Variety, Analogy, and Periodicity in Inductive Logic Rudolf Carnap Philosophy of Science, Vol. 30, No. 3. (Jul., 1963), pp. 222227. Stable URL: http://links.jstor.org/sici?sici=00318248%28196307%2930%3A3%3C222%3AVAAPII%3E2.0.CO%3B26 Philosophy of Science is currently published by The University of Chicago Press.
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DISCUSSION: VARIETY, ANALOGY, AND PERIODICITY IN
INDUCTIVE LOGIC.*
R U D O L F CARNAP1 University of California at Los Angeles
1. The Variety of Instances, Peter Achinstein gives in his papers [I] and [z] interesting analyses of some problems of inductive logic and of some approaches I have proposed. I shall discuss here some of these problems in order to clarify my present position. My comments will mainly concern the variety of instances, and only briefly the analogy influence, and the inductive methods for a coordinate language. I n his paper [2] (the ms. of which he kindly made available to me) Achinstein derives some formulas concerning my confirmation function c* for certain examples. He infers from the results that c* does not satisfy the requirement of the variety of instances (RVI). I had asserted in my paper ([3], $151 and again in my book ([5], $110 I] that c* satisfies RVI. He further generalizes his results by arguments intended to show that none of the confirmation functions in the lambdasystem, explained in my monograph [6],satisfies RVI. Many years ago, when I examined for the first time the question whether c* satisfies RVI, I used examples similar to those of Achinstein and found the same result: that the value of c* for the hypothesis was the same on each of two evidences, one with a variety of instances and the other with instances of one kind only. This led me too to the belief that the requirement was not fulfilled. However, on closer analysis I saw that the simple forms of evidence which I had used (and similarly now Achinstein) are not suitable for an examination with respect to RVI. T h e examination must rather be made in the following way. T h e observers X and Y start out with a common prior evidence e, which states the following about a class K of N objects: (a) N , specified objects of class I  have the property PI, N', have not; (b) N,, specified objects of those with P, have also the property P,, N,,. others have not. Both observers intend to test the law I: "All objects with P, have also P,". They make their tests separately, but they agree that each of them will test a sample of s specimens from the N , objects which were found to have PI. Now X takes his s specimens all from the N,, with P,, while Y, mindful of the requirement RVI, takes s,, specimens from those N,, and s  s,, others from the N,,. without P,. X's report el on his tests states that his s specimens have P,; and Y's report says the same of his s specimens. Thus either observer has found s confirming instances and no disconfirming ones. Now let h be the hypothesis saying that the next object (or the next n objects) found to have P, will also have P, and thus satisfy the law I. Then RVI demands that the confirmation function c be such that c(h, e.e,) > c(h, e.e,). This is the correct form of the requirement. And I shall show that this is fulfilled by my function c*. T h e essential point here is that the two observers have the same prior information about the observed frequencies of the various kinds of objects which come into consideration for a test of the law. This is necessary for a fair comparison. I n Achinstein's example, the evidence for each observer is restricted to those objects which he tests for P,. Thus the prior information of the first observer is restricted to those s ravens
* Received March, 1963.
' The author is indebted to the National Science Foundation for the support of research in inductive probability.
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which he has found to be young. This is then his total knowledge about ravens, since we must assume that he has fulfilled the requirement of total evidence ([j], $45 B), For all he knows there may be no old ravens. Under these conditions we cannot say that he violated a rnethodological principle by not choosing any old ravens for his test of the law.
