Extensions of the basic constructive logic for weak consistency BKc1 defined with a falsity constant

Gemma Robles

DOI: http://dx.doi.org/10.12775/LLP.2007.010


The logic BKc1 is the basic constructive logic for weak consistency (i.e., absence of the negation of a theorem) in the ternary relational semantics without a set of designated points. In this paper, a number of extensions of B Kc1 defined with a propositional falsity constant are defined. It is also proved that weak consistency is not equivalent to negation-consistency or absolute consistency (i.e., non-triviality) in any logic included in positive contractionless intermediate logic LC plus the constructive negation of BKc1 and the (constructive) contraposition axioms.


weak consistency; constructive falsity; ternary relational semantics; substructural logics, paraconsistent logics

Full Text:



Anderson, A.R., and N.D. Jr. Belnap, Entailment. The Logic of Relevance and Necessity, vol. I, Princeton University Press, 1975.

Andréka, H., J.X. Madarász, and I. Nemeti, “Mutual definability does not imply definitional equivalence, a simple example”, Mathematical Logic Quaterly 51 (2005), 591–597.

Dummett, M., “A propositional calculus with a denumerable matrix”, Journal of Symbolic Logic 24 (1959), 97–106.

Hacking, I., “What is strict implication?”, Journal of Symbolic Logic 28 (1963), 51–71.

Méndez, J.M., and G. Robles, “Relevance logics, paradoxes of consistency and the K rule II. A non-constructive negation”, Logic and Logical Philosophy 15 (2006), 175–191.

Odintsov, S.P., “‘Reductio ad absurdum’ and Łukasiewicz’s modalities”, Logic and Logical Philosophy 11 (2003), 149–166.

Priest, G., and K. Tanaka, “Paraconsistent Logic”, in: E.N. Zalta (ed.), The Standford Encyclopedia of Philosophy, Winter 2004 Edition, 2004. http://plato.stanford.edu/archives/win2004/entries/logic-paraconsistent/.

Robles, G., and J.M. Méndez, “The basic constructive logic for a weak sense of consistency”, Journal of Logic Language and Information, accepted, forthcoming, 2007.

Robles, G., and J.M. Méndez, “The basic constructive logic for a weak sense of consistency defined with a propositional falsity constant”, Logic Journal of the IGPL, accepted.

Routley, R., et al., Relevant Logics and their Rivals, vol. 1, Ridgeview Publishing Co., Atascadero, CA, 1982.

Slaney, J., MaGIC, Matrix Generator for Implication Connectives: Version 2.1, Notes and Guide, Canberra, Australian National University, 1995.

Print ISSN: 1425-3305
Online ISSN: 2300-9802

Partnerzy platformy czasopism