Journal of Philosophical Logic

, Volume 31, Issue 6, pp 591–612 | Cite as

Diamonds are a Philosopher's Best Friends

  • Heinrich Wansing
Article

Abstract

The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the unwanted disappearance of the diamond may be escaped. The emphasis is not laid on a discussion of the contentious premise of the knowability paradox, namely that all truths are possibly known, but on how from this assumption the conclusion is derived that all truths are, in fact, known. Nevertheless, the solution offered is in the spirit of the constructivist attitude usually maintained by defenders of the anti-realist premise. In order to avoid the paradoxical reasoning, a paraconsistent constructive relevant modal epistemic logic with strong negation is defined semantically. The system is axiomatized and shown to be complete.

constructive negation epistemic logic knowability paradox modal logic paraconsistent logic relevance logic 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. [1]
    Akama, S.: Curry's paradox in contractionless constructive logic, J. Philos. Logic 25 (1996), 135-150.Google Scholar
  2. [2]
    Almukdad, A. and Nelson, D.: Constructible falsity and inexact predicates, J. Symbolic Logic 49 (1984), 231-233.Google Scholar
  3. [3]
    Božíc, M. and Došen, K.: Models for normal modal intuitionistic logics, Studia Logica 43 (1984), 15-43.Google Scholar
  4. [4]
    Chellas, B.: Modal Logic. An Introduction, Cambridge University Press, Cambridge, 1980.Google Scholar
  5. [5]
    Dummett, M.: Victor's error, Analysis 61 (2001), 1-2.Google Scholar
  6. [6]
    Dunn, J. M.: Relevance logic and entailment, in F. Guenthner and D. Gabbay (eds.), Handbook of Philosophical Logic, Vol. 3, Reidel, Dordrecht, 1986, pp. 117-224.Google Scholar
  7. [7]
    Dunn, J. M.: Partiality and its dual, Studia Logica 66 (2000), 5-40.Google Scholar
  8. [8]
    Edgington, D.: The paradox of knowability, Mind 94 (1985), 557-568.Google Scholar
  9. [9]
    Fitch, F.: Symbolic Logic. An Introduction, Ronald Press, New York, 1952.Google Scholar
  10. [10]
    Fitch, F.: The system C△ of combinatory logic, J. Symbolic Logic 28 (1963), 87-97.Google Scholar
  11. [11]
    Fitch, F.: A logical analysis of some value concepts, J. Symbolic Logic 28 (1963), 135-142.Google Scholar
  12. [12]
    Fuhrmann, A.: Relevant logics, modal logics and theory change, Ph.D. thesis, Department of Philosophy, Australian National University, Canberra, 1988.Google Scholar
  13. [13]
    Fuhrmann, A.: Models for relevant modal logics, Studia Logica 49 (1990), 501-514.Google Scholar
  14. [14]
    Gettier, E.: Is justified true belief knowledge? Analysis 23 (1963), 121-123.Google Scholar
  15. [15]
    Gurevich, Y.: Intuitionistic logic with strong negation, Studia Logica 36 (1977), 49-59.Google Scholar
  16. [16]
    Hart,W.: The epistemology of abstract objects: Access and inference, Proc. Aristotelean Soc. 53 (1979), 152-165.Google Scholar
  17. [17]
    Johansson, I.: Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus, Composito Mathematicae 4 (1936), 119-136.Google Scholar
  18. [18]
    Kracht, M.: On extensions of intermediate logics by strong negation, J. Philos. Logic 27 (1998), 49-73.Google Scholar
  19. [19]
    von Kutschera, F.: Ein verallgemeinerter Widerlegungsbegriff für Gentzenkalküle, Arch. für Math. Logik Grundlag. 12 (1969), 104-118.Google Scholar
  20. [20]
    Lindström, S.: Situations, truth and knowability: A situation-theoretic analysis of a paradox by Fitch, in E. Ejerhed and S. Lindström (eds.), Logic, Action, and Cognition – Essays in Philosophical Logic, Kluwer Academic Publishers, Dordrecht, 1997, pp. 181-209.Google Scholar
  21. [21]
    López-Escobar, E.: Refutability and elementary number theory, Indag. Math. 34 (1972), 362-374.Google Scholar
  22. [22]
    Mackie, J.: Truth and knowability, Analysis 40 (1980), 90-92.Google Scholar
  23. [23]
    Melia, J.: Anti-realism untouched, Mind 100 (1991), 341-342.Google Scholar
  24. [24]