2. On the function c*, We consider the law I: '(x) (iMx 3 M'x)', interpreted, e.g., as "All metal bodies are good conductors of electricity". I shall use the concepts and notations of ([j], $$31, 32, 38). We take a finite language L with N individuals and K Qpredicates. We define 'M,' by ' M . M' ', and 'M,' by 'M.M1'. T h e law I says that M , is empty. Let the p predicates 'K,', ..., 'I(,' be pairwise Lexclusive, and their disjunction be Eequivalent to ' M y .Thus they designate p different, nonoverlapping kinds of metal (e.g., iron, copper, etc.). For each q, let K,,, be the conjunction of M , and K,, and K 2 ,,that of M , and K,. Let w,,,be the logical width of Kl,q, and w,,q that of K,,,. Each of the predicates mentioned is supposed to be a factual, molecular predicate in L, hence Lequivalent to a disjunction of (one or more) Qpredicates. We assume that X's prior information is expressed in three sentences k, k*, and j, as follows. Each of them is a conjunction of full sentences of certain predicates; all these full sentences contain distinct individual constants. k contains s' full sentences, to wit, for every q, s', full sentences of K,,,. Thus k is a report about earlier tests of the law I, with s' specimens of M, among them s', of the kind K,, all of which were found to have M,, and thus to confirm the law. k* contains s* full sentences of different predicates, each predicate Limplying nonM. This is a report on observations of s* objects which were all found to be nonmetals. j contains s, full sentences with Kpredicates, among them s,,, with K,. j is a description of s, metal specimens, each of them specified as belonging to one of the p subkinds K,, ..., K,. Now X makes new tests for the law I. He chooses s of the specimens described in j, among them, for every q (from 1 to p), s, specimens from those described in j as belonging to the kind K g . (Hence s , 5 s,,,; some of the numbers s, may be 0.) All these tests have positive results, i.e., these s specimens are found to be M, and thus M ' , and hence to confirm the law. Let i be the report of these results. Thus i is a conjunction of s full sentences of M , with distinct individual constants; among these constants there are, for every q, s, which occur in j with K,. T h e posterior information consists of the prior one together with i. I shall now give a theorem about c* involving the sentences just described. (I referred to this theorem in [3, $151 without stating it explicitly; its proof is too complicated to be given here.)

I have chosen here a language L with a finite number N of individuals, so that the law I is a finite conjunction of instances and has positive c*values. (For an infinite law, c* is 0 ; see ([5], $110 F , (12)). We define R as the ratio of the posterior confirmation of I to its prior confirmation:
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We see easily that i is Limplied by j.1, and hence also by k.k*.j.l. Therefore, by ([5], T613c):
We see from (3) and (1) that R is independent of the values of s*, s , , and s,,, (i.e., s,,,, ..., s,,,), although the two c*values in (2) are dependent on them.
3, Application to Nagel's Numerical Examples, Nagel has given some numerical examples ([9], pp. 6871) in order to illustrate the great difficulties for the construction of a cfunction that would be in accord with RVI. He makes use of two subkinds K , and K,. Hence we have p = 2. In order to make our numerical calculations easier, we make the following simplifying assumptions: (A) we omit k ; hence we have sf, = 0. This means that the prior information does not contain any results of tests of metal specimens with respect to conductivity; thus the only results of this kind are those described in i. (B) We assume that each of the widths w,,, w,,, w,,, w,, is 1 ; thus the width of M is 4. (We might take the width of nonM as 12; then K = 16; but these numbers are irrelevant for R, though relevant for the c*values.) For the intended examination of c* with respect to RVI, it is fortunately not necessary to calculate the posterior values of c* for the various cases. It is sufficient to determine R. And for this we obtain, under the simplifying conditions mentioned, an extremely simple formula (from (3) and (1)): Nagel discusses nine different possible cases P,,
..., P, with respect
to the total number
s of specimens tested, and the numbers s, and s, of the two kinds, as given in the sub
sequent table. Let us imagine nine observers X,, ..., X,, who have the same prior information k*.j, but then make different decisions with respect to s, s,, and s,, as specified in Nagel's nine cases. (In order to make possible the values of s, and s, in all nine cases, we must choose the numbers in j sufficiently high so that, in each case, s,,, 2 s,; hence we must take s,,, 2 200 and s,,, 2 100; but here we need not bother about these numbers, since they do not affect R.) T h e nine observers obtain different results i, say i,, ..., i,, and hence different values of the posterior confirmation of the law. Now we have from (2): For n = 1, ..., 9, c*(l, k*.j.i,) = R,.c*(l, k*.j). (5) I have added to Nagel's table the last line, giving the values of R,, ..., R,, determined by (4). Since the prior confirmation is the same in all nine cases, the posterior confirmation is, according to (5), proportional to R,.
