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The Problem of Probabilistic Justification of Enumerattve Induction

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Twenty-Five Years of Logical Methodology in Poland

Part of the book series: Synthese Library ((SYLI,volume 87))

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Abstract

For a long time logicians have had various doubts concerning the justification of inductive inference. Is inductive inference justifiable, and when? Is inductive inference a correct one? Do the premises of inductive inference justify the acceptance of a conclusion, and when? Such are the questions which logicians have to answer, concerning the justification of induction.1

First published in Studia Logica V (1957). Translated by S. Wojnicki.

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References

  1. This paper is a somewhat modified fragment of the doctoral dissertation prepared by the author.

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  2. Cf. M. Black ‘On the Justification of Induction’, Language and Philosophy, pp. 63–64.

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  3. The terms ‘induction’ resp. ‘inductive inference’ are used in so many different meanings that I think it an unfeasible task to find their analytical definition. In contemporary literature of logic the tendency may be observed to such understanding of these terms in which their range is very wide and embraces all the fallible inference methods which are usually approved in the empirical sciences. It is in this, very imprecise sense that I use the term ‘induction’ resp. ‘inductive inference’ in the present paper.

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  4. M. Black, op. cit., pp. 66–68.

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  5. It cannot be generally established what probability should be called high probability. There is always some convention in choosing the lower limit of high probabilities, as for example in adopting the level of significance in statistical tests. Undoubtedly our inclination to consider a given probability high depends in practice to a large extent to the loss connected with the acceptance of a false conclusion.

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  6. J. M. Keynes, A Treatise on Probability, 1929.

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  7. J. Hosiasson, ‘O prawomocnosci indukcji hipotetycznej’ (‘The Problem of Justification of Hypothetical Induction’), Fragmenty Filozoficzne, Warszawa 1934.

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  8. G. H. von Wright, The Logical Problem of Induction, 1941.

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  9. R. Carnap, Logical Foundations of Probability, 1951.

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  10. Constructing such theories is nowadays fashionable among western logicians (cf. e.g. Carnap’s Logical Foundations of Probability and the lively discussion following it which has been going on in logical and philosophical periodicals). Instead of the term ‘logical probability’ the term ‘degree of confirmation’ is used, or shortly ‘confirmation’. I think construction of such theories is unnecessary for the solution of the problem of justification of induction for the following reasons: the existing axioms for the concept of confirmation usually repeat the axioms given for the mathematical concept of probability. (Cf. the axioms given by Keynes and Hosiasson in the works mentioned or the axioms given by A. Shi-mony in his paper ‘Coherence and the Axioms of Confirmation’, Journal of Symbolic Logic 20.) On the other hand, attempts to give an explicit definition of the degree of confirmation usually reduce it to classical probability, defined not on events but on sentences (cf. Carnap, op cit.). I think this difference is not substantial. In my opinion it is obvious that classical probability may equally well be considered a function where arguments are events or sentences (cf. here i.a. K. R. Popper, Logik der Forschung, p. 188 etc., W. Glivenko, Kurs teorii veroyatnosti (Calculus of probability), Moscow — Leningrad 1939, pp. 231-231; R. Carnap, op. cit., pp. 29-30). Hence it seems that construction of special theories of confirmation as theories of classical probability defined on sentences is unnecessary. The confirmation theories which I know do not bring anything new, neither from the formal, nor from the substantial points of view, in comparison with mathematical theories of probability, and they only repeat what is already known from the mathematical theories. The construction of a special theory of logical probability becomes reasonable only when one holds the view that logical and mathematical probability differ not only in that the former is defined on sentences and the latter on events, but when there is a difference which makes impossible the adoption of the axioms of mathematical probability as ready made axioms for logical probability. This is, for instance, Popper’s opinion: according to him, logical probability (Hypothesenwahrscheinlichkeit), understood as the degree of justification of a hypothesis by facts, has some properties totally different from mathematical probability (Ereigniswahrscheinlichkeit), even if the latter were to be interpreted as function propositional arguments (cf. Logik der Forschung, pp. 188 ff, and POP-per’s articles in British Journal for the Philosophy of Science 5, 6). There is no place here to deal further with this interesting question.

