Journal of Logic, Language and Information

, Volume 25, Issue 2, pp 163–189 | Cite as

Strengthening Brady’s Paraconsistent 4-Valued Logic BN4 with Truth-Functional Modal Operators

  • José M. MéndezEmail author
  • Gemma Robles


Łukasiewicz presented two different analyses of modal notions by means of many-valued logics: (1) the linearly ordered systems Ł3,..., Open image in new window ,..., \(\hbox {L}_{\omega }\); (2) the 4-valued logic Ł he defined in the last years of his career. Unfortunately, all these systems contain “Łukasiewicz type (modal) paradoxes”. On the other hand, Brady’s 4-valued logic BN4 is the basic 4-valued bilattice logic. The aim of this paper is to show that BN4 can be strengthened with modal operators following Łukasiewicz’s strategy for defining truth-functional modal logics. The systems we define lack “Łukasiewicz type paradoxes”. Following Brady, we endow them with Belnap–Dunn type bivalent semantics.


Many-valued logics Modal many-valued logics Łukasiewicz many-valued logics Łukasiewicz 4-valued modal logic Brady’s 4-valued logic Bilattice logics Belnap–Dunn type bivalent semantics 

Mathematics Subject Classification




Work supported by research project FFI2014-53919-P financed by the Spanish Ministry of Economy and Competitiveness. We sincerely thank an anonymous referee of the JoLLI for his (her) comments and suggestions on a previous draft of this paper.


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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Universidad de SalamancaSalamancaSpain
  2. 2.Dpto. de Psicología, Sociología y FilosofíaUniversidad de LeónLeónSpain

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