 41 of Trends in Logic. Studia Logica Library, pp. 145–166. Heidelberg: Springer.

    Google Scholar 

  • Omori, H. and Waragai, T. (2011). Some observations on the systems LFI and LFI*, In F. Morvan, A. M. Tjoa, and R. R. Wagner (Eds.), Twentysecond International Workshop on Database and Expert System Applications (DEXA 2011). Toulouse, France, 29 August–2 September 2011. Proceedings, Los Alamitos, pp. 320–324. IEEE Computer Society Press.

    Google Scholar 

  • Parks, R. Z. (1972). A note on R-mingle and Sobociński’s three-valued logic, Notre Dame Journal of Formal Logic 13, 227–228.

    Google Scholar 

  • Pawlak, Z. (1982). Rough sets, International Journal of Computer and Information Sciences 11, 341–356.

    Google Scholar 

  • Pearce, D. (1999). From here to there: Stable negation in logic programming, In D. M. Gabbay and H. Wansing (Eds.), What is Negation?, Volume 13 of Applied Logic Series, pp. 161–181. Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Pei, D. (2003). On equivalent forms of fuzzy logic systems NM and IMTL, Fuzzy Sets and Systems. An international journal in Information Science and Engineering 138, 187–195.

    Google Scholar 

  • Pei, D. and Wang, G. (2002). The completeness and applications of the formal system L*, Science in China. Series F: Information Science 45, 40–50.

    Google Scholar 

  • Phillips, J. D. and Stanovský, D. (2010). Automated theorem proving in quasigroup and loop theory, AI Communications 23, 267–283.

    Google Scholar 

  • Pigozzi, D. (1975). Equational logic and equational theories of algebras, Technical report CSD-135, Computer Science Department, Purdue University, Indiana, http://docs.lib.purdue.edu/cstech/85, accessed 12 March 2016.

  • Pigozzi, D. (1991). Fregean algebraic logic, In H. Andréka, J. D. Monk, and I. Németi (Eds.), Algebraic Logic, Volume 54 of Colloquia Mathematica Societatis János Bolyai Budapest (Hungary), pp. 473–502. New York: North-Holland Publishing Company.

    Google Scholar 

  • Priest, G. (1989). Reasoning about truth, Artificial Intelligence. An International Journal 39, 231–244.

    Google Scholar 

  • Pynko, A. P. (1999). Definitional equivalence and algebraizability of generalised logical systems, Annals of Pure and Applied Logic 98, 1–68.

    Google Scholar 

  • Raftery, J. G. (2006). The equational definability of truth predicates, Reports on Mathematical Logic 41, 95–149.

    Google Scholar 

  • Rasiowa, H. (1958). -lattices and constructive logic with strong negation, Fundamenta Mathematicae 46, 61–80.

    Google Scholar 

  • Rasiowa, H. (1959). Algebraische Charakterisierung der intuitionistischen Logik mit starker Negation, In A. Heyting (Ed.), Constructivity in Mathematics. Proceedings of the Colloquium held at Amsterdam, 1957, Studies in Logic and the Foundations of Mathematics, Amsterdam, pp. 234–240. North-Holland Publishing Company.

    Google Scholar 

  • Rasiowa, H. (1974). An Algebraic Approach to Non-Classical Logics, Volume 78 of Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland Publishing Company.

    Google Scholar 

  • Restall, G. (1993). How to be really contraction free, Studia Logica 52, 381–391.

    Google Scholar 

  • Restall, G. (2000). An Introduction to Substructual Logics, London: Routledge.

    Google Scholar 

  • Restall, G. (2005). Łukasiewicz, supervaluations, and the future, L&PS — Logic and Philosophy of Science 3, 1–10.

    Google Scholar 

  • Restall, G. (2006). Relevant and substructural logics, In D. M. Gabbay and J. Woods (Eds.), Logic and the Modalities in the Twentieth Century, Volume 7 of Handbook of the History of Logic, pp. 289–398. Amsterdam: Elsevier.

    Google Scholar 

  • Rivieccio, U. (2010). An Algbraic Study of Bilattice-based Logics, Ph. D. thesis, University of Barcelona, arXiv:1010.2552[math.LO], accessed 12 March 2016.

  • Rivieccio, U. (2011). Paraconsistent modal logics, Electronic Notes in Theoretical Computer Science 278, 173–186.

    Google Scholar 

  • Rivieccio, U. (2014a). Algebraic semantics for bilattice public announcement logic, In A. Indrzejczak, J. Kaczmarek, and M. Zawidzki (Eds.), Trends in Logic XIII. Łódź, Poland, July 2014. Proceedings, Łódź, pp. 199–215. Łódź University Press.

