Journal of Logic, Language and Information

, Volume 18, Issue 3, pp 357–402 | Cite as

Strong Paraconsistency and the Basic Constructive Logic for an Even Weaker Sense of Consistency

  • Gemma RoblesEmail author
  • José M. Méndez


In a standard sense, consistency and paraconsistency are understood as the absence of any contradiction and as the absence of the ECQ (‘E contradictione quodlibet’) rule, respectively. The concepts of weak consistency (in two different senses) as well as that of F-consistency have been defined by the authors. The aim of this paper is (a) to define alternative (to the standard one) concepts of paraconsistency in respect of the aforementioned notions of weak consistency and F-consistency; (b) to define the concept of strong paraconsistency; (c) to build up a series of strongly paraconsistent logics; (d) to define the basic constructive logic adequate to a rather weak sense of consistency. All logics treated in this paper are strongly paraconsistent. All of them are sound and complete in respect a modification of Routley and Meyer’s ternary relational semantics for relevant logics (no logic in this paper is relevant).


Consistency Paraconsistent logics Constructive negation Substructural logics Ternary relational semantics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Anderson, A. R., & Belnap, N. D. Jr. (1975). Entailment. The logic of relevance and necessity, Vol. I. Princeton University Press.Google Scholar
  2. Andréka H., Madarász J.X., Nemeti I. (2005) Mutual definability does not imply definitional equivalence, a simple example. Mathematical Logic Quaterly 51: 591–597CrossRefGoogle Scholar
  3. Copeland B.J. (1979) On when a semantics is not a semantics: some reasons for disliking the Routley-Meyer semantics for relevance logics. Journal of Philosophical Logic 8: 299–343CrossRefGoogle Scholar
  4. Dummett M. (1959) A propositional calculus with a denumerable matrix. The Journal of Symbolic Logic 24: 97–106CrossRefGoogle Scholar
  5. Hacking I. (1963) What is strict implication?. Journal of Symbolic Logic 28: 51–71CrossRefGoogle Scholar
  6. Méndez J.M., Robles G. (2009) The basic constructive logic for absolute consistency. Journal of Logic Language and Information 18(2): 199–216CrossRefGoogle Scholar
  7. Méndez J.M., Salto F. (2000) Intuitionistic propositional Logic without ‘contraction’ but with ‘reductio’. Studia Logica 66: 409–418CrossRefGoogle Scholar
  8. Méndez J.M., Robles G., Salto F. (2007) The basic constructive logic for negation-consistency defined with a propositional falsity constant. Bulletin of the Section of Logic 36(1–2): 45–58Google Scholar
  9. Priest, G., & Tanaka, K. (2004). Paraconsistent logic. In E. N. Zalta (Ed.), The standford encyclopedia of philosophy/. Winter 2004 edition.
  10. Restall G. (1999) Negation in relevant logics. In: Gabbay D., Wansing H. (eds) What is negation?. Kluwer, Boston, pp 53–76Google Scholar
  11. Robles G. (2008) The basic constructive logic for negation-consistency. Journal of Logic Language and Information 17(2): 161–181CrossRefGoogle Scholar
  12. Robles, G., & Méndez, J. M. (2005a). On defining constructive negation in logics of entailment. In Paper presented at the First World Congress on Universal Logic, Montreux, Switzerland.Google Scholar
  13. Robles G., Méndez J.M. (2005b) Relational ternary semantics for a logic equivalent to Involutive Monoidal t-norm based logic IMTL. Bulletin of the Section of Logic 34(2): 101–116Google Scholar
  14. Robles G., Méndez J.M. (2008a) The basic constructive logic for a weak sense of consistency. Journal of Logic Language and Information 17(1): 89–107CrossRefGoogle Scholar
  15. Robles G., Méndez J.M. (2008b) The basic constructivec logic for a weak sense of consistency defined with a propositional falsity constant. Logic Journal of the IGPL 16(1): 33–41CrossRefGoogle Scholar
  16. Robles, G., Méndez, J. M., & Salto, F. (2007). Relevance logics, paradoxes of consistency and the K rule. Logique et Analyse, 198, 129–145. (An abstract of this paper was read at the Logic Colloquium 2006, Nijmegen, Holland, 27 July–2 August 2006).Google Scholar
  17. Routley, R. et al. (1982). Relevant logics and their rivals, Vol. 1. Atascadero, CA: Ridgeview Publishing Co.Google Scholar
  18. Slaney, J. (1995). MaGIC, matrix generator for implication connectives: Version 2.1, notes and guide. Canberra: Australian National University.

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Dpto. de Historia y Filosofía de la CC, la Ed. y el Leng.Universidad de La Laguna, Facultad de FilosofíaLa LagunaSpain
  2. 2.Universidad de SalamancaSalamancaSpain

Personalised recommendations