2016 | 25 | 2 | 173–201
Article title

Relational semantics for the 4-valued relevant logics BN4 and E4

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The logic BN4 was defined by R.T. Brady in 1982. It can be considered as the 4-valued logic of the relevant conditional. E4 is a variant of BN4 that can be considered as the 4-valued logic of (relevant) entailment. The aim of this paper is to define reduced general Routley-Meyer semantics for BN4 and E4. It is proved that BN4 and E4 are strongly sound and complete w.r.t. their respective semantics.
Physical description
  • Dpto. de Psicología, Sociología y Filosofía, Universidad de León, Campus de Vegazana, s/n 24071, León, Spain ,
  • Anderson, A.R., and N.D. Belnap, Jr., Entailment. The Logic of Relevance and Necessity, vol. I, Princeton University Press, 1975.
  • Anderson, A.R., N.D. Belnap, Jr., and J.M. Dunn, Entailment. The Logic of Relevance and Necessity, vol. II, Princeton University Press, 1992.
  • Belnap, N.D., Jr., “Entailment and relevance”, The Journal of Symbolic Logic, 25 (1960): 388–389.
  • Belnap, N.D., Jr., “How a computer should think”, pages 30–55 in G. Ryle (ed.), Contemporary Aspects of Philosophy, Oriel Press Ltd., Stocksfield, 1977.
  • Belnap, N.D., Jr., “A useful four-valued logic”, pages 8–37 in J.M. Dunn and G. Epstein (eds.), Modern Uses of Multiple-Valued Logic, D. Reidel Publishing Co., Dordrecht, 1977.
  • Brady, R.T., “Completeness Proofs for the Systems RM3 and BN4”, Logique et Analyse 25 (1982): 9–32.
  • Brady, R.T. (ed.), Relevant Logics and Their Rivals, vol. II, Ashgate, Aldershot, 2003.
  • Brady, R.T., Universal Logic, CSLI, Stanford, CA, 2006.
  • Dunn, J.M., “Partiality and its Dual”, Studia Logica, 65 (2000): 5–40. DOI: 10.1023/A:1026740726955
  • González, C., MaTest, 2012. Available at (Last access 23/03/2016)
  • Meyer, R.K., S. Giambrone, and R.T. Brady, “Where gamma fails”, Studia Logica, 43 (1984): 247–256. DOI: 10.1007/BF02429841
  • Odintsov, S.P., and H. Wansing, “Modal logics with Belnapian truth values”, Journal of Applied Non-Classical Logics, 20 (2010): 279–301. DOI: 10.3166/jancl.20.279-304
  • Robles, G., “A Routley-Meyer semantics for Gödel 3-valued logic and its paraconsistent counterpart”, Logica Universalis 7 (2013): 507–532. DOI: 10.1007/s11787-013-0088-7
  • Robles, G., and J.M. Méndez, “A Routley-Meyer semantics for truth-preserving and well-determined Łukasiewicz 3-valued logics”, Logic Journal of the IGPL 22 (2014): 1–23. DOI: 10.1093/jigpal/jzt017
  • Robles, G., and J. M. Méndez, “The non-relevant De Morgan minimal logic in Routley-Meyer semantics with no designated points”, Journal of Applied Non-Classical Logics, 24 (2014): 321–332. DOI: 10.1080/11663081.2014.972306
  • Robles, G., and J.M. Méndez, “A companion to Brady’s 4-valued relevant logic BN4: The 4-valued logic of entailment E4”, Logic Journal of the IGPL, First published online: April 11, 2016. DOI: 10.1093/jigpal/jzw011
  • Routley, R., R.K. Meyer, V. Plumwood, 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. DOI: 10.1007/978-3-540-30597-2_9
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