    Google Scholar 

  • Rivieccio, U. (2014b). Bilattice public announcement logic, In R. Goré, B. Kooi, and A. Kurucz (Eds.), Advances in Modal Logic. Proceedings, Volume 10, London, pp. 459–477. College Publications.

    Google Scholar 

  • Rivieccio, U. and Bou, F. and Jansana, R. (2011). Varieties of interlaced bilattices, Algebra Universalis 66, 115–141.

    Google Scholar 

  • Routley, R. (1974). Semantical analyses of propositional systems of Fitch and Nelson, Studia Logica 33, 283–298.

    Google Scholar 

  • Sano, K. and Omori, H. (2014). An expansion of first-order Belnap-Dunn logic, Logic Journal of the IGPL. Interest Group in Pure and Applied Logic 22, 458–481.

    Google Scholar 

  • Schütte, K. (1960). Beweistheorie, Berlin: Springer-Verlag.

    Google Scholar 

  • Sendlewski, A. (1984). Some investigations of varieties of N-lattices, Studia Logica 43, 257–280.

    Google Scholar 

  • Sendlewski, A. (1990). Nelson algebras through Heyting ones: I, Studia Logica 49, 105–126.

    Google Scholar 

  • Slaney, J. (1984). A metacompleteness theorem for contraction-free relevant logics, Studia Logica 43, 159–168.

    Google Scholar 

  • Slaney, J. (1991). The implications of paraconsistency, In J. Mylopoulos and R. Reiter (Eds.), Twelfth International Joint Conference on Artificial Intelligence (IJCAI-91). Sydney, Australia, August 1991. Proceedings, San Francisco, pp. 1052–1057. Morgan Kaufmann.

    Google Scholar 

  • Slaney, J. (2004). Relevant logic and paraconsistency, In L. Bertossi, A. Hunter, and T. Schaub (Eds.), Inconsistency Tolerance, Volume 3300 of Lecture Notes in Computer Science, pp. 270–293. Berlin: Springer-Verlag.

    Google Scholar 

  • Slaney, J. (2010). A logic for vagueness, Australasian Journal of Logic 8, 100–134.

    Google Scholar 

  • Slaney, J. and Surendonk, T. and Girle, R. (1989). Time, truth and logic, Technical Report TR-ARP-11/89, Automated Reasoning Project, Australian National University, Canberra, http://www.rsise.anu.edu.au/papers/slaney/TTL/TTL.ps.gz, accessed 12 March 2016.

  • Smiley, T. (1962). The independence of connectives, Journal of Symbolic Logic 27, 426–436.

    Google Scholar 

  • Smiley, T. (1963). Relative necessity, Journal of Symbolic Logic 28, 113–134.

    Google Scholar 

  • Spinks, M. (2004). Ternary and quaternary deductive terms for Nelson algebras, Algebra Universalis 51, 125–136.

    Google Scholar 

  • Spinks, M. and Bignall, R. J. and Veroff, R., Discriminator logics, In preparation.

    Google Scholar 

  • Spinks, M. and Bignall, R. J. and Veroff, R. (2014). Discriminator logics (Research announcement), Australasian Journal of Logic 11, 159–171.

    Google Scholar 

  • Spinks, M. and Veroff, R. (a). Paraconsistent constructive logic with strong negation is a contraction-free relevant logic. I. Term equivalence, In preparation.

    Google Scholar 

  • Spinks, M. and Veroff, R. (b). Paraconsistent constructive logic with strong negation is a contraction-free relevant logic. II. Definitional equivalence, In preparation.

    Google Scholar 

  • Spinks, M. and Veroff, R. (c). Paraconsistent constructive logic with strong negation is a contraction-free relevant logic. III. Extensions and expansions, In preparation.

    Google Scholar 

  • Spinks, M. and Veroff, R. (2008a). Constructive logic with strong negation is a substructural logic. I, Studia Logica 88, 325–348.

    Google Scholar 

  • Spinks, M. and Veroff, R. (2008b). Constructive logic with strong negation is a substructural logic. II, Studia Logica 89, 401–425.

    Google Scholar 

  • Spinks, M. and Veroff, R. (2010). Slaney’s logic F** is constructive logic with strong negation, Bulletin of the Section of Logic 39, 161–174.

    Google Scholar 

  • Suszko, R. (1968). Ontology in the Tractatus of L. Wittgenstein, Notre Dame Journal of Formal Logic 9, 7–33.

    Google Scholar 

  • Suszko, R. (1971). Identity connective and modality, Studia Logica 27, 9–39.

    Google Scholar 

  • Suszko, R. (1975). Abolition of the Fregean axiom, In R. Parikh (Ed.), Logic Colloquium. Symposium on Logic held at Boston, 1972–73, Volume 453 of Lecture Notes in Mathematics, pp. 169–236. Berlin: Springer-Verlag.

