Advertisement

Many-Valued Logic

  • Nicholas Rescher
Chapter
Part of the Synthese Library book series (SYLI, volume 17)

Abstract

Throughout the orthodox mainsteam of the development of logic in the West, the prevailing view was that every proposition is either true or else false (although which of these is the case may well neither be necessary as regards the matter itself nor determinable as regards our knowledge of it). This thesis, commonly called the ‘law of bivalence’ — constituting one key articulation of the ‘law of the excluded middle’ — was, however, already questioned in antiquity. In Ch. IX of his treatise On Interpretation (De interpretatione), Aristotle discussed of the truth-status of alternatives regarding ‘future-contingent’ matters, whose occurrence — like that of the sea-battle tomorrow — is not yet determinable by us, and may indeed actually be undetermined. His views on the matter are still disputed, but many commentators, both in antiquity and later, held him to assert that propositions about future contingents, like that asserting the occurrence of the sea-battle, are neither actually true nor actually false, but potentially either, thus having — at least prior to the event — a third, indeterminate truth-status. The acceptance of the principle of bivalence was, in antiquity, closely bound up with the doctrine of determinism. The Epicureans, who were indeterminists, rejected the law of bivalence; the Stoics (and above all Chrysippus) who were rigid determinists, insisted upon it.1

Keywords

Propositional Logic Philosophical Logic Axiom Scheme Propositional Connective Notre Dame Journal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

