A Philosophy of Many-Valued Logic. The Third Logical Value and Beyond

  • Grzegorz Malinowski
Part of the Logic, Epistemology, and the Unity of Science book series (LEUS, volume 16)

The roots of many-valued logic lie in Aristotle’s (4th century BC) discussion of future contingents and of tomorrow’s famous sea battle. Similar concerns can be found in medieval philosophy, in Duns Scot, William of Ockham and Peter de Rivo (Louvain). However, one needs to wait until the turn of the twentieth century to see the first attempts at creating non-classical – mainly three-valued – systems. By the late 1890s, Hugh MacColl had presented his so-called “three-dimensional logic”, Charles S. (1839–1914) was working on “trychotomic mathematics” and Nicolai A. Vasiliev was developing a system in which propositions can be “affirmative”, “negative” or “indifferent”.


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  1. Bolzano, B. (1837), Wissenschaftslehre, Sulzbach, Seidel.Google Scholar
  2. Gentzen, G. (1934), ‘Untersuchungen über das Logische Schliessen’, Mathematische Zeitschrift, 39, 176–210.CrossRefGoogle Scholar
  3. Gonseth, F. (ed.) (1941), Les entretiens de Zurich sur les fondements et la méthode des sciences mathématiques 6–9 décembre 1938, Zurich, S.A. Leemann frères & Cie.Google Scholar
  4. Jaśkowski S. (1934), ‘On the rules of suppositions in formal logic’, Studia Logica, 1, 5–32.Google Scholar
  5. Kotarbiński, T. (1913), ‘Zagadnienie istnienia przyszłości’ (The problem of existence of the future), Przegląd Filozoficzny, VI.1.Google Scholar
  6. Łukasiewicz, J. (1906), ‘Analiza i konstrukcja pojęcia przyczyny’ (An analysis and a construction of the concept of cause), Przegląd Filozoficzny, 9, (2–3), 105–179.Google Scholar
  7. Łukasiewicz, J. (1910), O zasadzie sprzeczności u Arystotelesa. Studium krytyczne (On the principle of contradiction), Kraków (PWN, Warszawa 1987).Google Scholar
  8. Łukasiewicz, J. (1913), Die logischen Grundlagen der Wahrscheinlichkeitsrechnung, Cracow. Eng. trans., in Borkowski, L. (ed.), Selected works, Amsterdam, North-Holland, 16–63.Google Scholar
  9. Łukasiewicz, J. (1920), ‘O logice trójwartościowej’, Ruch Filozoficzny, 5, 170–171. Eng. trans. On three-valued logic, Łukasiewicz 1970, 87–88.Google Scholar
  10. Łukasiewicz, J. (1930), ‘Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls’, Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Cl. III, 23, 51–77.Google Scholar
  11. Łukasiewicz, J. (1970) Selected works, L. Borkowski (ed.). Amsterdam, North-Holland.Google Scholar
  12. Malinowski, G. (1977), ‘Classical characterization of n-valued Łukasiewicz calculi’, Reports on Mathematical Logic, 9, 41–45.Google Scholar
  13. Malinowski, G. (1993), Many-Valued Logic, Oxford, Clarendon Press.Google Scholar
  14. Malinowski, G. (2001), ‘Many-valued logics’, in L. Goble (ed.), The Blackwell Guide to Philosophical Logic, Blackwell Publishers, Oxford, 309–335.Google Scholar
  15. Post, E. L. (1920), ‘Introduction to a general theory of elementary propositions’, Bulletin of the American Mathematical Society, 26, 437.Google Scholar
  16. Post, E. L. (1921), ‘Introduction to a general theory of elementary propositions’, American Journal of Mathematics, 43, 163–185.CrossRefGoogle Scholar
  17. Rosser, J., Turquette, A. (1952), Many-Valued Logics, North-Holland, Amsterdam.Google Scholar
  18. Rousseau, G. (1967), ‘Sequents in many-valued logic’, Fundamenta Mathematicae, 60, 23–33.Google Scholar
  19. Schröter K. (1955) ‘Methoden zur Axiomatisierung beliebiger Aussagen- und Prädikatenkalküle’ Zeitschrift für Mathematische Logik und Grunglagen der Mathematik, 1, 241–251.CrossRefGoogle Scholar
  20. Suszko, R. (1957), ‘Formalna teoria wartości logicznych’ (A formal theory of logical values). Studia Logica, VI, 145–320.CrossRefGoogle Scholar
  21. Suszko, R. (1972), ‘Abolition of the Fregean Axiom, in R. Parikh (ed.), Logic Colloquium, Symposium on Logic held at Boston, 1972–73. Lecture Notes in Mathematics, 453, 169–239.CrossRefGoogle Scholar
  22. Suszko, R. (1977), ‘The Fregean Axiom and Polish mathematical logic in the 1920s’, Studia Logica, 36/4, 377–380.CrossRefGoogle Scholar
  23. Takahashi, R. (1970), ‘Many-valued logics of extended Gentzen style II’, Journal of Symbolic Logic, 35 (231), 493–528.Google Scholar
  24. Urquhart, A. (1986), ‘Many-valued logic’, in D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, vol. III. Dordrecht, D. Reidel, 71–116.Google Scholar
  25. Zach, R. (1993), Proof theory of finite-valued logics. Master’s thesis, Institut für Algebra und Discrete Mathematik, TU Wien, 338–353.Google Scholar
  26. Zawirski, Z. (1934), ‘Znaczenie logiki wielowartościowej i związek jej z rachunkiem prawdopodobieństwa’ (Significance of many-valued logic for cognition and its connection with the calculus of probability), Przegląd Filozoficzny, 37, 393–398.Google Scholar

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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Grzegorz Malinowski
    • 1
  1. 1.Uniwersytet ŁódzkiŁodzPoland

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