Advertisement

Journal of Logic, Language and Information

, Volume 23, Issue 4, pp 493–526 | Cite as

Blocking the Routes to Triviality with Depth Relevance

Article

Abstract

In Rogerson and Restall’s (J Philos Log 36, 2006, p. 435), the “class of implication formulas known to trivialize (NC)” (NC abbreviates “naïve comprehension”) is recorded. The aim of this paper is to show how to invalidate any member in this class by using “weak relevant model structures”. Weak relevant model structures verify deep relevant logics only.

Keywords

Naive set theory Weak relevant model structures Depth relevance Deep relevant logics 

Notes

Acknowledgments

Work supported by research project FFI2011-28494 financed by the Spanish Ministry of Economy and Competitiveness. G. Robles is supported by Program Ramón y Cajal of the Spanish Ministry of Economy and Competitiveness. We sincerely thank a referee of the Journal of Logic, Language and Information for his/her comments and suggestions on a previous draft of this paper.

References

  1. Anderson, A. R., & Belnap, N. D, Jr. (1975). Entailment. The logic of relevance and necessity. I. Princeton: Princeton University Press.Google Scholar
  2. Belnap, N. D, Jr. (1960). Entailment and relevance. Journal of Symbolic Logic, 25, 144–146.CrossRefGoogle Scholar
  3. Brady, R. T. (1984). Depth relevance of some paraconsistent logics. Studia Logica, 43, 63–73.CrossRefGoogle Scholar
  4. Brady, R. T. (1992). Hierarchical semantics for relevant logics. Journal of Philosophical Logic, 25, 357–374.Google Scholar
  5. Brady, R. T. (1996). Relevant implication and the case for a weaker logic. Journal of Philosophical Logic, 25, 151–183.CrossRefGoogle Scholar
  6. Brady, R. T. (Ed.). (2003). Relevant logics and their rivals, Vol. II. Aldershot: Ashgate.Google Scholar
  7. Brady, R. T. (2006). Universal logic. Stanford, CA: CSLI.Google Scholar
  8. Brady, R. T. (2013). The simple consistency of Naive set theory using metavaluations. Journal of Philosophical Logic. doi: 10.1007/s10992-012-9262-2.
  9. Curry, H. B. (1942). The combinatory foundations of logic. Journal of Symbolic Logic, 7, 49–64.CrossRefGoogle Scholar
  10. González, C. (2012). MaTest. http://ceguel.es/matest. Last accessed 2 March 2014.
  11. Méndez, J. M. (1988). The compatibility of relevance and mingle. Journal of Philosophical Logic, 17, 279–297.CrossRefGoogle Scholar
  12. Méndez, J. M. (2010). Erratum to: The compatibility of relevance and mingle. Journal of Philosophical Logic, 39(3), 339.CrossRefGoogle Scholar
  13. Robles, G., & Méndez, J. M. (2012). A general characterization of the variable-sharing property by means of logical matrices. Notre Dame Journal of Formal Logic, 53(2), 223–244.CrossRefGoogle Scholar
  14. Robles, G., & Méndez, J. M. (2014a). Curry’s paradox, generalized modus ponens axiom and depth relevance. Studia Logica, 102(1), 185–217.Google Scholar
  15. Robles, G., & Méndez, J. M. (2014b). Generalizing the depth relevance condition. Deep relevant logics not included in R-Mingle. Notre Dame Journal of Formal Logic, 55(1), 107–127.Google Scholar
  16. Robles, G., & Méndez, J. M. Depth relevance and the factor and summation axioms (in preparation).Google Scholar
  17. Rogerson, S., & Restall, G. (2006). Routes to triviality. Journal of Philosophical Logic, 33, 421–436.CrossRefGoogle Scholar
  18. Routley, R., Meyer, R. K., Plumwood, V., & Brady, R. T. (1982). Relevant logics and their rivals, Vol. I. Atascadero, CA: Ridgeview Publishing Co.Google Scholar
  19. Weber, Z. (2012). Transfinite cardinals in paraconsistent set theory. Review of Symbolic Logic, 5(2), 269–293.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Dpto. de Psicología, Sociología y FilosofíaUniversidad de LeónLeónSpain
  2. 2.Universidad de SalamancaSalamancaSpain

Personalised recommendations