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Inductive Probability
J. P. Day
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Inductive Probability
J. P. Day
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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.
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1 THE PHILOSOPHICAL PROBLEM OF INDUCTIVE PROBABILITY
1.1 ATTRIBUTE-INDUCTIONS, VARIABLE-INDUCTIONS AND EVIDENTIAL-STATEMENTS
1.1.1 The Meaning of âInductionâ
By âinductive probabilityâ I mean the probability of inductions. What then are inductions? âInductionâ 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 âprobableâ or âevidenceâ; 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 âan inductionâ as âa generalization or a proposition derived from a generalizationâ. This requires an explanation of the meaning of âa generalizationâ; not a stipulated definition thereof, for âgeneralizationâ 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 âThe fact that q is evidence that pâ. 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 âIf q that is evidence that pâ. 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. âDark cloud is evidence of coming rainâ.
Take now a special case of formula (1), namely the formula, (2): E(fAg, fâČAg), read âThe fact that all observed f are g is evidence that all f are gâ. I use f, g, h, etc., as attribute-variables, and symbolize âobserved f, g, h, etc.â as fâČ, gâČ, hâČ, etc. A is the universal quantifier, read âallâ. 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 âa generalizationâ, 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: âThe fact that all observed balls in this urn are hollow is evidence that all the balls in this urn are hollowâ, and abbreviate this statement: âThe fact that all observed BU are H is evidence that all BU are Hâ.
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 âandâ. 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: âThe fact that all BU are H and that this is a BU is evidence that this is Hâ. It may be objected that the formula that this statement exemplifies is rather: E(gx, fâČAg . fx). But on this question I agree with Mill. Doubtless there is what he calls âinference from particulars to particularsâ; 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 â I(gx). â symbolizes implication, read âifâ. [Mill, 121 ff., 187 ff.]
My distinction between primitive and derivative inductions closely resembles Nicodâs distinction between primary and secondary inductions. â⊠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.â [Nicod, 212] Cp. also Maceâs distinction between âdirectâ and âindirectâ 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 âan inductionâ. Some say that one sort of induction is induction by complete enumeration. Our word âinductionâ is derived from Aristotleâs technical term Ï”âÏαγÏγΟ, 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 fâČAg and that fAfâČ, 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 âperfectâ inductions as one of the two species of inductions. â⊠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 particularsâ. [De Doctrinae Ordine Apologia, 1594; quoted in Venn, P, 343] And in recent times Johnson has recognized them under the name of âsummary inductionsâ 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 âinductionsâ 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 âan inductionâ 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 âgo beyond the evidenceâ or involve a âleapâ 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 âless knownâ, 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 âintuitive inductionâ. [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. âAll coloured things must be extendedâ. To establish such a statement by intuitive induction apparently means this. I âseeâ that this particular thing that is coloured must be extended; but I simultaneously also âseeâ that all things that are coloured must be extended. The operation is sometimes called âseeing the general rule in the particular caseâ; Aristotle himself speaks of âinduction exhibiting the universal as implicit in the clearly known particularâ. 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 âinductionsâ, for two reasons. The first is the same as one of those for refusing to recognize âsummary inductionsâ 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 âThis coloured thing must be extended, therefore all coloured things must be extendedâ; we âsee the general rule in the particular caseâ. The second reason is this: if intuitive inductions are to be called inductions, then some inductions are necessary truths. But on my definition of âinductionâ 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 âdeductive derivative inductionsâ 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. âAll men must be mortalâ is not like âAll men must be maleâ. In the latter, âmustâ does indeed signify that the statement is a necessary or analytic truth. But in the former, âmustâ signifies that the statement is a necessary consequence merely, say of âAll animals are mortalâ conjoined with the additional premiss âAll men are animalsâ. And since âAll animals are mortalâ is a contingent or synthetic truth, its necessary consequence âAll men must be mortalâ 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. âA deductionâ and âa valid deductionâ 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 ânot-pâ), 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 . fâČI is self-contradictory, but fâČAg . fAg is not. (, , etc. are read, ânon-fâ, ânon-gâ, etc.; the quantifier I is to be read âSomeâ.) Humeâs 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 âan inductionâ 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. â⊠Induction may be defined, the operation of discovering and proving general propositions.â [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 âgeneralizationsâ, not âgeneral propositionsâ, 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 âgeneralizations from experienceâ, in Millâs phrase; and is the expression âempirical generalizationâ 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, âall inductive reasoning is but an inverse application of deductive reasoningâ, specifically of âprobable deductive reasoningâ. [Jevons, I, 307 f.; 239 ff.] Jevons follows Laplace, who attempts to justify induction by an inversion of Bernoulliâs theorem, in the proof of which he uses Bayesâ 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 Williamsâ theory later [4.2]. There is an important difference between his account and the othersâ 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 Williamsâ theory. But the general question should be raised here, Is Jevonsâ formula an acceptable account of inductions as...