A Brief Bibliography of Many-Valued Logic

  1. Ackermann, Robert M.(1967) Introduction to Many-Valued Logics (London, 1967).Google Scholar
  2. Baudry, L. (1950) La Querelle des futurs contingents (Louvain 1465–1475), Textes inédits (Paris, 1950 ).Google Scholar
  3. Bernstein, B. A. (1928) Modular Representations of Finite Algebras, Proceedings of the International Mathematical Congress Held in Toronto, August 11–16, 1924, 1 (1928) 207–216.Google Scholar
  4. Birkhoff, G. (1936) The Logic of Quantum Mechanics (with J. von Neumann), Annals of Mathematics 37 (1936) 823–843.CrossRefGoogle Scholar
  5. Bochvar, D.A. (1939) Od odnom tréhznacnom iscislénii i égo priménénii k analizu paradoksov klassicéskogo rasgirénnogo funkcionalnogo iscislénija [On a three-valued logical calculus and its application to the analysis of contradictions], Matématiééskij sbornik 4 (1939) 287–308.Google Scholar
  6. Bochvar, D.A. (1943) K voprosu o neprotivorecivosti odnogo trehznacnogo iséislénija [On the consistency of a three-valued calculus], Matématiééskij sbornik 12 (1943) 353–369.Google Scholar
  7. Boehner, Philotheus (1945) The “Tractatus de praedestinatione et de praescientia Dei et de futuris contingentibus” of William Ockham, edited with a study on the medieval problem of a three-valued logic (St. Bonaventure, New York; Franciscan Institute Publications, 1945 ).Google Scholar
  8. Brouwer, L. E. J. (1925) Intuitionistische Zerlegung mathematischen Grundbegriffe, Jahresbericht der deutschen Mathematiker-Vereinigung 33 (1925) 251–256.Google Scholar
  9. Chang, C.C. (1958) Algebraic Analysis of Many-Valued Logics, Transactions of the American Mathematical Society 88 (1958) 467–490.CrossRefGoogle Scholar
  10. Chang, C.C. (1959) A New Proof of the Completeness of the Lukasiewicz Axioms, Transactions of the American Mathematical Society 93 (1959) 74–80.Google Scholar
  11. Chang, C.C. (1962) Model Theories with Truth-Values in a Uniform Space (with H. Jerome Keisler), Bulletin of the American Mathematical Society 68 (1962) 107–109.Google Scholar
  12. Chang, C.C. (1965) Infinite-Valued Logic as a Basis for Set Theory, in Y. Bar Hillel (ed.), Logic, Methodology and Philosophy of Science: Proceedings of the 1964 International Congress (Amsterdam, 1965 ) p. 93–100.Google Scholar
  13. Church, Alonzo (1953) Non-normal Truth-tables for the Propositional Calculus, Boletinde la SociedadMatemdtica Mexicana 10 (1953) 41–52. (1956) Introduction to Mathematical Logic (Princeton, 1956 ).Google Scholar
  14. Destouches-Février, P. (1951) La structure des théories physiques (Paris, 1951 ).Google Scholar
  15. Dugundji, James (1940) Note on a Property of Matrices for Lewis and Langfords Calculi of Propositions, The Journal of Symbolic Logic 5 (1940) 150–151.Google Scholar
  16. Evans, Trevor (1957) Sheffer Stroke Functions in Many-Valued Logics (with Lane Hardy), Portugaliae Mathematica 16 (1957) 83–93.Google Scholar
  17. Feyerabend, Paul (1958) Reichenbachs Interpretation of Quantum-Mechanics, Philosophical Studies 9 (1958) 49–59.CrossRefGoogle Scholar
  18. Feys, R. (1937–38) Les logiques nouvelles des modalités, Revue néoscolastique de philosophie 40 (1937) 517–533, and 41 (1938) 217–252.Google Scholar
  19. Foxley, Eric (1962) The Determination of all Sheffer Functions in 3-Valued Logic, Using a Logical Computer, Notre Dame Journal of Formal Logic 3 (1962) 41–50.CrossRefGoogle Scholar
  20. Gödel, Kurt (1932) Zum intuitionistischen Aussagenkalkill, Anzeiger der Akademie der Wissenschaften in Wien, mathematisch-naturwissenschaftliche Klasse 69 (1932) 65–66.Google Scholar
  21. Hardy, Lane (1957) Sheffer Stroke Functions in Many-valued Logics (with Trevor Evans), Portugaliae Mathematica 16 (1957) 83–93. HEMPEL, Carl G. (1937) A Purely Topological Form of Non-Aristotelian Logic, The Journal of Symbolic Logic 2 (1937) 97–112.Google Scholar
  22. Heyting, A. ( 1956 ) Intuitionism: An Introduction (Amsterdam 1956 ).Google Scholar
