Abstract
These two propositions are far from being the same “I have found that such an object has always been attended with such an effect” and ‘‘I farsee, that other objects, which are, in appearance, similar, will be attended with similar effects,” I shall allow, if you please, that the one proposition may justly be inferred from the other. But if you insist that the inference is made by a chain of reasoning, I desire you to produce that reasoning. The connection between these propositions is not intuitive. There is required a medium, which may enable the mind to draw such an inference, if indeed it be drawn by reasoning and argument. What the medium is, I must confess, passes my comprehension; and it is incumbent on those to produce it, who assert that it really exists, and is the origin of all our conclusions concerning matter of fact.1
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References
David Hume, An Enquiry Concerning Human Understanding, 1748, Section IV.
Russell, Introduction to Mathematical Philosophy, 1920, p.169.
“Hume’s scepticism rests entirely upon his rejection of the principle of induction. The principle of induction, as applied to causation, says that, if A has been found very often accompanied or followed by B, and no instance is known of A not being accompanied or followed by B, then it is probable that on the next occasion on which A is observed it will be accompanied or followed by B. If the principle is to be adequate, a sufficient number of instances must make the probability not far short of certainty. If this principle, or any other from which it can be deduced, is true, then the causal inferences which Hume rejects are valid, not indeed as giving certainty, but as giving a sufficient probability for practical purposes. If this principle is not true, every attempt to arrive at general scientific laws from particular observations is fallacious, and Hume’s scepticism is inescapable for an empiricist. The principle itself cannot, of course, without circularity, be inferred from observed uniformities, since it is required to justify any such inference. It must therefore be, or be deduced from, an independent principle not based upon experience. To this extent, Hume has proved that pure empiricism is not a sufficient basis for science. But if this one principle is admitted, everything else can proceed in accordance with the theory that all our knowledge is based on experience. It must be granted that this is a serious departure from pure empiricism, and that those who are not empiricists may ask why, if one departure is allowed, others are to be forbidden. These, however, are questions not directly raised by Hume’s arguments. What these arguments prove — and I do not think the proof can be controverted — is that induction is an independent logical principle, incapable of being inferred either from experience or from other logical principles, and that without this principle science is impossible.” Russell, A History of Western Philosophy,1945, pp. 673–674.
Induction is a non-demonstrative inference leading to a conclusion referring to some or all members of a class, based on some evidence of these members. This kind of logical (or incomplete) induction must by no means be confused with mathematical (or complete) induction nor with electric induction, neither of which is the subject of this chapter.
This example may disprove the erroneous belief that inductive inference must proceed from the particular to the general, (just as it is false to think a deductive argument must proceed from the general to the particular). The reader will notice that the first three premises above are indeed universal propositions and the conclusion is a particular one.
We should like to adhere throughout the book to this inductive agnosticism which seems to be justified in the face of the unresolved controversy between logical positivists and subjectivists (see Subsection 5.24,5.32 and 5.33). It is interesting to note that any inductive argument can be converted into a deductive argument by appropriate additional premises, e.g. in the above instance by adding the statement: “Balls of the same color are of the same material.”
G. H. von Wright asserts that it was Aristotle who made the first attempt of a systematic treatment of induction, and summarizes Aristotle’s contributions as follows: (i) Aristotle was the first to point out the non-demonstrative character of the type of inference which we treat under the name of induction, and to contrast it with conclusive reasoning. The contrast, however, was obscured by his own terminology which has become established in traditional logic. (ii) Aristotle was aware of the double aspect of inductive method as a process
of inference and as a process of definition (formation of concepts). A Treatise on Induction and Probability, 1951, pp. 151–152. Yet Aristotle in his Topics describes induction as “a passage from particulars to universals” which, as our first examples illustrate, need not be the case.
In his Novum Organum (1620) he points out that “the induction which is to be available for the discovery and demonstration of sciences and arts, must analyze nature by proper rejection and exclusions’, and then after a sufficient number of negatives, come to a conclusion on the affirmative instances” (our italics). Depending on which aspects of F. Bacon’s work one concentrates, it may be praised or criticised. Russell asserts (1) that F. Bacon “missed most of what was being done in science in his day… (and)… rejected the Copernican theory” and (2) that his “inductive method is faulty through insufficient emphasis on hypothesis… [hoping]… that mere orderly arrangement of data would make the right hypothesis obvious”, History of Western Philosophy (p. 544), whereas G. H. von Wright emphasizes that “it is the immortal merit of Bacon to have fully appreciated the importance of… asymmetry in the logical structure of laws” (op. cit., p. 152), namely that these laws are not verifiable but merely falsifiable, thus anticipating the core of Popper’s dogma by more than three centuries. Although Popper discusses Bacon’s work on induction at many places, we are not certain whether he would agree to such a strong assertion.
