Modal logics with Belnapian truth values

Journal of Applied Non-Classical Logics 20 (3):279-304 (2010)
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Various four- and three-valued modal propositional logics are studied. The basic systems are modal extensions BK and BS4 of Belnap and Dunn's four-valued logic of firstdegree entailment. Three-valued extensions of BK and BS4 are considered as well. These logics are introduced semantically by means of relational models with two distinct evaluation relations, one for verification and the other for falsification. Axiom systems are defined and shown to be sound and complete with respect to the relational semantics and with respect to twist structures over modal algebras. Sound and complete tableau calculi are presented as well. Moreover, a number of constructive non-modal logics with strong negation are faithfully embedded into BS4, into its three-valued extension B3S4, or into temporal BS4, BtS4. These logics include David Nelson's three-valued logic N3, the four-valued logic N4 bottom, the connexive logic C, and several extensions of bi-intuitionistic logic by strong negation.



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Heinrich Wansing
Ruhr-Universität Bochum

References found in this work

An Introduction to Non-Classical Logic: From If to Is.Graham Priest - 2008 - New York: Cambridge University Press.
A useful four-valued logic.N. D. Belnap - 1977 - In J. M. Dunn & G. Epstein (eds.), Modern Uses of Multiple-Valued Logic. D. Reidel.
An Introduction to Non-Classical Logic: From If to Is.Graham Priest - 2008 - Bulletin of Symbolic Logic 14 (4):544-545.
How a computer should think.Nuel Belnap - 1977 - In Gilbert Ryle (ed.), Contemporary aspects of philosophy. Boston: Oriel Press.

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