    Google Scholar 

  • Tarski, A. (1968). Equational logic and equational theories of algebras, In H. A. Schmidt, K. Schütte, and H.-J. Thiele (Eds.), Contributions to mathematical logic. Proceedings of the Logic Colloquium, Hannover 1966, Volume 50 of Studies in Logic and the Foundations of Mathematics, pp. 275–288. Amsterdam: North-Holland Publishing Company.

    Google Scholar 

  • Taylor, W. (Survey 1979). Equational logic, Houston Journal of Mathematics.

    Google Scholar 

  • Tokarz, M. (1975). Functions definable in Sugihara algebras and their fragments (I), Studia Logica 34, 295–304.

    Google Scholar 

  • Troelstra, A. S. (1992). Lectures on linear logic, Number 29 in CSLI Lecture Notes. Stanford: Center for the Study of Language and Information.

    Google Scholar 

  • Vakarelov, D. (1977). Notes on N-lattices and constructive logic with strong negation, Studia Logica 36, 109–125.

    Google Scholar 

  • Vakarelov, D. (2006). Non-classical negation in the works of Helena Rasiowa and their impact on the theory of negation, Studia Logica 84, 105–127.

    Google Scholar 

  • Veroff, R. and Spinks, M. (2006). Axiomatizing the skew Boolean propositional calculus, Journal of Automated Reasoning 37, 3–20.

    Google Scholar 

  • Viglizzo, I. D. (1999). Álgebras de Nelson (Nelson Aglebras), Master’s thesis, Universidad Nacional del Sur, BahÍa Blanca, http://sites.google.com/site/viglizzo/investigacion/viglizzo99nelson, accessed 12 March 2016.

  • Villadsen, J. (2001). Combinators for paraconsistent attitudes, In P. de Groote, G. Morrill, and C. Retoré (Eds.), Logical Aspects of Computational Linguistics. 4th International Conference, LACL 2001. Le Croisic, France, June 27–29, 2001. Proceedings, Volume 2099 of Lecture notes in Artificial Intelligence. Subseries of Lecture Notes in Computer Science, pp. 91–102.

    Google Scholar 

  • Wang, G. J. (1997). A formal deductive system for fuzzy propositional calculus, Chinese Science Bulletin 42, 1041–1045.

    Google Scholar 

  • Wang, G. J. (1999). On the logic foundation of fuzzy reasoning, Information Sciences 117, 47–88.

    Google Scholar 

  • Wang, G. J. (2000). Non-classical Mathematical Logic and Approximate Reasoning, Beijing: Science Press, (Chinese).

    Google Scholar 

  • Wang, G. J. (2002). MV-algebras, BL-algebras, R0-algebras, and multiplevalued logic, Fuzzy Systems and Mathematics 15, 1–15, (Chinese).

    Google Scholar 

  • Wang, G. J. and Wang, H. (2001). Non-fuzzy versions of fuzzy reasoning in classical logics, Information Sciences 138, 211–236.

    Google Scholar 

  • Wansing, H. (1993). The Logic of Information Structures, Volume 681 of Lecture notes in Artificial Intelligence. Subseries of Lecture Notes in Computer Science, Berlin: Springer-Verlag.

    Google Scholar 

  • Werner, H. (1970). Eine charakterisierung funktional vollständiger algebren, Archiv der Mathematik 21, 381–385.

    Google Scholar 

  • Werner, H. (1978). Discriminator-Algebras, Number 6 in Studien zur Algebra und ihre Anwendungen. Berlin: Akademie-Verlag.

    Google Scholar 

  • Wójcicki, R. (1988). Theory of Logical Calculi. Basic Theory of Consequence Operations, Volume 199 of Synthese Library. Studies in Epistemology, Logic, Methodology, and Philosophy of Science, Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Zhou, X. and Li, Q. (2010). Boolean products of R 0-algebras, Mathematical Logic Quarterly 56, 289–298.

    Google Scholar 

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  1. Department of Philosophy, University of Cagliari, 09123, Cagliari, Italy

    Matthew Spinks

  2. Department of Computer Science, University of New Mexico, Albuquerque, NM, 87131, USA

    Robert Veroff

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    Janusz Czelakowski

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Spinks, M., Veroff, R. (2018). Paraconsistent constructive logic with strong negation as a contraction-free relevant logic. In: Czelakowski, J. (eds) Don Pigozzi on Abstract Algebraic Logic, Universal Algebra, and Computer Science. Outstanding Contributions to Logic, vol 16. Springer, Cham. https://doi.org/10.1007/978-3-319-74772-9_13

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