Advertisement

Journal of Philosophical Logic

, Volume 48, Issue 6, pp 1119–1144 | Cite as

Negation on the Australian Plan

  • Francesco BertoEmail author
  • Greg Restall
Open Access
Article

Abstract

We present and defend the Australian Plan semantics for negation. This is a comprehensive account, suitable for a variety of different logics. It is based on two ideas. The first is that negation is an exclusion-expressing device: we utter negations to express incompatibilities. The second is that, because incompatibility is modal, negation is a modal operator as well. It can, then, be modelled as a quantifier over points in frames, restricted by accessibility relations representing compatibilities and incompatibilities between such points. We defuse a number of objections to this Plan, raised by supporters of the American Plan for negation, in which negation is handled via a many-valued semantics. We show that the Australian Plan has substantial advantages over the American Plan.

Keywords

Negation Compatibility semantics Kripke semantics Non-classical logics Many-valued logics Modal logics 

Notes

References

  1. 1.
    Barwise, J., & Etchemendy, J. (1990). Information, infons, and inference. In Cooper, R., Mukai, K., Perry, J. (Eds.) Situation theory and its applications, (Vol. 1, number 22 pp. 33–78). Stanford.Google Scholar
  2. 2.
    Barwise, J., & Perry, J. (1983). Situations and attitudes. Bradford books. MIT Press.Google Scholar
  3. 3.
    Beall, J.C. (2009). Spandrels of truth. Oxford: Oxford University Press.CrossRefGoogle Scholar
  4. 4.
    Beall, J.C., & Restall, G. (2000). Logical pluralism. Australasian Journal of Philosophy, 78, 475–493. http://consequently.org/writing/pluralism.CrossRefGoogle Scholar
  5. 5.
    Beall, JC, & Restall, G. (2006). Logical pluralism. Oxford: Oxford University Press.Google Scholar
  6. 6.
    Berto, F. (2008). Adynaton and material exclusion. Australasian Journal of Philosophy, 86, 165–90.CrossRefGoogle Scholar
  7. 7.
    Berto, F. (2014). Absolute contradiction, dialetheism, and revenge. Review of Symbolic Logic, 7, 193–207.CrossRefGoogle Scholar
  8. 8.
    Berto, F. (2015). A modality called “negation”. Mind, 124(495), 761–793.CrossRefGoogle Scholar
  9. 9.
    Birkhoff, G., & von Neumann, J. (1936). The logic of quantum mechanics. The Annals of Mathematics, 37(4), 823–843. http://www.jstor.org/stable/1968621.CrossRefGoogle Scholar
  10. 10.
    Brandom, R.B. (1994). Making it explicit. Harvard University Press.Google Scholar
  11. 11.
    Brandom, R.B. (2000). Articulating reasons: an introduction to inferentialism. Harvard University Press. ISBN 0674001583.Google Scholar
  12. 12.
    De, M., & Omori, H. (2017). There is more to negation than modality. Journal of Philosophical Logic, 1–19. ISSN 1573-0433,  https://doi.org/10.1007/s10992-017-9427-0.CrossRefGoogle Scholar
  13. 13.
    Dunn, J.M., & Zhou, C. (2005). Negation in the context of gaggle theory. Studia Logica, 80, 235–64.CrossRefGoogle Scholar
  14. 14.
    Dunn, J.M. (1993). Partial-gaggles applied to logics with restricted structural rules. In Schroeder-Heister, P., & Došen, K. (Eds.) Substructural logics: Oxford University Press.Google Scholar
  15. 15.
    Dunn, J.M. (1994). Star and perp: two treatments of negation. In Tomberlin, J.E. (Ed.) Philosophical perspectives. http://www.jstor.org/stable/2214128, (Vol. 7 pp. 331–357). Atascadero: Ridgeview Publishing Company.CrossRefGoogle Scholar
  16. 16.
    Dunn, J.M. (1996). Generalised ortho negation. In Wansing, H. (Ed.) Negation: a notion in focus (pp. 3–26). Berlin: Walter de Gruyter.Google Scholar
  17. 17.
    Novaes, C.D. (2007). Contradiction: the real philosophical challenge for paraconsistent logic. In Béziau, J.Y., Carnielli, W., Gabbay, D. (Eds.) Handbook of paraconsistency (pp. 477–92). London: College Publications.Google Scholar
  18. 18.
    Fine, K. (2009). The question of ontology. In Chalmers, D., Manley, D., Wasserman, R. (Eds.) Metametaphysics (pp. 157–77). Clarendon.Google Scholar
  19. 19.
    Fitting, M. (1991). Bilattices and the semantics of logic programming. ISSN 0743-1066, (Vol. 11 pp. 91–116), DOI  https://doi.org/10.1016/0743-1066(91)90014-G, http://www.sciencedirect.com/science/article/pii/074310669190014G.CrossRefGoogle Scholar
  20. 20.
    Goldblatt, R. (1974). Semantic analysis of orthologic. The Journal of Philosophical Logic, 3(1–2), 19–35.  https://doi.org/10.1007/BF00652069. Reprinted as Chapter 3 of Mathematics of Modality, Robert Goldblatt. Mathematics of Modality. Number 43 in CSLI Lecture Notes. CSLI Publications, 1993. http://standish.stanford.edu/bin/detail?fileID=458253745.CrossRefGoogle Scholar
  21. 21.
    Grim, P. (2004). What is a contradiction? In Priest, G., Beall, J.C., Armour-Garb, B. (Eds.) The law of non-contradiction (pp. 49–72). Clarendon.Google Scholar
  22. 22.
