Abstract
The Belnap–Dunn logic (also known as First Degree Entailment, or FDE) is a well-known and well-studied four-valued logic, but until recently little has been known about its extensions, i.e. stronger logics in the same language, called super-Belnap logics here. We give an overview of several results on these logics which have been proved in recent works by Přenosil and Rivieccio. We present Hilbert-style axiomatizations, describe reduced matrix models, and give a description of the lattice of super-Belnap logics and its connections with graph theory. We adopt the point of view of Abstract Algebraic Logic, exploring applications of the general theory of algebraization of logics to the super-Belnap family. In this respect we establish a number of new results, including a description of the algebraic counterparts, Leibniz filters, and strong versions of super-Belnap logics, as well as the classification of these logics within the Leibniz and Frege hierarchies.
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References
Adams, M. E., and W. Dziobiak, Lattices of quasivarieties of 3-element algebras, Journal of Algebra 166: 181–210, 1994.
Albuquerque, H., J. M. Font, and R. Jansana, The strong version of a sentential logic, Studia Logica, 2017. doi:10.1007/s11225-017-9709-0.
Albuquerque, H., J. M. Font, and R. Jansana, Compatibility operators in abstract algebraic logic, The Journal of Symbolic Logic 81(2): 417–462, 2016.
Albuquerque, H., J. M. Font, R. Jansana, and T. Moraschini, Assertional logics, truth-equational logics and the hierarchies of AAL, in J. Czelakowski, (ed.), Don Pigozzi on Abstract Algebraic Logic and Universal Algebra. To appear.
Anderson, A. R., and N. D. Belnap, Entailment: The Logic of Relevance and Necessity, vol. 1, Princeton University Press, 1975.
Asenjo, F. G., A calculus of antinomies, Notre Dame Journal of Formal Logic 7(1): 103–105, 1966.
Babyonyshev, S. V., Fully Fregean logics, Reports on Mathematical Logic 37: 59–77, 2003.
Beall, J. C., T. Forster, and J. Seligman, A note on freedom from detachment in the Logic of Paradox, Notre Dame Journal of Formal Logic 54(1): 15–20, 2013.
Belnap, N. D., How a computer should think, in G. Ryle, (ed.), Contemporary Aspects of Philosophy, Oriel Press, Boston, 1977, pp. 30–56.
Belnap, N. D., A useful four-valued logic, in J. M. Dunn, and G. Epstein, (eds.), Modern uses of multiple-valued logic, Reidel, Dordrecht. Episteme, Vol. 2, 1977, pp. 5–37.
Blok, W. J., and D. Pigozzi, Algebraizable logics, Memoirs of the American Mathematical Society 77, 396, vi+78, 1989.
Blok, W. J., and J. G. Raftery, Assertionally equivalent quasivarieties, International Journal of Algebra and Computation 18(4): 589–681, 2008.
Blok, W. J., and J. Rebagliato, Algebraic semantics for deductive systems, Studia Logica 74(1–2): 153–180, 2003.
Bou, F., Implicación estricta y lógicas subintuicionistas, Master’s thesis, Universitat de Barcelona, 2001.
Burris, S., and H. P. Sankappanavar, A Course in Universal Algebra, vol. 78 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1981.
Cintula, P., and C. Noguera, The proof by cases property and its variants in structural consequence relations, Studia Logica 101(4): 713–747, 2013.
Cornish, W. H., and P. R. Fowler, Coproducts of De Morgan algebras, Bulletin of the Australian Mathematical Society 16: 1–13, 1977.
Czelakowski, J., Protoalgebraic logics, vol. 10 of Trends in Logic—Studia Logica Library, Kluwer Academic Publishers, Dordrecht, 2001.
Dunn, J. M., The algebra of intensional logics, Ph.D. thesis, University of Pittsburgh, Ann Arbor.
Dunn, J. M., Intuitive semantics for first-degree entailments and ‘coupled trees’, Philosophical Studies 29(3): 149–168, 1976.
Dunn, J. M., A Kripke-style semantics for R-mingle using a binary accessibility relation, Studia Logica 35: 163–172, 1976.
