On a Claim by Skyrms concerning Lawlikeness and Confirmation

On a Claim by Skyrms concerning Lawlikeness and Confirmation Carl G. Hempel Philosophy of Science, Vol. 35, No. 3. (Sep., 1968), pp. 274-278. Stable URL: http://links.jstor.org/sici?sici=0031-8248%28196809%2935%3A3%3C274%3AOACBSC%3E2.0.CO%3B2-I Philosophy of Science is currently published by The University of Chicago Press.

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DISCUSSION ON A CLAIM BY SRYRMS CONCERNING LAWLIKENESS AND CONFIRMATION*

CARL G. HEMPELT Princetoii University I11 his article [ 5 ] , Brian Skyrms adduces (on p. 233) some generalizations which, he claims, receive no confirmatory support from their positive instances even though all the predicates they contain are well entrenched in Goodman's sense. Invoking the principle that "a generalization is lawlike if it is capable of receiving confirmatory support from its positive instances" (p. 232: actually, the converse is needed and, I will assume, intended by him), he claims that his examples "provide striking demonstration of the fact that the lawlikeness of a hypothesis is not a simple function of the projectibility of its constituent predicates." I think the claim is of great interest; but I will try to show that Skyrms's argument fails to establish it because it presupposes an unwarranted assumption which raises a problem of general importance for confirmation theory. Skyrms introduces his specimen hypotheses in the context of a discussion of my article [4], in which I proposed a definition of a purely qualitative concept of confirmation as invoked in sentences of the form 'observational evidence (sentence) E confirms hypothesis N'.According to that definition, the universal conditional hypothesis H I : '(x)(Fx =I G.4' is confirmed by all sentences of the type 'Fa Ga', which represent what are often called the "positive instances" of the hypothesis, but H, is confirmed also by evidence sentences of many other kinds. Skyrms objects that my definition does not properly characterize the concept of confirmation: "What Hempel actually does," he says, "is to define 'E is the positive instance of H', assuming in his discussion that positive instances are always confirmatory in the required sense (i.e., that if E is a positive instance of H, then the likelihood of H given E is higher than the likelihood of H on tautological eviden~e)."~ Actually, I did not make the explicit assumption imputed to me in the parenthetical explication, for at the time-perhaps regrettably-I had no idea of how

* Rcceived March, 1968. 1- Preparation of this note was supported by a research grant from the National Science Foundation. P. 243, paragraph 3. The three references made there to "pages 6 and 7" should be replaced by references to p. 233 of the article. [5], p. 233 ; emphasis cited. The phrase 'the positive instance' is infelicitous in several respects. First, the definite article makes no sense. Second, even 'rc positive instance' would be inadequate; for the hypothesis Hi is confirmed in the sense of my definition also, for example, by a conjunction of positive instance sentences of the type mentioned a b o ~ e :a nd such a conjunction does not represent what would normally be called a positive instance of HI. Finally, the notion of positive instance has a relatively clear application only in reference to hypotheses of purely universal farm, and perhaps to those of purely existential form; whereas the concept defined in [4] is applicable to quantified and unquantified (molecular) hypotheses of any form expressibie in first-order logic without identity.

qualitative confirmation might be related to the quantitative concept of degree of confirmation or logical probability, which I take Skyrms to have in mind when he speaks of likelihood. But it is true that I thought of evidence that confirms a hypothesis in my sense as lending inductive support to the hypothesis. That assumption was undermined by Goodman's discovery ([2], Part 111) of hypotheses, couched in terms of his weird predicates, that receive no inductive support from their positive instances. And some years earlier, Carnap had pointed out that the concept I had defined does not satisfy a condition which, on his view, every adequately defined qualitative (or "classificatory") concept of confirmation should meet, namely: If E confirms H, then the probability of H on E is greater than the probability of H on tautological evidence, for some adequately defined concept of inductive probability. ([I], p. 472, condition (1). Carnap also laid down a more general condition, which will be considered later.) It is this requirement, then-I will call it Carnap's condition-which Skyrms adopts as a criterion for "the required sense" of 'confirmation' or 'support'. To demonstrate his claim that a universal conditional hypothesis may fail to receive support from its positive instances even though it contains only well entrenched predicates, Skyrms adduces the following example, among others: (Hz)

(x)[(x is a blade of grass

v x is the cloth on a billiard table) 3 x is green]

and he asks: "Does the observation of a green blade of grass lend any support to the assertion that all cloths on billiard tables are green?' (p. 233) Spelled out in more detail, the argument here adumbrated comes to this: 1. The evidence statement E,: 'b is a blade of grass and b is green' represents a positive instance of Hz. 2. If E2did lend support to Hz then it would also have to lend support to the hypothesis (Hik) (x)(x is the cloth on a billiard table 3 x is green);

for H z is a consequence of Hz.