  23. Keisler, H. Jerome (1962) Model Theories with Truth Values in a Uniform Space (with C. C. Chang), Bulletin of the American Mathematical Society 68 (1962) 107–109.CrossRefGoogle Scholar
  24. Kline, George L. (1965) N.A. Vasilev and the Development of Many-Valued Logics, Contributions to Logic and Methodology in Honor of J. M. Bochenski, ed. by Anna-Teresa Tymieniecka in collaboration with Charles D. Parsons (Amsterdam, 1965 ) pp. 315–326.Google Scholar
  25. Kolmogorov, A.N.(1924–25)Oprinciptertium non datur, Matskij sbornik 32 (1924–25) 646–667.Google Scholar
  26. Kuznetzov, B.G. (1959) Ob osnovax kvantovo-reljativistakoj logiki [On Quantum and Relativistic Logic], Logicéskie Issledovanija (Moscow, Institut Filosofii, Izdatelstvo Akademii Nauk SSSR ) (1959) 99–112.Google Scholar
  27. Langford, Cooper H. (1932) Symbolic Logic (with C. I. Lewis) (New York, 1932 ).Google Scholar
  28. Levi, Isaac (1959) Putnams Three Truth-Values, Philosophical Studies 10 (1959) 65–69.CrossRefGoogle Scholar
  29. Lewis, C. I. (1932a) Alternative Systems of Logic, The Monist 42 (1932) 481–507. (1932b) Symbolic Logic (with Cooper H. Langford) (New York, 1932 ).Google Scholar
  30. Lukasiewicz, Jan (1920) O logice trbjwarto§ciowej [On three-valued logic], Ruch Filozoficzny 5 (1920) 169–171.Google Scholar
  31. Lukasiewicz, Jan (1930a) Untersuchungen über den Aussagenkalkül (with Alfred Tarski), Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Classe III, 23 (1930) 1–21, 30–50.Google Scholar
  32. Lukasiewicz, Jan (1930b) Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls, Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Classe III, 23 (1930) 51–77.Google Scholar
  33. Lukasiewicz, Jan (1952) On the Intuitionistic Theory of Deduction, Indagationes Mathematicae 14 (1952) 202–212.Google Scholar
  34. Maccoll, Hugh (1877–98) The Calculus of Equivalent Statements, Proceedings of the London Mathematical Society 9 (1877–78) 177–186; 10 (1878–79) 16–28; 11 (1879–80) 113–121; 28 (1896–97) 156183,555–579 ; 29 (1897–98) 98–109.Google Scholar
  35. Margenau, H. (1950) The Nature of Physical Reality (New York, 1950 ).Google Scholar
  36. Martin, Norman M. (1950) Some Analogues of the Sheffer Stroke Function in n-Valued Logic, Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings of the section of sciences, 53, 1100–1107; also in Indagationes Mathematicae 12 (1950) 393–400.Google Scholar
  37. Martin, Norman M. (1951) A Note on Sheffer Functions in n-Valued Logic, Methodos 3 (1951) 240–242.Google Scholar
  38. Martin, Norman M. (1954) The Sheffer Functions of Three-Valued Logic, The Journal of Symbolic Logic 19 (1954) 45–51.Google Scholar
  39. Meredith, C. A. (1958) The Dependence of an Axiom of Lukasiewicz, Transactions of the American Mathematical Society 87 (1958) 54.Google Scholar
  40. Meredith, C. A. (1963) Notes on the Axiomatics of Propositional Calculus (with A. N. Prior), Notre Dame Journal of Formal Logic 4 (1963) 171–187.CrossRefGoogle Scholar
  41. Michalski, Konstanty (1937) Le problème de la volonté à Oxford et à Paris au XIVe siècle, Studia Philosophica 2 (1937) 233–365.Google Scholar
  42. Moh Shaw-Kwel (1954) Logical Paradoxes for Many-Valued Systems, The Journal of Symbolic Logic 19 (1954) 37–40.Google Scholar
  43. Moisil, Gr. C. (1938) Sur la théorie classique de la modalité des jugements, Bulletin Mathématique de la Société Roumaine des Sciences 40 (1938) 235–240.Google Scholar
  44. Moisil, Gr. C. (1939) Les principes de la logique et lidée de chaîne, Revista de filosofie 24 (1939) 3–12.Google Scholar
  45. Moisil, Gr. C. (1941) Recherches sur la théorie des chaînes, Annales scientifiques de lUniversité de Jassy 27 (1941) 181–240.Google Scholar
  46. Moisil, Gr. C. (1945) Lalgebra e la logica, Atti del Congresso Matematico, tenuto in Roma, 8–12 November 1942 ( Rome, Reale Istituto Nazionale di Alta Matematica ) (1945) pp. 143–152.Google Scholar
