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


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.

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Méndez, J.M., Robles, G. A Strong and Rich 4-Valued Modal Logic Without Łukasiewicz-Type Paradoxes. Log. Univers. 9, 501–522 (2015) doi:10.1007/s11787-015-0130-z

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Mathematics Subject Classification

  • Primary 03B47
  • Secondary 03B45
  • 03B50
  • 03B53


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