Journal of Philosophical Logic

, Volume 41, Issue 5, pp 901–922 | Cite as

Real Analysis in Paraconsistent Logic

Article

Abstract

This paper begins an analysis of the real line using an inconsistency-tolerant (paraconsistent) logic. We show that basic field and compactness properties hold, by way of novel proofs that make no use of consistency-reliant inferences; some techniques from constructive analysis are used instead. While no inconsistencies are found in the algebraic operations on the real number field, prospects for other non-trivializing contradictions are left open.

Keywords

Paraconsistent logic Non-classical mathematics Compactness theorems 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Batens, D., Mortensen, C., Priest, G., & van Bendegem, J.-P. (Eds.) (2000). Frontiers of paraconsistent logic. Philadelphia: Research Studies Press.Google Scholar
  2. 2.
    Beall, J., & Murzi, J. (2011). Two flavors of curry paradox. Under review; see http://homepages.uconn.edu/~jcb02005/.
  3. 3.
    Brady, R. (Ed.) (2003). Relevant logics and their rivals, volume II: A continuation of the work of Richard Sylvan, Robert Meyer, Val Plumwood and Ross Brady. Ashgate. With contributions by: Martin Bunder, Andre Fuhrmann, Andrea Loparic, Edwin Mares, Chris Mortensen, and Alasdair Urquhart.Google Scholar
  4. 4.
    Brady, R. (2006). Universal logic. Stanford: CSLI.Google Scholar
  5. 5.
    Brady, R. T. (1989). The non-triviality of dialectical set theory. In G. Priest, R. Routley, & J. Norman (Eds.), Paraconsistent logic: Essays on the inconsistent (pp. 437–470). Munich: Philosophia Verlag.Google Scholar
  6. 6.
    Bridges, D., & Richman, F. (1987). Varieties of constructive mathematics. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
  7. 7.
    Brown, B., & Priest, G. (2004). Chunk and permeate i: The infinitesimal calculus. Journal of Philosophical Logic, 33, 379–388.CrossRefGoogle Scholar
  8. 8.
    Brunner, A. B., & Carnielli, W. A. (2005). Anti-intuitionism and paraconsistency. Journal of Applied Logic, 3, 161–184.CrossRefGoogle Scholar
  9. 9.
    Butchart, S. (2008). Binary quantifiers for relevant paraconsistent logic. Presented at the Australasian Association for Philosophy Conference, Melbourne (to appear).Google Scholar
  10. 10.
    da Costa, N. (2000). Paraconsistent mathematics. In D. Batens, C. Mortensen, G. Priest, & J.-P. van Bendegem (Eds.), Frontiers of paraconsistent logic (pp. 165–180). Philadelphia: Research Studies Press.Google Scholar
  11. 11.
    da Costa, N. C. A. (1974). On the theory of inconsistent formal systems. Notre Dame Journal of Formal Logic, 15, 497–510.CrossRefGoogle Scholar
  12. 12.
    Field, H. (2008). Saving truth fromcontradiction. London, UK: Oxford University Press.CrossRefGoogle Scholar
  13. 13.
    Goodman, N. D. (1981). The logic of contradiction. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 27(8–10), 119–126.CrossRefGoogle Scholar
  14. 14.
    Lewis, C., & Langford, C. (1959). Symbolic logic. New York: Dover.Google Scholar
  15. 15.
    Mares, E. (2004). Relevant logic. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
  16. 16.
    Meyer, R. K., Routley, R., & Dunn, J. M. (1978). Curry’s paradox. Analysis, 39, 124–128.CrossRefGoogle Scholar
  17. 17.
    Mortensen, C. (1990). Models for inconsistent and incomplete differential calculus. Notre Dame Journal of Formal Logic, 31(2), 274–285.CrossRefGoogle Scholar
  18. 18.
    Mortensen, C. (1995). Inconsistent mathematics. Dordrecht, New York: Kluwer.Google Scholar
  19. 19.
    Mortensen, C. (2000). Prospects for inconsistency. In D. Batens, C. Mortensen, G. Priest, & J.-P. van Bendegem (Eds.), Frontiers of paraconsistent logic (pp. 203–208). Philadelphia: Research Studies Press.Google Scholar
  20. 20.
    Mortensen, C. (2009). Inconsistent mathematics: Some philosophical implications. In A. Irvine (Ed.), Handbook of the philosophy of science volume 9: Philosophy of mathematics. North Holland: Elsevier.Google Scholar
  21. 21.
    Priest, G. (2006). In contradiction: A study of the transconsistent (2nd ed.). Oxford: Oxford University Press.Google Scholar
  22. 22.
    Priest, G. (2008). An introduction to non-classical logic (2nd ed.). Cambridge.Google Scholar
  23. 23.
    Priest, G., Routley, R., & Norman, J. (Eds.) (1989). Paraconsistent logic: Essays on the inconsistent. Munich: Philosophia Verlag.Google Scholar
  24. 24.
    Routley, R. (1977). Ultralogic as universal? Relevance Logic Newsletter, 2, 51–89. Reprinted in R. Routley. Exploring Meinong’s Jungle and Beyond. Philosophy Department, RSSS, Australian National University, Canberra, 1980. Departmental Monograph number 3.Google Scholar
  25. 25.
    Routley, R., Plumwood, V., Meyer, R. K., & Brady, R. T. (1982). Relevant logics and their rivals. Ridgeview.Google Scholar
  26. 26.
    Weber, Z. (2010). Transfinite numbers in paraconsistent set theory. Review of Symbolic Logic, 3(1), 71–92.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CanterburyChristchurchNew Zealand
  2. 2.School of Historical and Philosophical StudiesUniversity of MelbourneMelbourneAustralia

Personalised recommendations