  47. Moisil, Gr. C. (1956) Utilisation des logiques trivalentes dans la théorie des mécanismes automatiques. II. Equation caractéristique dun relai polarisé, Comunicärile Academieii Republici Populare Romîne (Bucharest) 6 (1956) 231–234.Google Scholar
  48. Moisil, Gr. C. (1957) Aplicatrile logicii trivalente in studiul functionarii reale a schemelor cu contacte si reele [Application of three-valued logic in the study of the real operation of networks with contacts and relays], Bulletin Mathématique de la Société des Sciences Mathématiques et Physiques de la République Populaire de Roumanie 1 (49) (1957) 145–194.Google Scholar
  49. Moisil, Gr. C. (1960) Sur les idéaux des algèbres lukasiewicziennes trivalentes, Analele Universitatii Bucuresti Seria Acta Logica (Bucharest) 3 (1960) 83–95.Google Scholar
  50. Mostowski, Andrej (1961a) Axiomatizability of Some Many-Valued Predicate Calculi, Fundamenta Mathematicae 50 (1961) 165–190.Google Scholar
  51. Mostowski, Andrej (1961b) An Example of a Non-axiomatizable Many-Valued Logic, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 7 (1961) 72–76.CrossRefGoogle Scholar
  52. VON Neumann, John (1927) Zur Hilbertschen Beweistheorie, Mathematische Zeitschrift 26 (1927) 1–46.CrossRefGoogle Scholar
  53. VON Neumann (1936) The Logic of Quantum Mechanics(with G. Birkhoff), Annuals of Mathematics 37 (1936) 823–843.Google Scholar
  54. Post, Emil (1921) Introduction to a General Theory of Elementary Propositions, American Journal of Mathematics 43 (1921) 163–185. PRIOR, Arthur Norman (1953) Three-Valued Logic and Future Contingents, The Philosophical Quarterly 3 (1953) 317–326.Google Scholar
  55. Prior, Arthur Norman (1955) Many-Valued and Modal Systems: An Intuitive Approach, The Philosophical Review 64 (1955) 626–630.Google Scholar
  56. Prior, Arthur Norman (1957) Many-Valued Logics: The Last of Three Talks on “The Logic Game”, The Listener 57 (1957) 717–719.Google Scholar
  57. Prior, Arthur Norman (1963) Notes on the Axiomatics of Propositional Calculus (with C. A. Meredith), Notre Dame Journal of Formal Logic 4 (1963) 171–187.CrossRefGoogle Scholar
  58. Putnam, Hilary (1957) Three-Valued Logic, Philosophical Studies 8 (1957) 73–80.CrossRefGoogle Scholar
  59. Rasiowa, Helen (1950) Zydziedziny logiki matematycznej. II. Logiki wielowarto gciowe Lukasiewicza [From the domain of mathematical logic. II. The many-valued logics of Lukasiewicz], Matematyka 3 (1950) 4–11.Google Scholar
  60. Reichenbach,Hans (1935)Wahrscheinlichkeitslehre(Leyden, 1935).Google Scholar
  61. Reichenbach, Hans (1944) Philosophical Foundations of Quantum Mechanics ( Berkeley, Los Angeles, 1944 ).Google Scholar
  62. Reichenbach, Hans (1951) Ober die erkenntnistheoretische Problemlage und den Gebrauch einer dreiwertigen Logik in der Quantenmechanik, Zeitschrift für Naturforschung 6a (1951) 569–575.Google Scholar
  63. Reichenbach, Hans (1952–54), Hans (1952–54) Les fondements logiques de la théorie des quanta: Utilisation dune logique à trois valeurs, Applications scientifiques de la logique mathématique (Actes du 2e Colloque International de Logique Mathématique, Paris, 1952) pp. 103–114 ( Paris, Gauthier-Villars and Louvain, E. Nauwelaerts, 1954 ).Google Scholar
  64. Reichenbach, Hans (1954) Les fondements logiques de la théorie des quanta: Utilisation dune logique à trois valeurs, Collection de Logique mathématique 1952 (1954) 103–114.Google Scholar
  65. Rescher, Nicholas (1962) Quasi-Truth-Functional Systems of Propositional Logic, The Journal of Symbolic Logic 27 (1962) 1–10.Google Scholar
  66. Rescher, Nicholas (1963a) Studies in the History of Arabic Logic (Pittsburgh, 1963 ) pp. 43–44.Google Scholar
  67. Rescher, Nicholas (1963b) A Probabilistic Approach to Modal Logic, Acta Philosophica Fennica 16 (1963) 215–226.Google Scholar
  68. Rescher, Nicholas (1964) Quantifiers in Many-Valued Logic, Logique et Analyse 7 (1964) 181–184.Google Scholar
