Inductive Independence and the Paradoxes of Confirmation
References
Carl G. Hempel, ‘Studies in the Logic of Confirmation’, Mind 54 (1945), 1–26 and 97–121; reprinted in Carl G. Hempel, Aspects of Scientific Explanation and Other Essays, The Free Press, New York, 1965, pp. 3–46. Although the problem had been raised before Hempel, virtually the whole subsequent discussion seems to have derived its impetus from his article. I shall not try to survey this subsequent discussion of the paradox. Excellent partial surveys are provided by Israel Scheffler in The Anatomy of Inquiry, A. A. Knopf, New York, 1963, pp. 241–242, 258–295, and by Max Black in `Notes on the Paradoxes of Confirmation’, in Aspects of Inductive Logic (ed. by Jaakko Hintikka and Patrick Suppes ), North-Holland Publ. Co., Amsterdam, 1966, pp. 175–197.
See Nelson Goodman, Fact, Fiction, and Forecast, University of London Press, London, 1955.
If we start from a number of primitive (monadic) predicates P1, P2, Ps, these cells are defined by the complex predicates which Carnap calls Q-predicates and which specify which primitive predicates belong to an individual. They are thus of the form where each can be replaced by a negation sign or it can also be omitted in all the different combinations. For Q-predicates and related matters, see Rudolf Carnap, Logical Foundations of Probability, University of Chicago Press, Chicago, 1950 ( second ed., 1962); The Continuum of Inductive Methods, University of Chicago Press, Chicago, 1952.
This holds inter alia of all the members Carnap’s.l-continuum of inductive methods. (See the works listed in reference 3.)
Cf. my paper, ‘Induction by Elimination and Induction by Enumeration’, in The Problem of Inductive Logic (ed. by I. Lakatos ), North-Holland Publ. Co., Amsterdam, 1968.
Hempel, op. cit. Cf. also Scheffler’s and Black’s discussions of Hempel’s point, op. cit., pp. 258–269 and pp. 187–188, respectively.
This kind of solution of the `paradox’ has been put forward repeatedly; see e.g., J. Hosiasson-Lindenbaum, `On Confirmation’, The Journal of Symbolic Logic 5 (1940), 138–148; D. Pears, ‘Hypotheticals’, Analysis 10 (1950), 49–63; J. L. Mackie, ‘The Paradoxes of Confirmation’, The British Journal for the Philosophy of Science 13 (1963),265–277; Patrick Suppes, ‘A Bayesian Approach to the Paradoxes of Confirmation’, in Aspects of Inductive Logic, op. cit., pp. 198–207.
One thought-experiment which illustrates the effects of relative frequencies might be the following: Suppose our `population’ consists of all patients suffering from a certain illness, and suppose that we are considering the generalization `All patients who are given the drug x are cured’. If relatively few patients are administered the drug, we are naturally interested in what happens to them. But assume that the use of the drug becomes almost universal, and yet no counter-examples are produced. Would we not become increasingly more and more interested in those relatively few cases in which a cure is known not to have obtained, and inquire into whether these patients had been given the drug or not?
My doubts are closely related to the objections which Max Black puts forward in op. cit., pp. 195–197 against the same line of thought (which he calls the `Bayesian solution’ to the paradox).
See Carnap’s works listed in reference 3.
See e.g., my paper ‘Towards a Theory of Inductive Generalization’, in Logic, Methodology, and Philosophy of Science, Proceedings of the 1964 International Congress (ed. by Yehoshua Bar-Hillel ), North-Holland Publ. Co., Amsterdam, 1965, pp. 274–288.
If this point is correct, then a satisfactory probabilistic treatment of such paradoxical situations as the raven case (supposing that it follows this pattern) must yield the result that the a posteriori probability of the generalization `All ravens are black’ depends only on the number of observed ravens. We shall outline one treatment of this kind.
A Two-Dimensional Continuum of Inductive Methods’, in Aspects of Inductive Logic, op. cit., pp. 113–132.
Goodman, Fact, Fiction, and Forecast, op. cit.
In A Continuum of Inductive Methods, op. cit., Carnap calls these functions characteristic functions, and in some of my earlier papers I have followed this practice. He now prefers to use the term `representative function’, however, in order to avoid clashes with the standard mathematical and statistical terminology, and I am glad to follow suit. (Carnap’s book should be consulted for the details of his.l-system.)
One way of seeing this naturalness is to recall the familiar `interpretation’ of Carnap’s A as the cardinality of a fictional sample of imaginary individuals evenly divided be- tween the Q-predicates. Each dichotomy will naturally cut this `sample’ into half.
If we follow the letter of `A Two-Dimensional Continuum of Inductive Methods’, op. cit., we have to speak of `truth is principle’ here instead of truth simpliciter. However, the approach sketched there is easily modified in this respect, and in any case the difference does not affect the case of an infinite domain.
Thus the simple suggestion we are following yields something essentially new only in connection with a treatment of induction in which generalizations receive non-zero a priori probabilities (even in an infinite universe). The fact that this does not happen to generalizations in Carnap’s t-system may have something to do with the fact that our treatment of the paradoxes has no precedent in the literature.
In case you are not familiar with it, think of it as a generalization of the factorial 1 x 2 x 3 x x n = n! In fact, for a natural number n, F(n)(n —1)! Many of the familiar results on factorials (e.g., Stirling’s formula) hold for F.
Cf. e.g., G. H. von Wright, `The Paradoxes of Confirmation’, in Aspects ofInductive Logic, op. cit., pp. 208–218.
Cf. e.g., P. Suppes, ‘A Bayesian Approach...’, op. cit.; M. Black, `Notes...’, op. cit.
J. M. Keynes, A Treatise on Probability, Macmillan, London, 1921, pp. 255–256.
Fact, Fiction, and Forecast, op. cit.
We can obtain an even clearer example by taking (instead of the raven generalization) a generalization concerning a recently discovered species which has not yet figured in many inductions.