T h e values of R , clearly bear out what I said about the nine cases in ([3], $15). I n particular, these values show that the posterior confirmation is increased not only
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if the total number s of confirming instances is increased (and none of the numbers s, and s, is decreased), but also, and indeed much more, if with the same s the distribution is improved, i.e., the smaller of the numbers s, and s, is increased from 0 to a positive value, or from a positive value to another one nearer to s/2. T h u s the function c* is in good accordance with RVI.
4. The Analogy Influence. T h e principle of analogy says roughly this: the probability that an object 6 has a certain property, is increased by the information that one or more other objects, which are similar to b in other respects, have this property. In very simple situations, the old way of applying the function c* leads to the required result, as I have shown in ([5], $110 D). This holds likewise for any other cfunction of the lambdasystem. However, this is not the case for more complicated situations, e.g. those is Achinstein's example with rhodium. This is due to the failure of the old methods to take account of the different degrees of similarity between Qpredicates. If (as in ([5], p. 125, Table A311)) Q, is defined by P,.P,.P,, Q, by P,.P,. P,, Q 4 by PI. PB. P,, and Q, by  P I . P,. P,, then an object with Q, is most dissimilar to one with Q,; one with Q, is somewhat more similar, and one with Q, still more. This defect of the old methods is overcome bv what we now call the method of several families. We now regard the formulas for functions of the lambdasystem, including c*, as exactly valid if they are applied to predicates of one family. This is explained in the new Preface to ([5], second edition). When predicates of two or more families are involved, the old methods are at best approximately valid, and new methods are required. For example, the old methods are appropriate for a law of the form: "For every x, if x is green or blue, then x is blue", which says in effect that the property Green is empty. On the other hand, for a law like I in $2, according to the new approach, we should use a new kind of cfunction for two families, one family of kinds of substance, including K,, ..., K,, and one family of three predicates: "good conductor of electricity", "bad conductor", and "nonconductor". Soon after writing [6], I found a cfunction appropriate for two families. Then in 1953, John G. Icemeny and I together worked out methods for any number n of families. T h e formula for two families is given in (171, Anhang B, sec. VIII). These methods take account of the various degrees of similarity among the Qpredicates. They would, I suppose, lead to intuitively plausible results in Achinstein's examples. N
N
 

5. Coordinatelanguages. I n his earlier paper [I], Achinstein analyzes the problem of adequate cfunctions for a coordinate language with natural numbers as coordinates (comp. [5], pp. 62 f., 74). He correctly rejects certain approaches as inadequate. I n his requirements he makes use of my distinction between qualitative, positional, and mixed predicates. However, I would not require that only qualitative predicates occur in a law or other hypothesis; any sentence of the language may be taken as a hypothesis. What I mean is only this: in certain axioms and theorems referring to predicates, these predicates (or some of them) are required to be qualitative. Achinstein states a very general Principle 1, which seems to him plausible, and which is supposed to justify his requirements. However, I have serious doubts about this principle. If I understand it correctly, it seems to me not valid in a coordinate language, not even if the hypothesis h contains only qualitative predicates. Let h be 'P(201)'; let e say that the positions from 1 to 110 have P, and those from 111 to 200 nonP. Thus e contains 110 instances of h, and 90 of nonh. Principle 1 requires (if I under