To mention the most important names and publications we draw attention to: Ramsey, Truth and Probability’, 1926 and ‘Further Considerations’, 1928, both contained in his collection The Foundations of Mathematics, Kegan Paul, Trench, Trubner & Co., London, 1931, pp. 156–211. Richard von Mises, Probability, Statistics and Truth, in German 1928; English translation, 1957. Reichenbach, The Theory of Probability, in German 1934; English translation, 1949 Jeffreys, Theory of Probability, 1939.
A Treatise on Induction and Probability, 1951, p. 11.
E.g. see Black, Problems of Analysis, 1954, Chapter 11,
Braithwaite, Scientific Explanation, 1953.
Karl Popper, although agreeing with Hume’s rejection of logical justification, disagrees with Hume’s formulation of the “psychological justification.” See Popper’s Conjectures and Refutations, 1962, pp. 42–46.
Unconditioned reflex: the salivation of a hungry dog when shown food. Conditioned reflex: the salivation of a dog when a bell is rung after the dog has been conditioned by previously ringing a bell every time when food was presented to him.
Ludwig von Wittgenstein, Tractatus Logico-Philosophicus, 1961, first ed., 1922, item 6.3631, p. 142.
Barker, Induction and Hypothesis, 1957, pp. 12–13.
Brody, Readings in the Philosophy of Science, 1970, ‘Introduction to Part 3’, p. 376.
Since this time Popper has vastly elaborated upon his system, but his magnum opus remains his Logik der Forschung, 1934, and its English version The Logic of Scientific Discovery, 1959.
Popper, Conjectures and Refutations, 1962; quotations from Basic Books edition, 1962, pp. 36–37. Originally presented under ‘Philosophy of Science: a Personal Report’, in British Philosophy in the Mid-Century, ed. C. A. Mace, 1957.
Cf. Harsanyi, ‘Popper’s Improbability Criterion for the Choice of Scientific Hypotheses’, 1960–61, pp. 41–56.
Kyburg, Probability and Inductive Logic, 1970, p.157.
Kyburg, Probability and Inductive Logic, 1970, p. 161.
Barker, Probability and Inductive Logic, 1970, p. 159.
Wellmer, Methodologie als Erkenntnisproblem — Zur Wissenschaftslehre Karl R. Poppers, 1967.
1947, p. 4.
Salmon, The Foundations of Scientific Inference, 1966, pp. 26–27.
Quinton, ‘Popper, Karl Raimund’, 1967, p. 399.
Maxwell ‘A critique of Popper’s Views on Scientific Method’, 1972, p. 137. 28a The situation is different for hypotheses, which according to Stegmüller (1975a, p. 510) are not structures, like theories, but provisionally accepted sentences, and thus are open to refutation. The notion of a mathematical structure can be defined by an axiomatic representation in which the predicate “is an 5” is introduced to represent a structure that belongs to the category of the S-structures (suppose S stands for “planetary system” then various phenomena can be described as planetary structures in the following way: e.g. “The atom is a planetary structure”, “the lunar system is a planetary structure”, “the solar system is a planetary structure”; or suppose S stands for “double-classification accounting structure”, then we can say “Leontief’s input-output system is an accounting structure” or “The U.S.-National Income and Product Statistic is an accounting structure”, etc. In both cases a fundamental mathematical structure is being attributed to structurally similar phenomena). Such structures refer to the core of a theory while hypotheses may refer to constraints, to extensions of a core, etc. (cf. Subsection 7.32.)
Leplin, ‘Contextual Falsification and Scientific Methodology’, 1972, p. 476.
Suppe (ed.), The Structure of Scientific Theory, 1974, ‘Introduction’, p. 168.
Popper, Conjectures and Refutations, 1962, pp. 28–29.
Kyburg therefore speaks in this connection of “demonstrative induction”.
The Logical Problem of Induction, 1945; 2nd rev. ed., 1957.
See also Simon, Models of Man, 1957, Chapters 1 – 3.
Such a “cause” need not be the presence of a property, but could be its absence, which then may be listed as the presence of a separate property.
In this case the disjunction does not need to be emphasized, because a disjunction v ⋁ w is a sufficient condition for d, if and only if v is a sufficient condition for d and w is a sufficient condition for d. Therefore in discovering all the sufficient conditions that are not disjunctions one necessarily attains all the sufficient conditions that are disjunctions. For an analogous reason we did not emphasize conjunctions in connection with the direct method above.
Often implicitly used in correlation analysis. Cf. Goode and Hatt, Methods in Social Research, 1952, pp. 86–87.
Cf. Herschei, A Preliminary Discourse on the Study of Natural Philosophy, 1830. Another contemporary of Mill concerned with inductive logic and scientific discovery was William Whewell. See his History of Inductive Science, Vols. I—III, 1837 and Philosophy of the Inductive Sciences, 1840.
