Many-valuedness from a universal logic perspective


Jean-Yves Béziau


We start by presenting various ways to define and to talk about many-valued logic(s). We make the distinction between on the one hand the class of many-valued logics and on the other hand what we call “many-valuedness”: the meta-theory of many-valued logics and the related meta-theoretical framework that is useful for the study of any logical systems. We point out that universal logic, considered as a general theory of logical systems, can be seen as an extension of many-valuedness. After a short story of many-valuedness, stressing that it is present since the beginning of the history of logic in Ancient Greece, we discuss the distinction between dichotomy and polytomy and the possible reduction to bivalence. We then examine the relations between singularity and universality and the connection of many-valuedness with the universe of logical systems. In particular, we have a look at the interrelationship between modal logic, 3-valued logic and paraconsistent logic. We go on by dealing with philosophical aspects and discussing the applications of many-valuedness. We end with some personal recollections regarding Alexander Karpenko, from our first meeting in Ghent, Belgium in 1997, up to our last meeting in Saint Petersburg, Russia in 2016.




Non-classical logics


Asenjo, 1966 – Asenjo, F.G. “A calculus of antinomies”, Notre Dame Journal of Formal Logic, 1966, Vol. 7, No. 103, pp. 103–105.
Avron et al., 2018 – Avron, A., Arieli, O., Zamansky, A. Theory of effective proposi- tional paraconsistent logics, College Publication, London, 2018.
Belnap, 1977 – Belnap, N.D. “A useful four-valued logic”, in: Modern uses of multiple- valued logic, ed. by M. Dunn. Reidel, Boston, 1977, pp. 8–37.
Béziau, 1994 – Béziau, J.-Y. “Universal logic”, in: Logica’94 – Proceedings of the 8th International Symposium, T. Childers and O. Majer (eds.), Prague, 1994, pp. 73– 93.
Béziau, 1997 – Béziau, J.-Y. “What is many-valued logic ?”, in Proceedings of the 27th International Symposium on Multiple-Valued Logic, IEEE Computer Society, Los Alamitos, 1997, pp. 117–121.
Béziau, 2000 – Béziau, J.-Y. “What is paraconsistent logic?”, in: Frontiers of para- consistent logic, Research Studies Press, Baldock, 2000, pp. 95–111.
Béziau, 2004 – Béziau, J.-Y. “Non truth-functional many-valued semantics”, in As- pects of Universal Logic, J.-Y. Béziau, A. Costa-Leite and A. Facchini (eds.), Uni- versity of Neuchaˆtel, Neuchaˆtel, 2004, pp. 199–218.
Béziau, 2006a – Béziau, J.-Y. “The paraconsistent logic Z – A possible solution to Ja ́skowski’s problem”, Logic and Logical Philosophy, 2006, Vol. 15, pp. 99–111.
Béziau, 2006b – Béziau, J.-Y. “13 Questions about universal logic”, Bulletin of the Section of Logic, 2006, Vol. 35, pp. 133–150.
Béziau, 2010 – Béziau, J.-Y. “Logic is not logic”, Abstracta, 2010, Vol. 6, pp. 73–102. B ́eziau, 2011 – B ́eziau, J.-Y. “A new four-valued approach to modal logic”, Logique et
Analyse, 2011, Vol. 54, pp. 109–121.
Béziau, 2012a – Béziau, J.-Y. “History of truth-values”, in: Handbook of the History of Logic, Vol. 11 – Logic: a history of its central concepts, D.M. Gabbay, F.J. Pelletier and J. Woods (eds.), Elsevier, Amsterdam, 2012, pp. 233–305.
Béziau, 2012b – Béziau, J.-Y. “Paralogics and the theory of valuation”, in: Universal Logic: an Anthology – From Paul Hertz to Dov Gabbay, Birkha ̈user, Basel, 2012, pp. 361–372.
Béziau, 2012c – Universal Logic: an Anthology – From Paul Hertz to Dov Gabbay, ed. by Béziau J.-Y., Birkha ̈user, Basel, 2012.
Béziau, 2012d – Logica Universalis, Special Double Issue on the Hexagon of Opposi- tion, ed. by B ́eziau J.-Y., 2012, Vol. 6, Issue 1–2.
Béziau, 2013 – Béziau, J.-Y. “The metalogical hexagon of opposition”, Argumentos, 2013, Vol. 10, pp. 111–122.
Béziau, 2016a – Béziau, J.-Y. “Trivial dialetheism and the logic of paradox”, Logic and Logical Philosophy, 2016, Vol. 25, pp. 51–56.
