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Logic and Logical Philosophy

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

Authors

  • Gemma Robles Universidad de León http://orcid.org/0000-0001-6495-0388
  • José M. Blanco Universidad de Salamanca
  • Sandra M. López Universidad de Salamanca
  • Jesús R. Paradela Universidad de Salamanca
  • Marcos M. Recio Universidad de Salamanca

DOI:

https://doi.org/10.12775/LLP.2016.006

Keywords

relevant logics, many-valued logics, 4-valued logics, Routley-Meyer semantics

Abstract

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.

Author Biographies

Gemma Robles, Universidad de León

Ramón y Cajal researcher (RYC-2010-06319) at the Dpto. de Psicología, Sociología y Filosofía

José M. Blanco, Universidad de Salamanca

Graduate student

Sandra M. López, Universidad de Salamanca

Graduate student

Jesús R. Paradela, Universidad de Salamanca

Graduate student

Marcos M. Recio, Universidad de Salamanca

Undergraduate student

References

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 http://ceguel.es/matest (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

Logic and Logical Philosophy

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Published

2016-04-25

How to Cite

1.
ROBLES, Gemma, BLANCO, José M., LÓPEZ, Sandra M., PARADELA, Jesús R. and RECIO, Marcos M. Relational semantics for the 4-valued relevant logics BN4 and E4. Logic and Logical Philosophy. Online. 25 April 2016. Vol. 25, no. 2, pp. 173-201. [Accessed 9 July 2025]. DOI 10.12775/LLP.2016.006.
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Vol. 25 No. 2 (2016): June

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