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Negation on the Australian Plan

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.

References

  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.

  2. Barwise, J., & Perry, J. (1983). Situations and attitudes. Bradford books. MIT Press.

  3. Beall, J.C. (2009). Spandrels of truth. Oxford: Oxford University Press.

    Book  Google Scholar 

  4. Beall, J.C., & Restall, G. (2000). Logical pluralism. Australasian Journal of Philosophy, 78, 475–493. http://consequently.org/writing/pluralism.

    Article  Google Scholar 

  5. Beall, JC, & Restall, G. (2006). Logical pluralism. Oxford: Oxford University Press.

    Google Scholar 

  6. Berto, F. (2008). Adynaton and material exclusion. Australasian Journal of Philosophy, 86, 165–90.

    Article  Google Scholar 

  7. Berto, F. (2014). Absolute contradiction, dialetheism, and revenge. Review of Symbolic Logic, 7, 193–207.

    Article  Google Scholar 

  8. Berto, F. (2015). A modality called “negation”. Mind, 124(495), 761–793.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  10. Brandom, R.B. (1994). Making it explicit. Harvard University Press.

  11. Brandom, R.B. (2000). Articulating reasons: an introduction to inferentialism. Harvard University Press. ISBN 0674001583.

  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.

    Article  Google Scholar 

  13. Dunn, J.M., & Zhou, C. (2005). Negation in the context of gaggle theory. Studia Logica, 80, 235–64.

    Article  Google Scholar 

  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.

  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.

    Article  Google Scholar 

  16. Dunn, J.M. (1996). Generalised ortho negation. In Wansing, H. (Ed.) Negation: a notion in focus (pp. 3–26). Berlin: Walter de Gruyter.

  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.

  18. Fine, K. (2009). The question of ontology. In Chalmers, D., Manley, D., Wasserman, R. (Eds.) Metametaphysics (pp. 157–77). Clarendon.

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

  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.

    Article  Google Scholar 

  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.

  24. Kripke, S. (1980). Naming and necessity. Oxford: Blackwell.

    Google Scholar 

  25. Lewis, D.K. (1973). Counterfactuals. Oxford: Blackwell.

    Google Scholar 

  26. Mares, E.D. (1995). A star-free semantics for R. Journal of Symbolic Logic, 60, 579–590.

    Article  Google Scholar 

  27. Mares, E.D. (2004). Relevant logic. A philosophical interpretation. Cambridge: Cambridge University press.

    Book  Google Scholar 

  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.

    Article  Google Scholar 

  29. Price, H. (1990). Why ‘not’? Mind, 99, 221–38.

    Article  Google Scholar 

  30. Priest, G. (2008). An introduction to non-classical logic, 2nd edn. Vol. 2008. Cambridge: Cambridge University Press.

  31. Restall, G. (1993). Four-valued semantics for relevant logics (and some of their rivals). Journal of Philosophical Logic, 24, 139–69.

    Article  Google Scholar 

  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).

  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.

  34. Restall, G. (2000). An introduction to substructural logics. Routledge.

  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.

    Article  Google Scholar 

  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. Restall, G. (2009). Truth values and proof theory. Studia Logica, 92(2), 241–264. http://consequently.org/writing/tvpt/.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  39. Routley, R., & Meyer, R.K. (1972). The semantics of entailment II. Journal of Philosophical Logic, 1, 53–73.

    Article  Google Scholar 

  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.

  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.

    Article  Google Scholar 

  42. Routley, R., & Routley, V. (1972). Semantics of first degree entailment. Noûs, 6(4), 335–359. http://www.jstor.org/stable/2214309.

    Article  Google Scholar 

  43. Stalnaker, R. (1968). A theory of conditionals. In Rescher, N. (Ed.) Studies in logical theory (pp. 98–112). Oxford: Blackwell.

    Chapter  Google Scholar 

  44. Tahko, T. (2009). The law of non-contradiction as a metaphysical principle. Australasian Journal of Logic, 7, 32–47.

    Article  Google Scholar 

  45. Tennant, N. (1999). Negation, absurdity and contrariety. In Gabbay, D., & Wansing, H. (Eds.) What Is negation? (pp. 199–222). Dordrecht: Kluwer.

  46. Tye, M. (1990). Vague objects. Mind, XCIX(396), 535. https://doi.org/10.1093/mind/XCIX.396.535.

    Article  Google Scholar 

  47. Wansing, H. (2001). Negation. In Goble, L. (Ed.) The Blackwell guide to philosophical logic (pp. 415–36). Oxford: Blackwell.

    Chapter  Google Scholar 

  48. Wansing, H. (2008). Constructive negation, implication, and co-implication. Journal of Applied Non-Classical Logics, 18(2-3), 341–364.

    Article  Google Scholar 

  49. Wiggins, D. (2001). Sameness and substance renewed. Cambridge: Cambridge University Press.

    Book  Google Scholar 

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Correspondence to Francesco Berto.

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Berto, F., Restall, G. Negation on the Australian Plan. J Philos Logic 48, 1119–1144 (2019). https://doi.org/10.1007/s10992-019-09510-2

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Keywords

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