Béziau, 2016b – Béziau, J.-Y. “Two Genuine 3-Valued Paraconsistent Logics”, in: To- wards Paraconsistent Engineering, ed. by S.Akama, Springer International Pub- lishing, Cham, 2016, pp. 35–47.
Béziau, 2018a – Béziau, J.-Y. “The Pyramid of Meaning”, in: A Coat of Many Col- ours — D60, J. Ceuppens, H. Smessaert, J. van Craenenbroeck and G. Vanden Wyngaerd (eds.), Brussels, 2018.
Béziau, 2018b – Béziau, J.-Y. “Universal Logic: Evolution of a Project”, Logica Uni- versalis, 2018, Vol. 12, pp. 1–8.
Béziau, 2019 – Béziau, J.-Y. “Cats that are not cats”, in: Natural Arguments — A Tribute to John Woods, D. Gabbay, L. Magnani, W. Park, and A.V. Pietar- inen (eds.), College Publications, London, 2019, pp. 49–71.
Béziau, 2020 – Béziau, J.-Y. “Metalogic, Schopenhauer and Universal Logic”, in: Mathematics, Logic and Language in Schopenhauer, ed. by Jens Lemanski, Birkha ̈user, Basel, 2020.
Béziau, Franceschetto, 2016 – Béziau, J.-Y., Franceschetto, A. “Strong three-valued paraconsistent logics”, in: New Directions in Paraconsistent Logic, J.-Y. Béziau, M. Chakraborty and S. Dutta (eds.), Springer, New Delhi, 2016, pp. 131–145.
Blanch ́e, 1966 – Blanch ́e, R. Structures intellectuelles. Essai sur l’organisation syst ́ematique des concepts, Vrin, Paris, 1966.
da Costa et al., 1996 – da Costa, N.C.A., Béziau, J.-Y., Bueno, O.A.S. “Malinowski and Suszko on many-valued logics: On the reduction of many-valuedness to two- valuedness”, Modern Logic, 1996, Vol. 6, pp. 272–299.
Cobreros et al., 2014 – Cobreros, P., Egré, P., Ripley D., van Rooij, R. “Foreword: Three-valued logics and their applications”, Journal of Applied Non-Classical Lo- gics, 2014, Vol. 24, pp. 1–11.
D’Ottaviano, da Costa, 1970 – D’Ottaviano, I.M.L., da Costa, N.C.A. “Sur un probléme de Jaśkowski”, Comptes Rendus de l’Académie des Sciences de Paris, 1970, Vol. 270, pp. 1349–1353.
Dugundji, 1940 – Dugundji, J. “Note on a property of matrices for Lewis and Lang- ford’s calculi of propositions”, The Journal of Symbolic Logic, 1940, Vol. 5, pp. 150– 151.
Epstein, 1993 – Epstein, G. Multiple-Valued Logic Design, Institute of Physics Pub- lishing, Bristol, 1993.
Février, 1937 – Février, P. “Les relations d’incertitude d’Heisenberg et la logique”, in: Travaux du IXe Congr`es International de Philosophie, Vol. 6, Hermann, Paris, 1937, pp. 88–94,
Karpenko, 1999 – Karpenko, A. “Jaśkowski’s criterion and three-valued paraconsist- ent logics”, Logc and Logical Philosophy, 1999, Vol. 7, pp. 81–86.
Karpenko, 1999 – Karpenko, A. “Jaśkowski’s criterion and three-valued paraconsist- ent logics”, Logc and Logical Philosophy, 1999, Vol. 7, pp. 81–86.
Karpenko, 2006 – Karpenko, A. Łukasiewicz Logics and Prime Numbers, Luniver Press, Beckington, 2006.
Łukasiewicz, 1920 – Łukasiewicz, J. “O logice tr ́jwartosciowej”, Ruch Filozoficny, 1920, Vol. 5, pp. 170–171.
Łukasiewicz, 1953 – Łukasiewicz, J. “A system of modal logic”, Journal of Computing Systems, 1953, Vol. 1, pp. 111–149.
Łukasiewicz, Tarski, 1930 – Łukasiewicz, J., Tarski, A. “Untersuchungen u ̈ber den Aussagenkalku ̈l”, Comptes Rendus des S ́eances de la Soci ́et ́e des Sciences et des Lettres des Varsovie, Classe III, 1930, Vol. 23, pp. 30–50.
Malinowski, 1990 – Malinowski, G. “Q-consequence operation”, Reports on mathem- atical logic, 1990, Vol. 24, pp. 49–54.
Malinowski, 2007 – Malinowski, G. “Many-valued logic and its philosophy”, in: Hand- book of the history of logic, Volume 8, D.M. Gabbay, J. Woods (eds.), North- Holland, Amsterdam, 2007, pp. 13–94.
Priest, 1979 – Priest, G. “The logic of paradox”, Journal of Philosophical Logic, 1979, Vol. 8, pp. 219–241.
Quine, 1960 – Quine, W.V.O. Word and Object, MIT Press, Cambridge MA, 1960. Rescher, 1969 – Rescher, N. Many-Valued Logic, McGraw Hill, New York, 1969. Shramko, Wansing, 2011 – Shramko, Y., Wansing, H. Truth and Falsehood – An In-quiry inot Generalized Logical Values, Springer, Dordrecht, 2011.