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stood it correctly) that c(h, e) > c( h, e). I n contrast, I would think that c( h, e) should be near to 1, and therefore c(h, e) near to 0. I n a coordinate language, the numbers of positive instances and of negative instances have sometimes less effect on the cvalue of a prediction h than the nearness of the instances to the position referred to in h. (I call this effect the proximity influence.) Achinstein studies chiefly laws of periodicity. He analyzes various approaches, rejects each of them, and comes finally to an impasse. He believes that a solution will require a much stronger language containing predicates and quantified variables of higher order. I do not share this belief. I will briefly indicate how I would approach the problem of a coordinate language in inductive logic. As an example, let us think of a family of five predicates for simple qualitative properties PI, ..., P,, say colors. An msegment is a series of m consecutive positions. I introduce Qmpredicates for the possible properties ("mspecies") of msegments. For example, I define: Thus the sentence 'Q3,,,,, (8)' ascribes to the 3segment beginning with position 8 the 3species consisting of P,, P,, P, in this order; but formally, 'Q3,,,,,' is a oneplace predicate of positions. With the help of these Qmpredicates, many kinds of regularity in the order in which the colors appear can be formulated. A universal sentence saying that p specified mspecies never occur is called a law of span m and strength p (in analogy to ([5], D376a)). For example, a law saying that, if P, is followed by P,, then P, always follows, followed in turn by either P, or P,, is a law of span 4 and strength 12. I should prefer to use the term "periodicity law" only for those laws which describe a regular recurrence with a fixed period length n. Such a law has a span n. For example, the sentence "P, is not empty and, for every i, P,(i) iff P,(i 10)" has a span 10; it excludes all and only those 10species in which P, does not occur exactly once. All laws with finite span, whether periodical or not, can be dealt with in terms of Qmpredicates. Inductive methods for coordinate languages are under investigation; they take into account both regularities of succession (of finite span) and the proximity influence. But there are difficult ~ r o b l e m still s to be solved. For the laws here involved no variables or predicates of higher order are necessary. If a law refers to absolute values of the coordinate, then it is quite different from laws of finite span (which involve only coordinate diferences, not absolute values). An example is the law: "For every n, P,(n) z . n is a prime number", discussed by Hilary Putnam [lo] and Achinstein [I]. As I have explained in my reply to Putnam (181, §29), I think that few physicists would even consider laws of this kind, and that it is hardly worthwhile to take account of such laws in adequacy conditions for cfunctions for a coordinate language. It would perhaps be preferable to take as coordinates all integers, and not to give a distinguished role to any one position. This might diminish the temptation to refer in a law to absolute values of the coordinate.
+
REFERENCES
[I] ACHINSTEIN, P., "Confirmation Theory, Order, and Periodicity", Philosophy of Science, 30 (1963): 1735. [2] ACHINSTEIN, P., "Variety and Analogy in Confirmation Theory", Ibid., 207221. [3] CARNAP, R., "On Inductive Logic", Philosophy of Science, 12 (1945): 7297.
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[4] CARNAP,R., "On the Application of Inductive Logic", Philosophy and Phwommological Research, 8 (1947): 133147. [5] CARNAP, R., Logical Foundations of Probability., Chicago, 1950; second edition, 1962. R., The Continuum of Inductive Methods, Chicago, 1952. [6] CARNAP, R. and STEGMULLER, W., Induktive Logik und Wahrscheinlichkeit, Wien, 1959. [7] CARNAP, [8] CARNAP, R., "Replies and Systematic Expositions", in [ll], ($29: Hilary Putnam on Degree of Confirmation and Inductive Logic). [9] NAGEL, E., Principles of the Theory of Probability, Chicago, 1939. H. "Degree of Confirmation and Inductive Logic", in [ll]. [lo] PUTNAM, Ell] SCHILPP,P. A. (ed.), The Philosophy of Rudolf Carnap. (Library of Living Philosophers), Open Court Publishing Co., Wilmette, Ill., 1963.