Journal of Philosophical Logic

, Volume 36, Issue 6, pp 735–771 | Cite as

A Note on Binary Inductive Logic

Article
First Online:

Abstract

We consider the problem of induction over languages containing binary relations and outline a way of interpreting and constructing a class of probability functions on the sentences of such a language. Some principles of inductive reasoning satisfied by these probability functions are discussed, leading in turn to a representation theorem for a more general class of probability functions satisfying these principles.

Key words

exchangeability inductive logic probability logic uncertain reasoning 
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References

  1. 1.
    Carnap, R.: Logical Foundations of Probability, University of Chicago Press, Chicago, IL, Routledge & Kegan Paul Ltd, 1950.Google Scholar
  2. 2.
    Carnap, R.: The Continuum of Inductive Methods, University of Chicago Press, Chicago, IL, 1952.Google Scholar
  3. 3.
    Carnap, R.: A basic system of inductive logic, in R. C. Jeffrey (ed.), Studies in Inductive Logic and Probability, Volume II, University of California Press, Chicago, IL, pp. 7–155, 1980.Google Scholar
  4. 4.
    Carnap, R.: On the application of inductive logic, Philosophy and Phenomenological Research 8, (1947), 133–147.CrossRefGoogle Scholar
  5. 5.
    Carnap, R.: Reply to Nelson Goodman, Philosophy and Phenomenological Research 8, (1947), 461–462.CrossRefGoogle Scholar
  6. 6.
    Coletti, G. and Scozzafava, R.: Probabilistic logic in a coherent setting, Trends in Logic, 15, Kluwer, London, Dordrecht, 2002.Google Scholar
  7. 7.
    de Finetti, B.: On the condition of partial exchangeability, in R.C. Jeffrey (ed.), Studies in Inductive Logic and Probability, Volume II, University of California Press, Berkley, CA, 1980.Google Scholar
  8. 8.
    de Finetti, B.: Theory of Probability, Vol. 1, Wiley, New York, 1974.Google Scholar
  9. 9.
    Fitelson, B.: Inductive Logic: http://fitelson.org/il.pdf
  10. 10.
    Gaifman, H.: Concerning measures on first order calculi, Israel Journal of Mathematics 2 (1964), 1–18.Google Scholar
  11. 11.
    Gaifman, H. and Snir, M.: Probabilities over rich languages, Journal of Symbolic Logic 47(3) (1982), 495–548.CrossRefGoogle Scholar
  12. 12.
    Glaister, S.: Inductive logic, in D. Jacquette (ed.), A Companion to Philosophical Logic, Blackwell, London, 2001, pp. 565–581.Google Scholar
  13. 13.
    Goodman, N.: Fact, Fiction and Forecast, 4th edition, Harvard UP, Cambridge MA, 1983.Google Scholar
  14. 14.
    Goodman, N.: A query on confirmation, Journal of Philosophy 43 (1946), 383–385.CrossRefGoogle Scholar
  15. 15.
    Goodman, N.: On infirmities in confirmation-theory, Philosophy and Phenomenological Research 8 (1947), 149–151.CrossRefGoogle Scholar
  16. 16.
    Hill, M.J., Paris, J.B., and Wilmers, G.M.: Some observations on induction in predicate probabilistic reasoning, Journal of Philosophical Logic 31(1) (2002), 43–75.CrossRefGoogle Scholar
  17. 17.
    Hoover, D.N.: Relations on probability spaces and arrays of random variables. Preprint, Institute of Advanced Study, Princeton, 1979.Google Scholar
  18. 18.
    Jeffrey, R.C.: Goodman’s query, Journal of Philosophy 63(11) (1966), 281–288.CrossRefGoogle Scholar
  19. 19.
    Johnson, W.E.: Probability: The deductive and inductive problems, Mind 41(164) (1932), 409–423.CrossRefGoogle Scholar
  20. 20.
    Kallenberg, O.: Probabilistic Symmetries and Invariance Principles, Springer, New York, ISBN-10: 0-387-25115-4, 2005.Google Scholar
  21. 21.
    Kallenberg, O., The Ottawa Workshop, http://www.mathstat.uottawa.ca/~givanoff/wskallenberg.pdf
  22. 22.
    Kawalec, P., Back to green as perspectives on confirmation, Justification, Truth and Belief, http://www.jtb-forum.pl, January, 2002.
  23. 23.
    Kemeny, J.G.: Carnap’s theory of probability and induction, in P.A. Schilpp (ed.), The Philosophy of Rudolf Carnap, Open Court, La Salle, IL, 1963, pp. 711–738.Google Scholar
  24. 24.
    Krauss, P.H.: Representation of symmetric probability models, Journal of Symbolic Logic 34(2) (1969), 183–193.CrossRefGoogle Scholar
  25. 25.
    Landes, J.: Doctorial Thesis, Manchester University, UK, to appear.Google Scholar
  26. 26.
    Maher, P.: Probabilities for two properties, Erkenntnis 52 (2000), 63–91.CrossRefGoogle Scholar
  27. 27.
    Maher, P.: Probabilities for multiple properties: The models of Hesse, Carnap and Kemeny, Erkenntnis 55 (2001), 183–216.CrossRefGoogle Scholar
  28. 28.
    Matúš, F.: Block-factor fields of Bernoulli shifts, Proceedings of Prague Stochastics’98, Vol.II, 1998, pp. 383–389.Google Scholar
  29. 29.
    Miller, D.: Popper’s qualitative theory of versimilitude, British Journal for the Philosophy of Science 25 (1974), 166–177.CrossRefGoogle Scholar
  30. 30.
    Nix, C.J., Probabilistic Induction in the Predicate Calculus Doctorial Thesis, Manchester University, Manchester, UK, 2005. See http://www.maths.man.ac.uk/~jeff/#students.
  31. 31.
    Nix, C.J. and Paris, J.B.: A Continuum of inductive methods arising from a generalized principle of instantial relevance, Journal of Philosophical Logic, Online First Issue, DOI  10,1007/s 10992-005-9003x, ISSN 0022-3611 (Paper) 1573–0433, 2005.
  32. 32.
    Paris, J.B.: The Uncertain Reasoner’s Companion, Cambridge University Press, Cambridge, UK, 1994.Google Scholar
  33. 33.
    Russell, B., Principles of Mathematics 2nd edition, Allen, London, UK, 1937.Google Scholar
  34. 34.
    Scott, D. and Krauss, P.: Assigning probabilities to logical formulas, in J. Hintikka and P. Suppes (eds.), Aspects of Inductive Logic, North-Holland, Amsterdam, 1966, pp. 219–264.Google Scholar
  35. 35.
    Stalker, D. (ed.), Grue! The New Riddle of Induction, Open Court, La Salle, IL, 1994.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.School of MathematicsUniversity of ManchesterManchesterUK

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