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  11. Op. cit., pp. 235-236.

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  12. This condition can be replaced with the following one: there is an Δ>0 such that P(vn, Kn-1 · ឥ) ⫍ 1-Δ. Keynes proved his theorems on the grounds of his own axiomatic theory of probability. They can however be as easily demonstrated on the grounds of the mathematical theory of probability.

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  13. Op. cit., pp. 251 f.

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  14. For instance, J. Nicod, Le problùme logique de l’induction, pp. 76 ff.

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  15. If they are not finite, from the practical point of view it is important that the inductive generalization would hold for the subset of the examined set the elements of which we encounter in practical activities and this subset is always finite.

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  16. Cf. i.a. B. Russel, Human Knowledge, G. Allen, London 1948, p. 424.

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  17. As P(Vn, g) = l.

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  18. The assumption of finiteness of the set A does not exclude the possibility of an unlimited increase of the number of verifications, if only it is assumed that the examined elements of the set A are — statistically speaking — sampled with replacement. If we assumed that we are dealing with sampling without replacement, we would have to substitute the assumption that n tends to infinity by the assumption that n tends to N. In both cases the conclusions concerning the probabilistic justification of enumerative induction will be the same.

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  19. This measure of the degree of confirmation of hypothesis h by empirical data e corresponds to what Carnap calls ‘relevance quotient’ (Logical Foundations of Probability, p. 567). We might just as well take as measure of the degree of confirmation of h by e any function strictly increasing with m(h, e). This has been done by I. J. Good, who takes as measure of the degree of confirmation of h by e (Good uses the term ‘weight of evidence for h, given e’) the value of log P (h, e) + log P (e) + const. Cf. Probability and the Weighing of Evidence, p. 63.

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  20. Professor M. KokoszyƄska, in her contribution presented at the same conference and published in the same volume of Studia Logica, introduces among other things the non-relativized and the relativized concept of inductive fallacy. It is easy to see that my expression ‘e probabilistically confirms h’ has the same meaning as prof. KokoszyƄska’s expression ‘h is correctly inferred from e (in the absolute sense)’, as well as the sentence ‘e probabilistically confirms h1 more powerfully than h2’ means the same as prof. KokoszyƄska’s: ‘the inference of h1from e is less fallacious (more correct) than the inference of h2 from e’.

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  21. Statisticians call such statistical tests parametric tests.

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  22. Cf. the definition of statistical test given by J. Neyman in First Course in Prob ability and Statistics, p. 258. The description of the procedure of verifying statistical parametric hypotheses has been taken from M. Fisz’s Calculus of Probability and Mathematical Statistics, in particular Chap. 13.

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  23. I assume here that elements of the set A examined with respect to the property B are sampled with replacement. The whole reasoning might be based on the assumption of sampling without replacements and lead to a similar result.

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  24. Cf. Bacon’s well known opinions: “Inductio enim quae procedit per enumeration nem simplicem res peurilis est; precario concluda et periculo exponitur ab instantia contradictoria”
 And: “Inductio mala est quae per enumerationem simplicem Principia concludit Scientiarum
”(Novum Organum, Amsterdam 1964, I, 105, and I, 69). Among the contemporary logicians, Professor Kotarbinski says of enumerative induction: “A non-exhaustive induction, which proceeds by simple enumeration
 is a very weak method of justifying generalization”, Kurs logiki (A Course in Logic), 2nd ed., p. 135.

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  25. I owe this information to Professor S. Romahnowa.

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Authors
  1. Zbigniew CzerwiƄski
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Marian PrzeƂęcki Ryszard Wójcicki

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© 1977 PWN - Polish Scientific Publishers - Warszawa

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CzerwiƄski, Z. (1977). The Problem of Probabilistic Justification of Enumerattve Induction. In: PrzeƂęcki, M., Wójcicki, R. (eds) Twenty-Five Years of Logical Methodology in Poland. Synthese Library, vol 87. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1126-6_5

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  • DOI: https://doi.org/10.1007/978-94-010-1126-6_5

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  • Print ISBN: 978-94-010-1128-0

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