Kyburg, Probability and Inductive Logic, 1970, pp. 114, 120.
Another, more sophisticated, introduction to Carnap’s theory and original language system may be found in Tintner, ‘Foundations of Probability and Statistical Inference’, 1949.
2nd rev. edition, 1962; first ed. 1950.
A forthcoming posthumous publication by Rudolf Carnap, Basic Systems of Inductive Logic, is announced in the bibliography of Kyburg’s recent (1971) book.
See, e.g. Hintikka, ‘A Two-Dimensional Continuum of Inductive Methods’, 1966, pp. 113–132.
See, e.g. Hintikka, ‘A Two-Dimensional Continuum of Inductive Methods’, 1966, pp. 1–20, ‘Knowledge, Acceptance, and Inductive Logic’, by Hintikka and Hilpinen.
Levi, Gambling with Truth: An Essay on Induction and The Aims of Science, 1967.
Kyburg, Probability and Inductive Logic, 1970, pp. 77–93, 180–198.
This is a deviation from Carnap’s viewpoint. He deemed it rather to be a mistake to “regard the result of inductive reasoning as the acceptance of a new position” whereas the objective of inductive logic ought to be above all, the assignment of a degree of confirmation to this new position. Indeed, a serious argument against theories of acceptance, is the fact that people, whose behavior cannot be declared as “irrational”, do buy lottery tickets although realizing that their chance is so small (even where they deal with a fair lottery system), that no acceptance theory could ever recommend such a purchase.
Popper and other philosophers not subscribing to positivism would deny that there are incorrigible protocol sentences forming the ultimate empirical basis of all science.
In contrast to this posterior confirmation c(h, e • i), the expression c(h, e) would then be regarded as the prior confirmation of h. Cf. Carnap (1962, pp. 326 ff).
The c-functions often become in Carnap’s system measure-functions m.
See Carnap and Jeffrey (eds.), Studies in Inductive Logic and Probability, Vol. 1, 1971.
Raiffa, Decision Analysis, 1968, pp. 284–285.
Cf. Fellner, Probability and Profit, 1965, p. 37: “I share the belief that the subjectivists’ view will continue to gain pound over the frequentist (objective) view, and I expect distinct benefits from this change.” A more recent and most impressive defence of the personalistic approach has been advanced by a leading epistemologist: Stegmüller (1973b and c).
Cf. Fellner, Probability and Profit, 1965, p. 274 (degree of confidence should not be confused with degree of confirmation).
See Savage, The Foundations of Statistics, 1954. He confesses that at the time this work was created, although interested in personal probabilities, he was not yet a subjectivist, or “personalistic Bayesian”, as he prefers to say.
Richard von Mises, Probability, Statistics and Truth (original German ed., 1928; 2nd rev. English ed., 1957).
Reichenbach, The Theory of Probability (original German ed., 1935; English ed., 1949).
Reichenbach, Experience and Prediction, 1938.
Salmon, The Foundations of Scientific Inference, 1966.
Richard von Mises, Probability, Statistics and Truth 1928, p. 103.
A recent innovative defence of the frequentisi point of view can be found in several articles by G. Menges (ed.), Information, Inference and Decision, (1974).
Herbert Feigl introduced the seemingly important distinction between “justification” and “vindication” of induction. See his frequently quoted article ‘De Principiis non Disputandum…’, 1950, pp. 113–147.
A statistical hypothesis may be defined as “an assumption about the frequency function of a random variable.” Cf. Hoel, Introduction to Mathematical Statistics, 1954, p. 30.
Braithwaite, Scientific Explanation, 1960, pp. 116,119.
Neyman and Pearson, ‘On the Use and Interpretation of Certain Test Criteria for Purposes of Statistical Inference’, 1928; and idem, ‘On the problem of the most efficient tests of statistical hypotheses’, 1933.
The type I error (a) expresses the probability that a sample point will fall in the critical region when the hypothesis to be tested is actually true. The type II error (ß) expresses the probability that a sample point will fall in the non-critical region when the alternative hypothesis is actually true.
Pioneering work in this area was done by Markoff, Calculus of Probability (in Russian; 1913)
Fisher, ‘On Mathematical Foundations of Theoretical Statistics’, 1921;
Fisher, ‘Hotelling The Consistency and Ultimate Distribution of Optimum Statistics’, 1930
Dobb, ‘Probability and Statistics’ 1934.
Developed by Neyman, ‘Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability’, 1937.
“A maximum likelihood estimator θ̂ of the parameter θ in the frequency function f(x; θ) is an estimator that maximizes the likelihood function L (x1,…, xn θ) as a function of θ”. Hoel, Introduction to Mathematical Statistics, 1954, p. 40. As Kyburg (1970, p. 133) remarks, the estimators or estimation functions of the statisticians correspond to the inductive rules of the philosophers.
