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Subprevarieties versus extensions. Application to the logic of paradox

Published online by Cambridge University Press:  12 March 2014

Alexej P. Pynko*
Affiliation:
Department of Digital Automata Theory, V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Academik Glushkov Prosp. 40, Kiev 252022, Ukraine, E-mail: pynko@d105.icyb.kiev.ua

Abstract

In the present paper we prove that the poset of all extensions of the logic defined by a class of matrices whose sets of distinguished values are equationally definable by their algebra reducts is the retract, under a Galois connection, of the poset of all subprevarieties of the prevariety generated by the class of the algebra reducts of the matrices involved. We apply this general result to the problem of finding and studying all extensions of the logic of paradox (viz., the implication-free fragment of any non-classical normal extension of the relevance-mingle logic). In order to solve this problem, we first study the structure of prevarieties of Kleene lattices. Then, we show that the poset of extensions of the logic of paradox forms a four-element chain, all the extensions being finitely many-valued and finitely-axiomatizable logics. There are just two proper consistent extensions of the logic of paradox. The first is the classical logic that is relatively axiomatized by the Modus ponens rule for the material implication. The second extension, being intermediate between the logic of paradox and the classical logic, is the one relatively axiomatized by the Ex Contradictione Quodlibet rule.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2000

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References

[1]Balbes, R. and Dwinger, P., Distributive lattices, University of Missouri Press, Columbia (Missouri), 1974.Google Scholar
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[5]Pynko, A. P., Characterizing Belnap's logic via De Morgan's laws, Mathematical Logic Quarterly, vol. 41 (1995), pp. 442454.CrossRefGoogle Scholar
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[8]Pynko, A. P., Functional completeness and axiomatizability within Belnap's four-valued logic and its expansions, Journal of Applied Non-Classical Logics, vol. 9 (1999), pp. 61105, Special Issue on Multiple-Valued Logics.CrossRefGoogle Scholar
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