Logica Universalis

, Volume 9, Issue 4, pp 501–522 | Cite as

A Strong and Rich 4-Valued Modal Logic Without Łukasiewicz-Type Paradoxes

  • José M. Méndez
  • Gemma Robles


The aim of this paper is to introduce an alternative to Łukasiewicz’s 4-valued modal logic Ł. As it is known, Ł is afflicted by “Łukasiewicz (modal) type paradoxes”. The logic we define, PŁ4, is a strong paraconsistent and paracomplete 4-valued modal logic free from this type of paradoxes. PŁ4 is determined by the degree of truth-preserving consequence relation defined on the ordered set of values of a modification of the matrix MŁ characteristic for the logic Ł. On the other hand, PŁ4 is a rich logic in which a number of connectives can be defined. It also has a simple bivalent semantics of the Belnap–Dunn type.


Many-valued logics modal logics paraconsistent logics paracomplete logics 4-valued modal logics Łukasiewicz 4-valued modal logic Belnap–Dunn type semantics 

Mathematics Subject Classification

Primary 03B47 Secondary 03B45 03B50 03B53 


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Copyright information

© Springer Basel 2015

Authors and Affiliations

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

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