Strong Paraconsistency and the Basic Constructive Logic for an Even Weaker Sense of Consistency
First Online: 07 March 2009 Received: 05 September 2008 Accepted: 16 February 2009 DOI:
10.1007/s10849-009-9085-x Cite this article as: Robles, G. & Méndez, J.M. J of Log Lang and Inf (2009) 18: 357. doi:10.1007/s10849-009-9085-x Abstract
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). Keywords Consistency Paraconsistent logics Constructive negation Substructural logics Ternary relational semantics References
Anderson, A. R., & Belnap, N. D. Jr. (1975).
Entailment. The logic of relevance and necessity, Vol. I. Princeton University Press.
Andréka H., Madarász J.X., Nemeti I. (2005) Mutual definability does not imply definitional equivalence, a simple example. Mathematical Logic Quaterly 51: 591–597
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: 299–343
Dummett M. (1959) A propositional calculus with a denumerable matrix. The Journal of Symbolic Logic 24: 97–106
Hacking I. (1963) What is strict implication?. Journal of Symbolic Logic 28: 51–71
Méndez J.M., Robles G. (2009) The basic constructive logic for absolute consistency. Journal of Logic Language and Information 18(2): 199–216
Méndez J.M., Salto F. (2000) Intuitionistic propositional Logic without ‘contraction’ but with ‘reductio’. Studia Logica 66: 409–418
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): 45–58
Priest, G., & Tanaka, K. (2004). Paraconsistent logic. In E. N. Zalta (Ed.),
The standford encyclopedia of philosophy/
. Winter 2004 edition.
Restall G. (1999) Negation in relevant logics. In: Gabbay D., Wansing H. (eds) What is negation?. Kluwer, Boston, pp 53–76
Robles G. (2008) The basic constructive logic for negation-consistency. Journal of Logic Language and Information 17(2): 161–181
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.
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): 101–116
Robles G., Méndez J.M. (2008a) The basic constructive logic for a weak sense of consistency. Journal of Logic Language and Information 17(1): 89–107
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): 33–41
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).
Routley, R. et al. (1982).
Relevant logics and their rivals, Vol. 1. Atascadero, CA: Ridgeview Publishing Co.
Slaney, J. (1995).
MaGIC, matrix generator for implication connectives: Version 2.1, notes and guide
. Canberra: Australian National University.
© Springer Science+Business Media B.V. 2009