Journal of Philosophical Logic

, Volume 43, Issue 2–3, pp 303–332 | Cite as

Dual Equivalent Two-valued Under-determined and Over-determined Interpretations for Łukasiewicz’s 3-valued Logic Ł3

  • Gemma Robles
  • Francisco Salto
  • José M. Méndez


Łukasiewicz three-valued logic Ł3 is often understood as the set of all 3-valued valid formulas according to Łukasiewicz’s 3-valued matrices. Following Wojcicki, in addition, we shall consider two alternative interpretations of Ł3: “well-determined” Ł3a and “truth-preserving” Ł3b defined by two different consequence relations on the 3-valued matrices. The aim of this paper is to provide (by using Dunn semantics) dual equivalent two-valued under-determined and over-determined interpretations for Ł3, Ł3a and Ł3b. The logic Ł3 is axiomatized as an extension of Routley and Meyer’s basic positive logic following Brady’s strategy for axiomatizing many-valued logics by employing two-valued under-determined or over-determined interpretations. Finally, it is proved that “well determined” Łukasiewicz logics are paraconsistent.


Many-valued logic Łukasiewicz 3-valued logic Two-valued under-determined and over-determined interpretations Paraconsistent logics 


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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Gemma Robles
    • 1
  • Francisco Salto
    • 1
  • José M. Méndez
    • 2
  1. 1.Dpto. de Psicología, Sociología y FilosofíaUniversidad de LeónLeónSpain
  2. 2.Universidad de SalamancaSalamancaSpain

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