  69. Rescher, Nicholas (1965)An Intuitive Interpretation of Systems of Four-Valued Logic, Notre Dame Journal of Formal Logic 6 (1965)154–156.Google Scholar
  70. Rose, Alan (1950a) ‘A Lattice-Theoretic Characterization of Three-Valued Logic’, Journal of the London Mathematical Society 25 (1950) 255–259.CrossRefGoogle Scholar
  71. Rose, Alan(1950b) ‘Completeness of Lukasiewicz—Tarski Propositional Calculi’, Mathematische Annalen 122 (1950) 296–298.CrossRefGoogle Scholar
  72. Rose, Alan(1951a) Conditional Disjunction as a Primitive Connective for the m-Valued Propositional Calculus, Mathematische Annalen 123 (1951) 76–78.CrossRefGoogle Scholar
  73. Rose, Alan(1951 b) ‘Systems of Logic Whose Truth-Values Form Lattices’, Mathematische Annalen 123 (1951) 152–165.CrossRefGoogle Scholar
  74. Rose, Alan(1951c) A Lattice-Theoretic Characterization of the n-Valued Propositional Calculus, Mathematische Annalen 123 (1951) 285287.Google Scholar
  75. Rose, Alan(1951d) The Degree of Completeness of Some Lukasiewicz—Tarski Propositional Calculi, The Journal of the London Mathematical Society 26 (1951) 41–49.Google Scholar
  76. Rose, Alan(1952) The Degree of Completeness of the m-Valued Lukasiewicz Propositional Calculus, The Journal of the London Mathematical Society 27 (1952) 92–102.CrossRefGoogle Scholar
  77. Rose, Alan(1953) Some Self-dual Primitive Functions for Propositional Calculi, Mathematische Annalen 126 (1953) 144–148.CrossRefGoogle Scholar
  78. Rose, Alan(1958a) Sur les définitions de limplication et de la négation dans certains systèmes de logique dont les valeurs forment des treillis, Comptes rendus hebdomadaires des séances de lAcadémie des Sciences 246 (1958) 2091–2094.Google Scholar
  79. Rose, Alan(1958b) Fragments of Many-Valued Statement Calculi (with J. B. Rosser), Transactions of the American Mathematical Society 87 (1958) 1–53.CrossRefGoogle Scholar
  80. Rose, Alan(1961) Self-dual Binary and Ternary Connectives for in-Valued Propositional Calculi, Mathematische Annalen 143 (1961) 448–462.CrossRefGoogle Scholar
  81. Rose, Alan (1965) A Formalisation of Posts m-Valued Propositional Calculus with Variable Functors, Zeitschriftfür mathematische Logik and Grundlagen der Mathematik 11 (1965) 221–226.CrossRefGoogle Scholar
  82. Rosser, J. Barkley (1939) The Introduction of Quantification into a Three-Valued Logic, abstracted in The Journal of Symbolic Logic 4 (1939) 170–171.Google Scholar
  83. Rosser, J. Barkley (1948) Axiom Schemes for m-Valued Functional Calculi of First Order. Part I. Definition of Axiom Schemes and Proof of Plausibility (with A. R. Turquette), The Journal of Symbolic Logic 13 (1948) 177–192.CrossRefGoogle Scholar
  84. Rosser, J. Barkley (1951) Axiom Schemes for m-Valued Functional Calculi of First Order. Part II. Deductive Completeness (with A. R. Turquette), The Journal of Symbolic Logic 16 (1951) 22–34.CrossRefGoogle Scholar
  85. Rosser, J. Barkley (1952) Many-Valued Logics (with A. R. Turquette) (Amsterdam, 1952 ).Google Scholar
  86. Rosser, J. Barkley (1958) Fragments of Many-Valued Statement Calculi (with Alan Rose), Transactions of the American Mathematical Society 87 (1958) 1–53.CrossRefGoogle Scholar
  87. Schrter, Karl (1955) Methoden zur Axiomatisierung beliebiger Aussagen-and Prädikatenkalküle, Zeitschrift für Mathematische Logik and Grundlagen der Mathematik 1 (1955) 241–251.CrossRefGoogle Scholar
  88. Shannon, Claude E. (1938) A Symbolic Analysis of Relay and Switching Circuits, Transactions of the American Institute of Electrical Engineers 57 (1938) 713–723.CrossRefGoogle Scholar
  89. Shestakov, V.I. (1946) Prédstavlénié haraktéristicéskih funkcij prédlozenij posrédstvom vyrazénij, realizuémyh réléjno-kontaktnymi shé mami [Representation of characteristic functions of propositions by expressions realizable by relay-contact circuits], Izvéstia Akadémii Nauk SSSR [Bulletin de lAcadémie des Sciences delURSS],Sériâmatematieéskaja 10 (1946) 529–554.Google Scholar
