Journal of Philosophical Logic 45 (4):451-465 (2016)

Authors
Stefan Wintein
Tilburg University
Reinhard Muskens
University of Amsterdam
Abstract
In their paper Nothing but the Truth Andreas Pietz and Umberto Rivieccio present Exactly True Logic, an interesting variation upon the four-valued logic for first-degree entailment FDE that was given by Belnap and Dunn in the 1970s. Pietz & Rivieccio provide this logic with a Hilbert-style axiomatisation and write that finding a nice sequent calculus for the logic will presumably not be easy. But a sequent calculus can be given and in this paper we will show that a calculus for the Belnap-Dunn logic we have defined earlier can in fact be reused for the purpose of characterising ETL, provided a small alteration is made—initial assignments of signs to the sentences of a sequent to be proved must be different from those used for characterising FDE. While Pietz & Rivieccio define ETL on the language of classical propositional logic we also study its consequence relation on an extension of this language that is functionally complete for the underlying four truth values. On this extension the calculus gets a multiple-tree character—two proof trees may be needed to establish one proof.
Keywords Belnap-Dunn logic  Gentzen calculi  Multiple tree calculi  Exactly true logic
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DOI 10.1007/s10992-015-9385-3
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References found in this work BETA

A Useful Four-Valued Logic.N. D. Belnap - 1977 - In J. M. Dunn & G. Epstein (eds.), Modern Uses of Multiple-Valued Logic. D. Reidel.
Logical Pluralism, Meaning-Variance, and VerbalDisputes.Ole Thomassen Hjortland - 2013 - Australasian Journal of Philosophy 91 (2):355-373.
Reasoning with Logical Bilattices.Ofer Arieli & Arnon Avron - 1996 - Journal of Logic, Language and Information 5 (1):25--63.

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