Inductive Probability

J. P. Day

• Publisher: Routledge
• Year: 2021
• ISBN:
  9781000504323
• Edition: 1
• Topic: Philosophy
• Subtopic: Logic in Philosophy

First published in 1961, Inductive Probability is a dialectical analysis of probability as it occurs in inductions. The book elucidates on the various forms of inductive, the criteria for their validity, and the consequent probabilities. This survey is complemented with a critical evaluation of various arguments concerning induction and a consideration of relation between inductive reasoning and logic. The book promises accessibility to even casual readers of philosophy, but it will hold particular interest for students of Philosophy, Mathematics and Logic.

1 THE PHILOSOPHICAL PROBLEM OF INDUCTIVE PROBABILITY

1.1 ATTRIBUTE-INDUCTIONS, VARIABLE-INDUCTIONS AND EVIDENTIAL-STATEMENTS

1.1.1 The Meaning of ‘Induction’

By ‘inductive probability’ I mean the probability of inductions. What then are inductions? ‘Induction’ is a technical term of Philosophy, so that there cannot be a problem of explaining its meaning in the same way as there is a problem of explaining the meaning of non-philosophical terms such as ‘probable’ or ‘evidence’; its meaning is what philosophers stipulatively define it to be. Nevertheless, there is a problem of a sort, since philosophers do not all agree about the stipulated definition of the word.

Conformably with the practice of at any rate some philosophers, I define ‘an induction’ as ‘a generalization or a proposition derived from a generalization’. This requires an explanation of the meaning of ‘a generalization’; not a stipulated definition thereof, for ‘generalization’ is not a philosophical term. The explanation must be given, in my view, in terms of the notion of an evidential-statement. Consider the formula, (1) :E(p, q), read ‘The fact that q is evidence that p’. I use p, q, r, etc., as proposition-variables; in this formula it is convenient to call q the evidencing-statement-formula and p the evidenced-statement-formula. I call this formula the categorical evidential-statement-formula. It would also be possible to construct an hypothetical evidential-statement-formula, to be read ‘If q that is evidence that p’. But in this essay it will be convenient to confine our attention in the main to categorical formulas. Notice, however, that indicative sentences are often used to make hypothetical, not categorical, evidential-statements; e.g. ‘Dark cloud is evidence of coming rain’.

Take now a special case of formula (1), namely the formula, (2): E(fAg, f′Ag), read ‘The fact that all observed f are g is evidence that all f are g’. I use f, g, h, etc., as attribute-variables, and symbolize ‘observed f, g, h, etc.’ as f′, g′, h′, etc. A is the universal quantifier, read ‘all’. I call (2) an inductive primitive evidential-statement-formula. Its evidenced-statement-formula is a generalization-formula or primitive induction-formula, and statements substituted on it are generalizations or primitive inductions. This is an illustration designed to explain the meaning of ‘a generalization’, not a definition. For it is too narrow to serve as a definition. For, as will be shown, there are variable-generalizations as well as attribute-generalizations. Moreover, it illustrates only one type of attribute-generalization, namely, the universal subject-predicate type; but, as will also be shown, there are other types of attribute-generalization. I take as paradigm exemplification of formula (2) the evidential-statement: ‘The fact that all observed balls in this urn are hollow is evidence that all the balls in this urn are hollow’, and abbreviate this statement: ‘The fact that all observed BU are H is evidence that all BU are H’.

Consider next the formula, (3): E(gx, I(fAg). fx). I use x, y, z, etc., as individual-variables; fx and gx are to be read respectively as x is f and x is g.. symbolizes conjunction, read ‘and’. As for the constituent formula I(fAg), this indicates that fAg is an induction-formula and not e.g. an observation-statement-formula. I call (3) an inductive derivative evidential-statement-formula. The difference between a derivative and a primitive evidential-statement-formula is that the evidencing-statement-formula of the former contains as constituent an evidenced-statement-formula, whereas that of the latter does not. Then, the evidenced-statement-formula of (3) is a derivative induction-formula. This again is an illustration, not a definition, being too narrow to serve for the latter. An exemplification of formula (3) is: ‘The fact that all BU are H and that this is a BU is evidence that this is H’. It may be objected that the formula that this statement exemplifies is rather: E(gx, f′Ag . fx). But on this question I agree with Mill. Doubtless there is what he calls ‘inference from particulars to particulars’; i.e. inductive inference in which we move from fx to gx without first making the generalization I(fAg). But, though they need not be so regarded, such inferences may always be regarded as passing through the generalization; and it is convenient so to regard them. I.e., we have first an inductive inference in accordance with formula (2); and second a deductive inference in accordance with the formula: I(fAg). fx ⊃ I(gx). ⊃ symbolizes implication, read ‘if’. [Mill, 121 ff., 187 ff.]