    Nelson, D.: Constructible falsity, J. Symbolic Logic 14 (1949), 16-26.Google Scholar
  25. [25]
    Nelson, D.: Negation and separation of concepts in constructive systems, in A. Heyting (ed.), Constructivity in Mathematics, North-Holland, Amsterdam, 1959, pp. 208-225.Google Scholar
  26. [26]
    Pearce, D.: n Reasons for choosing N, Technical Report 14/91, Gruppe für Logik, Wissenstheorie und Information, Free University of Berlin, 1991.Google Scholar
  27. [27]
    Percival, P.: Fitch and intuitionistic knowability, Analysis 50 (1990), 182-187.Google Scholar
  28. [28]
    Rabinowicz, W.: Intuitionistic truth, J. Philos. Logic 14 (1985), 191-228.Google Scholar
  29. [29]
    Rabinowicz, W. and Segerberg, K.: Actual truth, possible knowledge, Topoi 13 (1994), 101-115.Google Scholar
  30. [30]
    Rasiowa, H.: An Algebraic Approach to Non-classical Logic, North-Holland, Amsterdam, 1974.Google Scholar
  31. [31]
    Rautenberg, W.: Klassische und nichtklassische Aussagenlogik, Vieweg Verlag, Braunschweig, 1979.Google Scholar
  32. [32]
    Routley, R.: Semantical analyses of propositional systems of Fitch and Nelson, Studia Logica 33 (1974), 283-298.Google Scholar
  33. [33]
    Routley, R. and Meyer, R. K.: The semantics of entailment II, J. Philos. Logic 1 (1972), 53-73.Google Scholar
  34. [34]
    Routley, R. and Meyer, R. K.: The semantics of entailment III, J. Philos. Logic 1 (1972), 192-208.Google Scholar
  35. [35]
    Routley, R. and Meyer, R. K.: The semantics of entailment, in H. Leblanc (ed.), Truth, Syntax and Modality, North-Holland, Amsterdam, 1973, pp. 194-243.Google Scholar
  36. [36]
    Rückert, H.: A solution to Fitch's paradox of knowability, in D. Gabbay, S. Rahman, J. Torres and J. P. van Bendegem (eds.), Logic, Epistemology and the Unity of Science, Kluwer Academic Publishers, Dordrecht, 2002, to appear.Google Scholar
  37. [37]
    Shramko, Y., Dunn, M., and Takenaka, T.: The tri-lattice of constructive truth values, J. Logic Comput. 11 (2001), 761-788.Google Scholar
  38. [38]
    Tennant, N.: The Taming of the True, Clarendon Press, Oxford, 1997.Google Scholar
  39. [39]
    Tennant, N.: Victor Vanquished, Analysis, to appear.Google Scholar
  40. [40]
    Thomason, R.: A semantical study of constructible falsity, Z. Math. Logik Grundlag. Math. 15 (1969), 247-257.Google Scholar
  41. [41]
    Wagner, G.: Logic programming with strong negation and unexact predicates, J. Logic Comput. 1 (1991), 835-859.Google Scholar
  42. [42]
    Wansing, H.: The Logic of Information Structures, Lecture Notes in AI 681, Springer-Verlag, Berlin, 1993.Google Scholar
  43. [43]
    Wansing, H.: Tarskian structured consequence relations and functional completeness, Math. Logic Quart. 41 (1995), 73-92.Google Scholar
  44. [44]
    Wansing, H.: Semantics-based nonmonotonic inference, Notre Dame J. Formal Logic 36 (1995), 44-54.Google Scholar
  45. [45]
    Wansing, H.: Negation as falsity: A reply to Tennant, in D. Gabbay and H. Wansing (eds.), What is Negation?, Kluwer Academic Publishers, Dordrecht, 1999, pp. 223-238.Google Scholar
  46. [46]
    Wansing, H.: Negation, in L. Goble (ed.), The Blackwell Guide to Philosophical Logic, Basil Blackwell, Cambridge, MA, 2001, pp. 415-436.Google Scholar
  47. [47]
    Williamson, T.: Intuitionism disproved? Analysis 42 (1982), 203-207.Google Scholar
  48. [48]
    Williamson, T.: On the paradox of knowability, Mind 96 (1987), 256-261.Google Scholar
  49. [49]
    Williamson, T.: Knowability and constructivism, Philos. Quart. 38 (1988), 422-432.Google Scholar
  50. [50]
    Williamson, T.: On intuitionistic modal epistemic logic, J. Philos. Logic 21 (1992), 63-89.Google Scholar
  51. [51]
    Williamson, T.: Never say never, Topoi 13 (1994), 135-145.Google Scholar
  52. [52]
    Williamson, T.: Tennant on knowable truth, Ratio 31 (2000), 99-114.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Heinrich Wansing
    • 1
  1. 1.Dresden University of TechnologyInstitute of PhilosophyDresdenGermany

Personalised recommendations