See Russell, Human Knowledge, Its Scope and Limits, 1948.
Wisdom, Foundations of Inference in Natural Science, 1952.
Burks, The Presupposition Theory of Induction,’ 1953.
Burks, ‘On the Presuppositions of Induction’, 1954/55.
In the sense of leading to the cognition of lawlike statements or hypotheses based on the uniformity of the universe of discourse.
Skyrms, Choice & Chance: An Introduction to Inductive Logic, 1966.
Reprinted from Skyrms (1966).
Russell, Human Knowledge, Its Scope and Limits, 1949, p. 377. Copyright © 1948 by Bertrand Russell; © 1975 by the estate of Bertrand Russell.
Kyburg (1970, p. 31) for example points out that “defining probability as a ratio of equiprobable alternatives” need not be circular “as long as there is some way, not involving reference to probability, for identifying the equiprobable alternatives. That is,… if we can define equiprobability in general terms of equiprobable alternatives, and then define equiprobability of alternatives in terms of something that does not involve reference to probability.” The principle of indifference fulfills this requirement of individuating the alternatives to be counted. This principle assumed equal probabilities for events which do not indicate any reason to believe otherwise.
For illustrations of such contradictions see Kyburg (1970, pp. 34–38), and Salmon (1966, pp. 67–69).
Richard von Mises, Probability, Statistics and Truth, 1957, p. vii.
Russell, An Inquiry into Meaning and Truth, 1940; quoted from the Pelican edition, 1962, p. 272.
Cf. Russell, Human Knowledge, p. 69 ff.
Cf. Braithwaite, Scientific Explanation, 1953; reference to 1960, p. 191.
For further details see Hacking, Logic of Statistical Inference, 1965, and
Popper, ‘The Propensity Interpretation of Probability’, 1960. See also Stegmüller (1973c).
Fellner, Probability and Profit, 1965, p. 46.
Thus Carnap recognizes both kinds of probability and occasionally recommends probability2 as an estimate of probability1 or vice versa.
Adapted from Carnap (1960, p. 297); cf. Salmon (1966, p. 69). These rectangles represent sentences and not sets, thus the figures are not Venn-diagrams. In Figure 5.2 (a), therefore e ⊃ h (if e and h were sets, one would have to write e ⊂ h).
Although Jeffreys, whose Theory of Probability did not appear before 1939, raises some claim of priority, referring to two articles which he published in co-authorship with Dorothy Winch. Cf. Essays in Biography, 1933. Cf. Raiffa (1968, p. 275).
This is manifested in Carnap’s The Continuum of the Inductive Methods, 1952, where the availability of a continuum of infinitely many inductive methods and possibilities to measure the degree of confirmation is revealed.
Kyburg, Probability and the Logic of Rational Belief, 1961.
Some authors regard the “degree of confirmation” problem a dead issue, as the following colorful titles of recent publications indicate, e.g. Settle, ‘Confirmation as a Probability: Dead, but it Won’t Lie Down’, 1970, p. 200; and Schlesinger, ‘On Irrelevant Criteria of Confirmation’, Kyburg, Probability and the Logic of Rational Belief, 1961 p. 282.
Savage, The Foundations of Statistics, 1954, p. 57.
See Bunge, Scientific Research, I-The Search for System, 1967, p. 428. Interpretations 1 and 2 yield conceptual models, while the other interpretations yield factual models.
It is not necessary that a proposition must manifest itself in a sentence (in the narrow linguistic) sense; that it can also manifest itself in an action-configuration of mechanical, electrical and chemical events, has been discussed by this author elsewhere: see Mattessich, ‘Epistemologica’ Consequences of Artificial Intelligence and Systems Research’, Methodology and Philosophy of Science, London, Ont, 1975b, pp. V-85 to V-86.
This outline is based on the discussion by Bunge, 1967, Vol. 2, pp. 291–294.
Under “inductive normative arguments” we understand arguments as illustrated in Table 5–6, something that must not be confused with what Stegmüller (1973b, pp. 389–543), calls “Normative Theorie des induktiven Räsonierens” (normative theory of inductive reasoning). By this somewhat unfortunate expression Stegmüller does not refer to inductive arguments in which a premise and the conclusion are imperatives or ought-sentences, but refers to a reconstruction of Carnap’s inductive logic (which of course is based on some norms or value judgements in the meta-language). Thus there does exist literature (even extensive one) in the area of inductive reasoning with norms in the meta-language but hardly in the area of inductive reasoning where the norms refer to the object language, that is to say to the premises and the conclusion.
Black, M. Induction, 1967, pp. 178–179.
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Mattessich, R. (1978). The Controversy Around Inductive Logic. In: Instrumental Reasoning and Systems Methodology. Theory and Decision Library, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-9431-3_5
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