    Incurvati, L., & Schlöder, J.J. (2017). Weak rejection. Australasian Journal of Philosophy, 95(4), 741–760.  https://doi.org/10.1080/00048402.2016.1277771.CrossRefGoogle Scholar
  23. 23.
    Kripke, S. (1965). Semantical analysis of intuitionistic logic. In Crossley, J., & Dummett, M.A.E. (Eds.) Formal Systems and Recursive Functions (pp. 92–130). Amsterdam: North-Holland Publishing.Google Scholar
  24. 24.
    Kripke, S. (1980). Naming and necessity. Oxford: Blackwell.Google Scholar
  25. 25.
    Lewis, D.K. (1973). Counterfactuals. Oxford: Blackwell.Google Scholar
  26. 26.
    Mares, E.D. (1995). A star-free semantics for R. Journal of Symbolic Logic, 60, 579–590.CrossRefGoogle Scholar
  27. 27.
    Mares, E.D. (2004). Relevant logic. A philosophical interpretation. Cambridge: Cambridge University press.CrossRefGoogle Scholar
  28. 28.
    Meyer, R.K., & Martin, E.P. (1986). Logic on the australian plan. The Journal of Philosophical Logic, 15(3), 305–332.  https://doi.org/10.1007/BF00248574.CrossRefGoogle Scholar
  29. 29.
    Price, H. (1990). Why ‘not’? Mind, 99, 221–38.CrossRefGoogle Scholar
  30. 30.
    Priest, G. (2008). An introduction to non-classical logic, 2nd edn. Vol. 2008. Cambridge: Cambridge University Press.Google Scholar
  31. 31.
    Restall, G. (1993). Four-valued semantics for relevant logics (and some of their rivals). Journal of Philosophical Logic, 24, 139–69.CrossRefGoogle Scholar
  32. 32.
    Restall, G. (1995). Information flow and relevant logics. In Seligman, J., & Westerståhl, D. (Eds.) Logic, language and computation: the 1994 Moraga proceedings (pp. 463–477).Google Scholar
  33. 33.
    Restall, G. (1999). Negation in relevant logics (how i stopped worrying and learned to love the Routley star). In Gabbay, D., & Wansing, H. (Eds.) What is negation? (pp. 53–76). Dordrecht: Kluwer.Google Scholar
  34. 34.
    Restall, G. (2000). An introduction to substructural logics. Routledge.Google Scholar
  35. 35.
    Restall, G. (2000). Defining double negation elimination. The Logic Journal of the IGPL, 8(6), 853–860. http://jigpal.oxfordjournals.org/cgi/content/abstract/8/6/853.CrossRefGoogle Scholar
  36. 36.
    Restall, G. (2005). Multiple conclusions. In Hájek, P., Valdés-Villanueva, L., Westerståhl, D. (Eds.) Logic, methodology and philosophy of science: proceedings of the twelfth international congress (pp. 189–205): KCL Publications. http://consequently.org/writing/multipleconclusions.
  37. 37.
    Restall, G. (2009). Truth values and proof theory. Studia Logica, 92(2), 241–264. http://consequently.org/writing/tvpt/.CrossRefGoogle Scholar
  38. 38.
    Ripley, D. (2013). Paradoxes and failures of cut. Australasian Journal of Philosophy, 91(1), 139–164.  https://doi.org/10.1080/00048402.2011.630010.CrossRefGoogle Scholar
  39. 39.
    Routley, R., & Meyer, R.K. (1972). The semantics of entailment II. Journal of Philosophical Logic, 1, 53–73.CrossRefGoogle Scholar
  40. 40.
    Routley, R., & Meyer, R.K. (1973). The semantics of entailment I. In Leblanc, H. (Ed.) Truth, syntax, and semantics (pp. 194–243). North-Holland.Google Scholar
  41. 41.
    Routley, R. (1984). The American plan completed: alternative classical-style semantics, without stars, for relevant and paraconsistent logics. Studia Logica, 43(1–2), 131–158.  https://doi.org/10.1007/BF00935746.CrossRefGoogle Scholar
  42. 42.
    Routley, R., & Routley, V. (1972). Semantics of first degree entailment. Noûs, 6(4), 335–359. http://www.jstor.org/stable/2214309.CrossRefGoogle Scholar
  43. 43.
    Stalnaker, R. (1968). A theory of conditionals. In Rescher, N. (Ed.) Studies in logical theory (pp. 98–112). Oxford: Blackwell.CrossRefGoogle Scholar
  44. 44.
    Tahko, T. (2009). The law of non-contradiction as a metaphysical principle. Australasian Journal of Logic, 7, 32–47.CrossRefGoogle Scholar
  45. 45.
    Tennant, N. (1999). Negation, absurdity and contrariety. In Gabbay, D., & Wansing, H. (Eds.) What Is negation? (pp. 199–222). Dordrecht: Kluwer.Google Scholar
  46. 46.
    Tye, M. (1990). Vague objects. Mind, XCIX(396), 535.  https://doi.org/10.1093/mind/XCIX.396.535.CrossRefGoogle Scholar
  47. 47.
    Wansing, H. (2001). Negation. In Goble, L. (Ed.) The Blackwell guide to philosophical logic (pp. 415–36). Oxford: Blackwell.CrossRefGoogle Scholar
  48. 48.
    Wansing, H. (2008). Constructive negation, implication, and co-implication. Journal of Applied Non-Classical Logics, 18(2-3), 341–364.CrossRefGoogle Scholar
  49. 49.
    Wiggins, D. (2001). Sameness and substance renewed. Cambridge: Cambridge University Press.CrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity of St AndrewsSt AndrewsUK
  2. 2.Institute for Logic, Language and Computation (ILLC)University of AmsterdamAmsterdamThe Netherlands
  3. 3.Department of PhilosophyUniversity of MelbourneMelbourneAustralia

Personalised recommendations