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Pure three-valued Łukasiewiczian implication

Published online by Cambridge University Press:  12 March 2014

Storrs McCall  and
R. K. Meyer
Storrs McCall
Affiliation:
University of Pittsburgh and Makarere University College, Kampala, Uganda
R. K. Meyer
Affiliation:
University of Pittsburgh and Makarere University College, Kampala, Uganda
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Extract

The matrix defining Łukasiewicz's three-valued logic, constructed in 1920 and described at length in [1], is the following:

This matrix was axiomatized in 1931 by Wajsberg (see [6]), who showed that the following axioms together with the rules of substitution and modus ponens were sufficient:

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 31 , Issue 3 , 02 September 1966 , pp. 399 - 405
Copyright
Copyright © Association for Symbolic Logic 1996

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References

[1]Łukasiewicz, J., Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls, C.r. Soc. Sci. Lett. de Varsovie, Cl. III, 23 (1930), pp. 51–77. English translation to appear in Polish Logic, ed. S. McCall, to be published by the Clarendon Press.Google Scholar
[2]Meredith, C. A., The dependence of an axiom of Łukasiewicz, Transactions of the American Mathematical Society, 87 (1958), p. 54.Google Scholar
[3]Rose, A. and Rosser, J. B., Fragments of many-valued statement calculi, Transactions of the American Mathematical Society, 87 (1958), pp. 1–53.CrossRefGoogle Scholar
[4]Schröter, K., Methoden zur Axiomatisierung beliebiger Aussagen- und Prādikatenkalküle, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 1 (1955), pp. 241–251.CrossRefGoogle Scholar
[5]Tarski, A., Logic, Semantics, Metamathematics, Oxford 1956.Google Scholar
[6]Wajsberg, M., Aksjomatyzacja trójwartościowego rachunku zdán, C. r. Soc. Sci. Lett. de Varsovie, Cl. III, 24 (1931), pp. 126–145. English translation to appear in Polish Logic.Google Scholar