My distinction between primitive and derivative inductions closely resembles Nicod’s distinction between primary and secondary inductions. ‘… suppose that an induction has among its premises the conclusion of another induction. We shall call it then a secondary induction. Primary inductions are those whose premises do not derive their certainty or probability from any induction.’ [Nicod, 212] Cp. also Mace’s distinction between ‘direct’ and ‘indirect’ inductions. [Mace, 259, 285] All generalizations are primitive inductions and conversely; and since generalizations are of course general, all singular inductions are derivative. This is not to deny that there are general derivative inductions; consider e.g. statements exemplifying the evidenced-statement-formula of the formula, (4): E(fAg, I(hAg). fAh). But it is to deny that general derivative inductions are (rightly called) generalizations.

It will now be convenient to comment on some other definitions or explanations of the meaning of ‘an induction’. Some say that one sort of induction is induction by complete enumeration. Our word ‘induction’ is derived from Aristotle’s technical term ϵ’παγωγή, by which he means the establishing of universal statements by a consideration of particular cases falling under them. One, but not the only, way in which he says that this is done may be illustrated as follows. [Aristotle, 512 ff.; Kneale, 7 ff.; Mace, 245 ff.] I know by observation that f′Ag and that fAf′, and assert that fAg. Then, a statement substituted upon this last formula is an induction by complete enumeration. This doctrine has endured. E. g., in the 16th century Zabarella represents inductions by complete enumeration or ‘perfect’ inductions as one of the two species of inductions. ‘… there is no one who does not know that Induction is a logical instrument by which from particular notions a less known universal is demonstrated; and that this is of two kinds, perfect, which concludes necessarily because it embraces all the particulars, and imperfect which does not conclude necessarily because it does not embrace all the particulars’. [De Doctrinae Ordine Apologia, 1594; quoted in Venn, P, 343] And in recent times Johnson has recognized them under the name of ‘summary inductions’ as one of the four species of induction that he distinguishes. [Johnson, III, xiv] But I do not propose to call universal statements so established ‘inductions’ for the following reason. If, knowing that all the BU I have observed are H and that the BU I have observed are all the BU there are, I say that all BU are H, my statement is a description of BU. But my account of the meaning of ‘an induction’ is in terms of a generalization; and descriptions, as I shall argue more fully later, must be clearly distinguished from generalizations. The essential feature of the latter is that they assert something about unobserved instances of a kind on the evidence of observed instances of that kind, that they ‘go beyond the evidence’ or involve a ‘leap’ as it is often put; whereas the former logically must be about observed instances only. Hence, too, an induction is a conclusion of an inference or argument, whereas a description is not.

The passage from Zabarella raises two further points that deserve mention. He states that inductive inference is always from particulars to a universal conclusion. This doctrine also derives from Aristotle and is widely held; it is used to distinguish induction from deduction, the former being said to proceed from particular cases to a universal conclusion, the latter conversely. [Aristotle, Posterior Analytics, 81b] On my account of inductions, this is false; cp. formula (3). It is not true even of primitive inductions either, for we shall see that some generalizations are not universal, i.e. not of the form fAg. Again, Zabarella says that an induction is always ‘less known’, i.e. less certain, than its evidencing-statement; but we shall see that this too is false, since sometimes both have the same degree of certainty, namely, when both are true.

The other way in which Aristotle says that universal statements can be established by observation of particular cases is by what Johnson calls ‘intuitive induction’. [Aristotle, Posterior Analytics, 71a, 81b; Kneale, 30 ff.; Mace, 248ff.] The kind of universal statements that can be so established are necessary ones, e.g. ‘All coloured things must be extended’. To establish such a statement by intuitive induction apparently means this. I ‘see’ that this particular thing that is coloured must be extended; but I simultaneously also ‘see’ that all things that are coloured must be extended. The operation is sometimes called ‘seeing the general rule in the particular case’; Aristotle himself speaks of ‘induction exhibiting the universal as implicit in the clearly known particular’. I do not myself believe that we ever establish universal necessary truths in this way. And even if we do, I do not propose to call universal necessary truths so established ‘inductions’, for two reasons. The first is the same as one of those for refusing to recognize ‘summary inductions’ as a sort of inductions; namely, that on my definition an induction is a conclusion of an inference, whereas intuitive induction involves no inference. We do not argue or reason ‘This coloured thing must be extended, therefore all coloured things must be extended’; we ‘see the general rule in the particular case’. The second reason is this: if intuitive inductions are to be called inductions, then some inductions are necessary truths. But on my definition of ‘induction’ no inductions are necessary truths, since generalizations are not necessary truths. This needs explaining.