  90. Shestakov, V.I. (1953) Modelirovanie operacij iscislenija prédlozenij posrédstvom prostejsih cetyrehpolusnyh shem [Modeling the operations of the propositional calculus by means of the simplest four-pole networks], Vyéislitelnaja matematika i vyeislitel naja texnika 1 (1953) 56–89.Google Scholar
  91. Shestakov, V.I. (1960) O dvojnaj arifineticeskoj interpretacii trexznacnogo iscislenija vyskazyvanij, Primenenie logiki v nauke i texnike (Moscow, Institut Filosofii, Izdatélstvo Akademii Nauk SSSR ), (1960) pp. 341–376.Google Scholar
  92. Skolem, Thoralf (1957) Mengenlehre gegründet auf einer Logik mit unendlich vielen Wahrheitswerten, Sitzungsberichte der Berliner Mathematischen Gesellschaft (1957) 41–56.Google Scholar
  93. Sugihara, Takeo (1951) Brouwer ronrigaku no tachi-ronrigaku-teik tokusei [Many-valued logical characteristics of Brouwerian logic], Kagaku 21 (1951) 294–295.Google Scholar
  94. Tarski, Alfred (1930) Untersuchungen über den Aussagenkalkül (with J. Lukasiewicz), Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Classe III, 23 (1930) 1–21, 30–50.Google Scholar
  95. Tarski, Alfred (1935–36) Wahrscheinlichkeitslehre und mehrwertige Logik, Erkenntnis 5 (1935–36) 174–175.Google Scholar
  96. Trnebohm, Hakan (1957) On Two Logical Systems Proposed in the Philosophy of Quantum Mechanics, Theoria 23 (1957) 84–101.Google Scholar
  97. Turquette A. R. (1948) Axiom Schemes for m-Valued Functional Calculi of First Order. Part I. Definition of Axiom Schemes and Proof of Plausibility (with J. B. R.sser), The Journal of Symbolic Logic 13 (1948) 177–192.CrossRefGoogle Scholar
  98. Tarski, Alfred (1951) Axiom Schemes for m-Valued Functional Calculi of First Order. Part II. Deductive Completeness (with J. B. Rosser), The Journal of Symbolic Logic 16 (1951) 22–34.CrossRefGoogle Scholar
  99. Tarski, Alfred (1952) Many-Valued Logics (with J.B. Rosser) (Amsterdam, 1952 ).Google Scholar
  100. Vasilév, N. A. (1910) O castnyh suzdéniâh, o tréugolniké protivopoloznostéj, o zakoné iskli énn ago cétvértago [On particular propositions, the triangle of opposition, and the law of excluded fourth], Ucënié zapiski Kanzanskogo Universitéte (1910), 47 pp.Google Scholar
  101. Vasilv, N. A. (1912) Voobrazaémaâ (néaristotéléva) logika [Imaginary (Non-Aristotelian) logic], Zurnal Ministérstva Narodnogo Prosvéséenia 40 (1912) 207–246.Google Scholar
  102. Wajsberg, Mordchaj (1931) ‘Aksjomatyzacja trbjwartgciowego rachunku zdan’ [Axiomatization of the 3-Valued propositional calculus], Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Classe III, 24 (1931) 126–148.Google Scholar
  103. Weiss, Paul (1933) ‘On Alternative Logics’, ThePhilos. Review42(1933) 520–525.Google Scholar
  104. Zawirski,Zygmunt (1931) ‘Logikatrjwartosciowa Jans Lukasiewicza. O logice L. E. J. Brouwera. Pr6by stosowania logiki wielowartogciowej do wsp lczesnego przyrodoznawstwa’ [Jan Lukasiewicz three-valued logic. On the logic of L. E. J. Brouwer. Attempts at application of many-valued logic to contemporary natural science], Sprawozdania Poznanskiego Towarzystwa Przyjaciol Nauk 2 (1931) nos. 2–4.Google Scholar
  105. Zawirski, Zygmunt (1934) ‘Znaczenie logiki wielowartogciowej dla poznania i zwiqzek jej z rachunkiem prawdopodobienstwa’ [Significance ofm any-valued logic for cognition and its connection with the calculus of probability], Przeglad Filozoficzny 37 (1934) 393–398.Google Scholar
  106. Zawirski, Zygmunt (1935) Über des Verhältnis mehrwertigen Logik zur Wahrschein lichkeitsrechnung, Studia Philosophia 1 (1935) 407–442.Google Scholar
  107. Zinovev, A. A. (1963) Philosophical Problems of Many-Valued Logic, ed. and tr. By G. Kiang and D. D. Comey (Dordrecht, 1963 ).Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1968

Authors and Affiliations

  • Nicholas Rescher

There are no affiliations available

Personalised recommendations