Journal of Philosophical Logic

, Volume 13, Issue 2, pp 213–232 | Cite as

Paradox, truth and logic part I: Paradox and truth

  • Peter W. Woodruff

Conclusion

The discussion of the semantics of inconsistent truth theories now comes to a pause. The preceding is of course but a sketch; many interesting questions remain to be answered. The second part of this essay, however, will not seek to answer them. Rather, I will turn to the discussion of the proof theory of truth theory: the local and global logic of truth.

Under the first heading, I show how to replace the inductive construction of models with an appropriate infinitary proof theory, and relate this on the one hand to the so-called “dependence” approach to inductive truth theories (Davis, 1979; Yablo, 1982) and on the other to van Fraassen's “fact” semantics for relevance logic.

Under the second heading, I offer formals systems which capture the inferences valid in all approximate models. Not surprisingly, these turn out to be relevant logics.

With formalism in hand, I discuss finally the extent to which the gap and/or glut approach can in fact be said to “solve” the paradoxes; that is, to allow us to say that the very language we are speaking is of the sort described in our theory.

Bibliography

  1. Belnap, N., ‘A Useful Four-valued Logic’, Modern Uses of Multiple-valued Logic, Reidel, Dordrecht, 1976, pp. 5–37.Google Scholar
  2. Bialynicki-Birula, A. and Rasiowa, H., ‘On the Representation of Quasi-Boolean Algebras’, Bulletin of the Polish Academy of Sciences, C1 III, 5, 1957, pp. 259–261.Google Scholar
  3. Birkhoff, G., Lattice Theory, 3rd edition, American Mathematical Society, Providence, 1976.Google Scholar
  4. Davis, L., ‘An Alternate Formulation of Kripke's Theory of Truth’, Journal of Philosophical Logic 8 (1976), 298–296.Google Scholar
  5. Dunn, J. M., The Algebra of Intensional Logicas, PhD. Dissertation University of Pittsburgh, 1966.Google Scholar
  6. Dunn, J. M., ‘Natural Language versus Formal Language’, Symposium talk, Association for Symbolic Logic, December, 1969.Google Scholar
  7. Dunn, J. M., ‘Intuitive Semantics for First-degree Entailments and “Coupled Trees”’, Philosophical Studies 29 (1976), 149–168.Google Scholar
  8. Dunn, J. M., ‘A Relational Representation of Quasi-Boolean Algebras’, Notre Dame Journal of Formal Logic 23 (1982), 353–357.Google Scholar
  9. Feferman, S., ‘Non-extensional Type-free Theories of Partial Operations and Classifications I’, Proof Theory Symposium, Kiel 1974, J. Diller and G.-H. Müller (s.), Springer Lecture Notes in Mathematics, Vol. 500, Springer, Berlin, 1974, pp. 73–118.Google Scholar
  10. Feferman, S., ‘Toward Useful Type-free Theories, I’, paper delivered to the Conference on Self-refrence in Semantics, University of Claifornia, Irvine, June, 1982; to appear in Martin (1983).Google Scholar
  11. Gabbay, D. and Guenthner, F. (eds.), Handbook of Philosophical Logic, Volume IV, Reidel, Dordrecht, 1985 (to appear).Google Scholar
  12. Gupta, A., ‘Truth and Paradox’, Journal of Philosophical Logic 11 (1982), 1–60.Google Scholar
  13. Herzberger, H., ‘Notes on Naive Semantics’, Journal of Philosophical Logic 11 (1982), 61–102.Google Scholar
  14. Kleene, S. C., Introduction to Metamathematics, Van Nostrand, Princeton, 1952.Google Scholar
  15. Kripke, S., ‘Outline of a Theory of Truth’, Journal of Philosophy, 72 (1975), 690–716.Google Scholar
  16. Martin, R. L. (ed.), New Essays on Truth and The Liar Paradox, Oxford, 1983.Google Scholar
  17. Martin, R. L. and Woodruff, P. W., ‘On Representing “true-in-L” in L’, Philosophia (Israel) 5 (1975), 213–217.Google Scholar
  18. Moschovakis, Y. L., Elementary Induction on Abstract Structures, North-Holland, Amsterdam, 1974.Google Scholar
  19. Priest, G., ‘The Logic of Paradox’, Journal of Philosophical Logic, 8 (1979), 219–241.Google Scholar
  20. Routley, R. and Routley, V., ‘The Semantics of First-degree Entailment’, Nous 6 (1972), 335–358.Google Scholar
  21. Scott, D., ‘Continuous Lattices’, Toposes, Algebraic Geometry and Logic, Springer Lecture Notes in Mathematics, 274 (1972), 97–136.Google Scholar
  22. Skyrms, B., ‘Intensional Aspects of Semantical Self-reference’, Talk delivered to the Conference on Semantical Self-reference, University of California, Irvine, June, 1982; to appear in Martin (1983).Google Scholar
  23. Tarski, A., ‘A Lattice-theoretical Fixpoint Theorem and its Applications’, Pacific Journal of Mathematics, 5 (1955), 285–309.Google Scholar
  24. Visser, A., ‘Sketchy Notes on a Four-valued Logic and a “Solution” of the Liar’, manuscript, Stanford University, 1982.Google Scholar
  25. Yablo, S., ‘Grounding, Dependence and Paradox’, Journal of Philosophical Logic 11 (1982) 117–137.Google Scholar

Copyright information

© D. Reidel Publishing Company 1984

Authors and Affiliations

  • Peter W. Woodruff
    • 1
  1. 1.Department of PhilosophyUniversity of California, IrvineIrvineU.S.A.

Personalised recommendations