3. But E2does not lend support to Hz. 4. Hence, If2 receives no support from its positive instance Ez.

The argument is valid, and the first premiss is true. But what of the other two? The second premiss reflects the assumption that the relation of support satisfies a condition which in [4], section 8, I had set down, under the name of (special) consequence condition, for the concept of confirmation: If E confirms H, then E also confirms every consequence of H. This condition has considerablc intuitive plausibility, and it can be supported also by the observation that the probability E confers on a logical consequence of H is always at least as great as the probability E confers on H. But while the consequence condition is unexceptionally satisfied by the concept of confirmation defined in [4],3 Carnap has shown (El], pp. 474-475) See Theorem 8.21 and its proof in [3], p. 142. And since on my definition, E2 confirms Hz, it follows that Ez also confirms Hz. Skyrms adduces this consequence and some analogous ones as difficulties for my definition of confirination and adds: "There seems no way Wempel can rule [them] out" (p. 243). But having insisted on the consequence condition, I had no intention

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that it is violated by any concept of confirmation that accords with his condition. This is true, therefore, also of Skyrms's notion of inductive support. Hence, premiss 2 of the argument above is not established and, consequently, neither is the conclusion. Skyrms cites several other hypotheses containing well-entrenched predicates only; but, with one exception, his argument that they receive no support from their positive instances presupposes the consequence condition, which his concept of support does not satisfy. The one exception is the hypothesis 'Everyone has a large nose', for which Skyrms correctly cites the sentence 'Jimmy Durante has a large nose' as representing a positive instance. He seems to think it clear that this instance lends no support to the generalization; but he offers no reasons for this view. Indeed, short of being supplemented by the specification of an appropriate measure function for logical probability, the criterion of support invoked by Skyrms does not, of course, determine whether or not a given evidence sentence supports a given h y p ~ t h e s i s It . ~ cannot be shown, for example, that the sentence 'Jimmy Durante has a large nose' fails to confer upon the hypothesis just mentioned a greater probability than does a tautological evidence sentence. Nor can premiss 3 in the argument above be justified by means of that criterion. In both cases, the judgement of non-support has been made on other grounds, presumably on grounds of what Carnap has called inductive intuition. For the reasons here stated, I think that Skyrms's thesis is not established by his argument. It seems to me of general interest for confirmation theory that, in much the same way as in Skyrms's argument, the validity of the consequence condition for the concept of confirmatory support is presupposed also in Goodman's argument that the hypothesis 'All emeralds are grue' is not "genuinely confirmed" by its positive instances established before the critical time t. The gist of the argument is that statements to the effect that each of the emeralds examined before t has been grue (i.e. green) lend no support to the prediction, implied by the hypothesis, that all emeralds subsequently examined will be grue (and thus, blue) ([2], pp. 74-75). In fact, somewhat earlier, Goodman explicitly says: "We naturally assume further of ruling them out; indeed, they are just illustrations of what I had explicitly discussed, in [3] and [4], as the "paradoxes" of confirmation-which, I had argued, are paradoxical in appearance only. I n the context under discussion, Skyrms denies, without proof, another consequence of my definition of confirmation, namely, that a self-contradictory evidence sentence confirms every hypothesis ([4], second footnote in section 8). But he is mistaken: If E is self-contradictory, then the class of individual constants that E contains essentially (cf. 141, last footnote in section 9) is empty, and the "development" of any hypothesis N with respect to the null class is H itself ([3], p. 131, Definition 4.1). Being contradictory, E implies this development of H and thus confirms H. Moreover. on the basis of measures of logical vrobabilitv such as those develoved in Carnap's inductive logic, a universal hypothesis about a n infinite-range of individuals cannot be confirmed by any (molecular) observation sentence, for on any such evidence, the hypothesis has zero confirmation (cf. [I], pp. 570-571). T o make it possible for a hypothesis to be supported by at least some of its positive instances or by some other evidence, therefore, a different kind of probability measure would have to be utilized. The consequence condition, however, could not be re-established in that fashion since it is incompatible with the basic principles that govern all probability measures.