Dunn, J. M., Partiality and its dual, Studia Logica 66(1): 5–40, 2000.
Font, J. M., Belnap’s four-valued logic and De Morgan lattices, Logic Journal of the IGPL 5(3): 1–29, 1997.
Font, J. M., Abstract Algebraic Logic – An Introductory Textbook, vol. 60 of Studies in Logic, College Publications, London, 2016.
Font, J. M., and R. Jansana, Leibniz filters and the strong version of a protoalgebraic logic, Archive for Mathematical Logic 40: 437–465, 2001.
Font, J. M., and R. J., A general algebraic semantics for sentential logics, vol. 7 of Lecture Notes in Logic, second revised edition, Association for Symbolic Logic, 2009. Electronic version freely available through Project Euclid at http://projecteuclid.org/euclid.lnl/1235416965. First edition published by Springer-Verlag, 1996.
Gaitán, H., and M. H. Perea, A non-finitely based quasi-variety of De Morgan algebras, Studia Logica 78: 237–248, 2004.
Hell, P., and J. Nešetřil, Graphs and homomorphisms, vol. 28 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2004.
Jansana, R., Selfextensional logics with a conjunction, Studia Logica 84: 63–104, 2006.
Kalman, J. A., Lattices with involution, Transactions of the American Mathematical Society 87(2): 485–491, 1958.
Kleene, S. C., On notation for ordinal numbers, The Journal of Symbolic Logic 3(4): 150–155, 1938.
Kleene, S. C., Introduction to Metamathematics, vol. 1 of Bibliotheca Mathematica, North-Holland Publishing Co., Amsterdam, 1952.
Kripke, S. A., Outline of a theory of truth, Journal of Philosophy 72(19): 690–716, 1975.
Makinson, D. C., Topics in Modern Logic, Methuen, London, 1973.
Marcos, J., The value of the two values, in M. E. Coniglio, and J.-Y. Béziau, (eds.), Logic without Frontiers: Festschrift for Walter Alexandre Carnielli on the occasion of his 60th birthday, College Publications, London, 2011, pp. 277–294.
Pietz, A., and U. Rivieccio, Nothing but the truth, Journal of Philosophical Logic 42(1): 125–135, 2013.
Priest, G., The Logic of Paradox, Journal of Philosophical Logic 8(1): 219–241, 1979.
Přenosil, A., The lattice of super-Belnap logics, 201x. Submitted manuscript.
Pynko, A. P., Characterizing Belnap’s logic via De Morgan’s laws, Mathematical Logic Quarterly 41(4): 442–454, 1995.
Pynko, A. P., On Priest’s Logic of Paradox, Journal of Applied Non-Classical Logics 5(2): 219–225, 1995.
Pynko, A. P., Implicational classes of De Morgan lattices, Discrete Mathematics 205(1–3): 171–181, 1999.
Pynko, A. P., Subprevarieties versus extensions. Application to the logic of paradox, The Journal of Symbolic Logic 65(2): 756–766, 2000.
Raftery, J. G., The equational definability of truth predicates, Reports on Mathematical Logic 41: 95–149, 2006.
Rivieccio, U., An infinity of super-Belnap logics, Journal of Applied Non-Classical Logics 22(4): 319–335, 2012.
Wójcicki, R., Lectures on propositional calculi, Ossolineum Publishing Co., Wrocław, 1984.
Wójcicki, R., Theory of logical calculi: Basic theory of consequence operations, vol. 199 of Synthese Library, Kluwer Academic Publishers Group, Dordrecht, 1988.
Acknowledgements
Adam Přenosil gratefully acknowledges the support of the Grant P202/12/G061 of the Czech Science Foundation. The research of Umberto Rivieccio has been supported by EU FP7 Marie Curie PIRSES-GA-2012-318986 project and by CNPq process 482809/2013-2 of the Brazilian National Council for Scientific and Technological Development.
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Albuquerque, H., Přenosil, A. & Rivieccio, U. An Algebraic View of Super-Belnap Logics. Stud Logica 105, 1051–1086 (2017). https://doi.org/10.1007/s11225-017-9739-7
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DOI: https://doi.org/10.1007/s11225-017-9739-7