Functional Completeness and Axiomatizability within Belnap's Four-Valued Logic and its Expansions

Journal of Applied Non-Classical Logics 9 (1):61-105 (1999)
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Abstract

In this paper we study 12 four-valued logics arisen from Belnap's truth and/or knowledge four-valued lattices, with or without constants, by adding one or both or none of two new non-regular operations—classical negation and natural implication. We prove that the secondary connectives of the bilattice four-valued logic with bilattice constants are exactly the regular four-valued operations. Moreover, we prove that its expansion by any non-regular connective (such as, e.g., classical negation or natural implication) is strictly functionally complete. Further, finding axiomatizations of the quasi varieties generated by the 12 logics involved (that prove to be varieties), we find naturell equational axiomatizations of these logics. Finally, applying Pynko's general theory of algebraizable sequential consequence operations, we also find equivalent natural sequentiell axiomatizations of the logics under consideration that expand either of two Pynko's sequential calculi for the constant-free truth-lattice four-valued logic.

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Citations of this work

A Gentzen Calculus for Nothing but the Truth.Stefan Wintein & Reinhard Muskens - 2016 - Journal of Philosophical Logic 45 (4):451-465.
An infinity of super-Belnap logics.Umberto Rivieccio - 2012 - Journal of Applied Non-Classical Logics 22 (4):319-335.
An expansion of first-order Belnap-Dunn logic.K. Sano & H. Omori - 2014 - Logic Journal of the IGPL 22 (3):458-481.

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References found in this work

Introduction to metamathematics.Stephen Cole Kleene - 1952 - Groningen: P. Noordhoff N.V..
Characterizing Belnap's Logic via De Morgan's Laws.Alexej P. Pynko - 1995 - Mathematical Logic Quarterly 41 (4):442-454.
On Priest's logic of paradox.Alexej P. Pynko - 1995 - Journal of Applied Non-Classical Logics 5 (2):219-225.
Universal Algebra.George Grätzer - 1982 - Studia Logica 41 (4):430-431.

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