DISCUSSION

: CLAIM BY

SKYRMS CONCERNING LAWLIKENESS AND CONFIRMATION

277

that whatever confirms a given statement confirms also whatever follows from that statement" ([2], p. 68).5 The same assumption underlies, of course, the many arguments by other writers that are patterned on Goodman's model. Clearly, then, the consequence condition is widely taken for granted in philosophical discussions of qualitative confirmation. And the use made of it in Goodman's, Skyrms's, and many similar arguments does seem very plausible : indeed, one might say, whatever gives support or credibility to a given hypothesis surely does so also to any part of its content, to anything that the hypothesis asserts implicitly, i.e. by way of its logical consequences. This idea also appears to provide the rationale for the reliance placed on predictions derived from hypotheses that have good evidential support. Against considerations of this kind, Carnap ([I], pp. 475-476) has suggested that the concept of confirmation I had in mind in setting down the consequence condition might be the relation that E bears to H when the probability which E confers on H exceeds some fixed critical value, such as 0 or %. This construal does guarantee fulfillment of the consequence condition6; but, Carnap argues, it does not afford an adequate explication of confirmation. For even if the probability of H on E exceeds the specified critical value, it may still be less than the probability of H on the evidence available prior to the establishment of E. In this case, E has negative probabilistic relevance for H a n d should not, therefore, count as confirming H. I n effect, Carnap thus regards confirmation, properly conceived, as relative to the knowledge situation in which the relevance of E to H i s to be appraised : if the total evidence available in that situation is I, then, on his construal, a new piece of evidence, E, counts as confirming H only if y(H, I. E) > p(H, I).This is, in fact, precisely the more general condition, referred to earlier, that Carnap imposes on the qualitative concept of confirmation ([I], p. 472). And what, above, was called "Carnap's condition" follows from it for the special case where the prior evidence I is tautological. But the concept I had in mind was not intended to be relativized in Carnap's manner. Its definition was proposed as a tentative answer to the question: Given a But Goodman nowhere endorses Carnap's condition and, thus, is not making incompatible assumptions. Moreover, in a footnote to the passage just quoted, Goodman disclaims the intention to declare the consequence condition "an indispensable requirement upon a definition of confirmation," and he leaves open the matter of adjudicating conflicting requirements of this general kind ([2], p. 68). The construal is incompatible, however, with some of the other conditions that I formulated for the concept of confirmation I had in mind (cf. [4], section 8). Specifically, the (general) "consequence condition," which implies that E should confirm the conjunction of any two hypotheses it confirms individually, rules out the possibility that the critical value, say r, in Carnap's construal, might be non-zero: for the conjunction of two hypotheses each of which has a probability exceeding a given positive u may well have a probabi!ity that is less than r. But neither can v be zero; for otherwise, a hypothesis would be confirmed by E if it has a positive probability on E: and this condition can be met, for given E, by incompatible hypotheses, whereas the "consistency condition" imposed by me requires that the class of all hypotheses confirmed by a consistent E be logically consistent. This last condition, however, has been widely held to be too restrictive, and, as pointed out already in [4], section 8, it might well be weakened or abandoned. But even then, the choice of the critical value r=O seems clearly inappropriate because a consistent evidence sentence E would then confirm, among others, any non-quantified hypothesis with which it is logically consistent.

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hypothesis H a n d a body of evidence represented by a sentence E, under what condition would it be reasonable to say that E as a whole accords with, or conforms to, H, and that in this sense E confirms H? In order to specify the intended sense more precisely, I formulated some general "criteria of adequacy" for any proper explication of the idea in question: the consequence condition is one of those criteria. The conception thus suggested appears to inform many of our ideas concerning confirmation, as is illustrated by the reliance that is so extensively, if not always explicitly, placed on the consequence condition; hence, a conception of this kind may well be analytically clarifying and theoretically fruitful in its own right, in addition to Carnap's relativized construal of qualitative confirmation. In any event, however, those who, like Skyrms, adopt Carnap's condition in either its narrower or its wider form, but who also wish to make some use of the consequence condition, will have to impose suitable restrictions on the latter. In this context, it remains an important desideratum to formulate a restricted version of the consequence condition which is both logically adequate and conceptually ill~rninating.~ REFERENCES

[I] Carnap, R., Logical Fo~rndationsof Probability, The University of Chicago Press, 1950. [2] Goodman, N., Fact, Fiction and Forecast, Harvard University Press, Cambridge, Mass., 1955. Second edition, The Bobbs-Merrill Company, Inc., Indianapolis, 1965. [3] Hempel, C. G., "A Purely Syntactical Definition of Confirmation," The Journal of Symbolic Logic, vol. 8, 1943, pp. 122-143. [4] Hempel, C. G., "Studies in the Logic of Confirmation," Mind, vol. 54, 1945, pp. 1-26 and 97-121 ; reprinted with some addenda in Hempel, C. G., Aspects of Scientific Explanation, The Free Press, New York, 1965, pp. 3-51. [5] Skyrms, B., "Nomological Necessity and the Paradoxes of Confirmation," Philosophy of Science, vol. 33, 1966, pp. 230-249.

' A logically adequate, but unilluminating version is readily stated: If E confirms H, then E confirms all and only those consequences of H which satisfy Carnap's condition.