Necessary truths must be distinguished from necessary consequences. What I call ‘deductive derivative inductions’ are by definition necessary consequences of primitive inductions or generalizations. But they are not on this account necessary truths; they would be so only if the generalizations of which they are necessary consequences were necessary truths. However, it may be urged that they sometimes are. It may be said: We say that all men must die; this primitive induction is therefore a necessary truth; and when we deduce from it and the additional premiss that Tom is a man that Tom must die, this derivative induction is therefore a necessary truth too. But this still confuses necessary consequences with necessary truths. ‘All men must be mortal’ is not like ‘All men must be male’. In the latter, ‘must’ does indeed signify that the statement is a necessary or analytic truth. But in the former, ‘must’ signifies that the statement is a necessary consequence merely, say of ‘All animals are mortal’ conjoined with the additional premiss ‘All men are animals’. And since ‘All animals are mortal’ is a contingent or synthetic truth, its necessary consequence ‘All men must be mortal’ is not a necessary truth either.

It may also be thought that mathematical induction is a species of induction in my sense. [Kneale, 37 ff.] But this is not so. The essential difference is this. Mathematical inductions are deductions or theorems, mathematical induction being a process of deduction. But primitive inductions are not deductions, as Hume points out. [Hume, E, 35] His point is this. ‘A deduction’ and ‘a valid deduction’ mean the same thing. To say that p is a deduction is therefore to say that the conjunction of the contradictory of p, ~p (read ‘not-p’), with the premiss from which it is inferred, q, is self-contradictory. It is to say that q. ~p is self-contradictory. But to say that p is a primitive induction is not to say that the conjunction of its contradictory with the evidencing-statement from which it is inferred, q, is self-contradictory. It is not to say that q. ~p is self-contradictory. Thus e.g. fAg . f′I$\overline{g}$ is self-contradictory, but f′Ag . fAg is not. ($\overline{f}$, $\overline{g}$, etc. are read, ‘non-f’, ‘non-g’, etc.; the quantifier I is to be read ‘Some’.) Hume’s distinction differentiates deductions from primitive inductions, but not from derivative inductions, since some of these are deductions from primitive inductions. Nevertheless, mathematical inductions are not derivative inductions either. For the axioms or premisses from which they and the theorems of Pure Mathematics generally are deduced are not generalizations. Mill, indeed, asserts the opposite. [Mill, 147 ff.] According to him, the axioms of Pure Mathematics are precisely empirical generalizations of the widest scope; so that by his account mathematical inductions and mathematical theorems generally are derivative inductions. But his account is not acceptable.

The question whether mathematical inductions are a sort of inductions in my sense of ‘an induction’ must be distinguished from the question whether inductions in my sense are allowable in Pure Mathematics. The answer to the latter question seems to me to be as follows. When the process of induction is applied to empirical objects, it is both a method of discovery and a method of proof. ‘… Induction may be defined, the operation of discovering and proving general propositions.’ [Mill, 186] Actually, this definition is both too narrow and too wide; too narrow in that it excludes singular (derivative) inductions, and too wide in that it admits e.g. deductions as well as inductions; Mill really means ‘generalizations’, not ‘general propositions’, as indeed he himself later says. [Mill, 200] When I establish that all observed BU are H and generalize from this fact that all BU are H, I simultaneously discover the composition of the population and prove that it has that composition. But in Pure Mathematics, where the process of induction is applied to non-empirical objects, say numbers, it is allowable as a method of discovery only. The reason is that in this field induction is not regarded as the right sort of proof; here, deductive proof or demonstration is alone accepted, one species of which is mathematical induction. But that induction is a common method of mathematical discovery is not disputed; thus, by his own account Newton discovered the binomial theorem in this way, though he left it to others to provide a satisfactory demonstration of it.

In the light of these considerations we can answer a connected question, Are there non-empirical inductions? Or are all generalizations necessarily ‘generalizations from experience’, in Mill’s phrase; and is the expression ‘empirical generalization’ consequently pleonastic? Granted that numbers are not empirical objects, it appears from the foregoing discussion that there are nonempirical inductions. But in this essay I shall discuss empirical inductions only. For my topic is inductive probability, and this relates to the probative, not to the heuristic, aspect of induction. But we have seen that in non-empirical domains, such as Pure Mathematics, the probative aspect of induction does not apply. Consequently, philosophical problems about the inductive probability of mathematical theorems do not arise.

According to Jevons, ‘all inductive reasoning is but an inverse application of deductive reasoning’, specifically of ‘probable deductive reasoning’. [Jevons, I, 307 f.; 239 ff.] Jevons follows Laplace, who attempts to justify induction by an inversion of Bernoulli’s theorem, in the proof of which he uses Bayes’ inversion formula, otherwise called the principle of inverse probability. [Laplace, 15 f.] Somewhat similar attempts are made by e.g. Keynes and Broad, Jeffreys and Williams. I shall consider Williams’ theory later [4.2]. There is an important difference between his account and the others’ in that he makes no use of the principle of inverse probability, and indeed claims it as a leading merit of his theory that it dispenses with this principle and so avoids the difficulties that it involves. [Williams, G, 99, 192; Keynes, 148 f., 174ff., 367 ff.; Kneale, 201 ff.] Detailed discussion of argument along these lines is therefore deferred to my examination of Williams’ theory. But the general question should be raised here, Is Jevons’ formula an acceptable account of inductions as...