Belnap-Dunn Semantics for the Variants of BN4 and E4 which Contain Routley and Meyer’s Logic B

Sandra M. López



The logics BN4 and E4 can be considered as the 4-valued logics of the relevant conditional and (relevant) entailment, respectively. The logic BN4 was developed by Brady in 1982 and the logic E4 by Robles and Méndez in 2016. The aim of this paper is to investigate the implicative variants (of both systems) which contain Routley and Meyer’s logic B and endow them with a Belnap-Dunn type bivalent semantics.


4-valued logics; many-valued logics; Belnap-Dunn semantics; Routley and Meyer’s logic B; relevant logics

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Anderson, A.R., and N.D. Belnap, Jr., Entailment. The Logic of Relevance and Necessity, vol. I, Princeton University Press, 1975.

Arieli, O., and A. Avron, “Reasoning with logical bilattices”, Journal of Logic, Language and Information 5 (1996): 25–63.

Arieli, O., and A. Avron, “The value of the four values”, Artificial Intelligence 102 (1998): 97–141.

Belnap, N.D., Jr., “Entailment and relevance”, The Journal of Symbolic Logic 25, 2 (1960): 144–146.

Belnap, N.D., Jr., “A useful four-valued logic”, pages 8–37 in G. Epstein and J.M. Dunn (eds.),Modern Uses of Multiple-Valued Logic, D. Reidel Publishing Co., Dordrecht, 1977.

Belnap, N.D., Jr., “How a computer should think”, pages 30–35 in G. Ryle (ed.), Contemporary Aspects of Philosophy, Oriel Press Ltd., Stocksfield, 1977.

Brady, R.T., “Completeness proofs for the systems RM3 and BN4”, Logique et Analyse 25 (1982): 9–32.

Brady, R.T., “Rules in relevant logic — II: Formula representation”, Studia Logica 52 (1993): 565–585. DOI: 1

Dunn, J.M., “Intuitive semantics for first degree entailments and ‘couple trees’”, Philosophical Studies 29 (1976): 149–168.

Dunn, J.M., “Partiality and its dual”, Studia Logica 66, 1 (2000): 5–40. DOI:

Ginsberg, M.L., “Multi-valued logics”, pages 243–247 in Proceedings of the Fifth AAAI National Conference on Artificial Intelligence (AAAI’86), AAAI Press, 1986.

González, C., “MaTest”, 2012. Available at (last accessed 21/02/2020).

Méndez, J.M., and G. Robles, “Strengthening Brady’s paraconsistent 4-Valued logic BN4 with truth-functional modal operators”, Journal of Logic Language and Information 25, 2 (2016): 163–189. DOI:

Meyer, R.K., S. Giambrone and R.T. Brady, “Where gamma fails”, Studia Logica 43 (1984): 247–256.

Omori, H., and H. Wansing, “40 years of FDE: An introductory overview”, Studia Logica 105 (2017): 1021–1049. DOI:

Petrukhin, Y., and V. Shangin, “Correspondence analysis and automated proof-searching for first degree entailment”, European Journal of Mathematics 6 (2020): 1452–1495. DOI:

Robles, G., J.M. Blanco, S.M. López, J.R. Paradela and M.M. Recio, “Relational semantics for the 4-valued relevant logics BN4 and E4”, Logic and Logical Philosophy 25, 2 (2016): 173–201. DOI:

Robles, G., J.M. Méndez, “A companion to Brady’s 4-valued relevant logic BN4: The 4-valued logic of entailment E4”, Logic Journal of the IGLP 24, 5 (2016): 838–858.

Robles, G., J.M. Méndez, “Belnap-Dunn semantics for natural implicative expansions of Kleene’s strong three-valued matrix with two designated values”, Journal of Applied Non-Classical Logics 29, 1 (2019): 37–63. DOI:

Robles, G., S.M. López, J.M. Blanco, M.M. Recio and J.R. Paradela, “A 2-set-up Routley-Meyer semantics for the 4-valued relevant logic E4”, Bulletin of the Section of Logic 45, 2 (2016): 93–109. DOI:

Routley, R., V. Plumwood, R.K. Meyer and R.T. Brady, Relevant Logics and their Rivals, vol. 1, Atascadero, CA: Ridgeview Publishing Co., 1982.

Slaney, J.K., “Relevant logic and paraconsistency”, pages 270–293 in L. Bertossi, A. Hunter and T. Schaub (eds.), Inconsistency Tolerance, vol. 3300 of Lecture Notes in Computer Science, 2005.

Tomova, N., “A Lattice of implicative extensions of regular Kleene’s logics”, Reports on Mathematical Logic 47 (2012): 173–182. DOI:

ISSN: 1425-3305 (print version)

ISSN: 2300-9802 (electronic version)

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