Abstract
It is easy to resolve a contradiction. All you have to do is reject or reconfigure one of the premises of the argument that leads to it. What makes paradoxes so difficult to resolve is that the assumptions that generate them are so intuitive that they resist rejection or reconfiguration. The “paradoxes of confirmation” have been especially difficult to resolve. As much is indicated by the vast literature to which they have given rise. The “raven” and “grue” paradoxes are associated with, and often thought to cause problems for, the so-called “positive instance” account of confirmation. The “old evidence” paradox arises in connection with traditional Bayesian accounts of confirmation and, in the minds of some, is a decisive objection to it. These two accounts differ in a number of important ways. What they share is the assumption that the notions of confirmation and evidence are inter-definable, an assumption so deeply embedded that it has altogether escaped notice. Our object in this chapter is to show, once again, why confirmation and evidence should be distinguished, this time because their conflation is one root of the paradoxes. The work done by many others on the paradoxes, much of it technical, has thrown a great deal of light on our inductive practices. In providing a unified, if admittedly rather general treatment of them, we hope to indicate a new direction for this work.
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Notes
- 1.
The classic account is in Carl Hempel, “Studies in the Logic of Confirmation,” in Hempel (1965).
- 2.
And thereby fails to include hypotheses that do not take this form, including the individual diagnostic hypotheses on which we have focused attention and, more generally and importantly, all statistical hypotheses.
- 3.
In Hempel’s words, these are “the conditions under which a body of evidence can be said to confirm or disconfirm a hypothesis of empirical character.” (Hempel 1965), p. 9 (our italics).
- 4.
Goodman (1983, p. 70).
- 5.
Hempel’s original discussion of the Raven Paradox in Sect. 5 of Hempel (1965) both makes explicit and depends on running confirmation and evidence together. “This implies that any non-raven represents confirming evidence for the hypothesis that all ravens are black” (p. 15, our italics).
- 6.
Like us, error-statisticians reject the idea that non-black non-ravens are evidence for the raven hypothesis, but for a very different reason: examining non-black non-ravens would not constitute a severe test of the raven hypothesis. The probability of finding non-black non-ravens on the raven hypothesis is very low. The test would have the same result if ravens came in a variety of colors. Error-statisticians would presumably also discount finding black ravens for the hypothesis that all ravens are black unless the testing procedures would be likely to turn up non-black non-ravens if the raven hypothesis were false. Among other things, this might take the form of gathering data adjusted for gender and geography. On our approach, black ravens do constitute evidence for the hypothesis that all ravens are black as against the hypothesis that they come in different colors. How strong this evidence might be is measured in terms of their respective likelihoods (which in a real-world test would include background information concerning gender and geography. See Giere (1970, p. 354).
- 7.
Royall in his commentary on the Raven Paradox (in an Appendix to his 1997) observes that how one got the white shoes is inferentially important. If you grabbed a non-raven object at random, then it does not bear on the question of whether all ravens are black. If on the other hand you grabbed a random non-black object, and it turned out to be a pair of shoes, then it provides a very tiny amount of evidence for the hypothesis that all ravens are black because the first sample estimates the proportion of non-black objects that are ravens as 0. Quantifying that evidence would be difficult without knowing how many non-black objects there are. In other words, the problem is one of confusing sample spaces. If you divide the world into two bins, ravens and non-ravens, no amount of sampling in the non-raven bin will give you any information about colors in the raven bin. However, if you divide the world into a bin of black things and a bin of non-black things, then finding a shoe in the non-black bin is evidence that all ravens are black (although fantastically weak evidence). The white shoe has increased the likelihood (albeit infinitesimally) that the proportion of items in the non-bland bin that are ravens is 0.
- 8.
(Hempel 1965, p. 18).
- 9.
It needs to be emphasized that our characterization of evidence, quite apart from the requisite distinction from confirmation that it makes possible, rules out ab initio any attempt to characterize “evidence” as such, for example in terms of the form of the sentences which represent it (as in Hempel) or its alleged incorrigibility or its “givenness” (as in Lewis), and on. Evidence has no intrinsic characteristic which identifies it, but is constituted by context-relative data whose strength is measured by the likelihood ratio (or something comparable).
- 10.
- 11.
We earlier indicated the necessity of including background information in our calculation of posterior probabilities and likelihoods, but since it had little bearing on the main theme of the monograph, have not made much of it. When one gets to the details of the applications of most of the main theories of confirmation and evidence to particular cases, background information is very important, although difficult to delimit in a principled way.
- 12.
Again see Royal (1997).
- 13.
(Goodman 1983), p. 74 (our italics).
- 14.
In his “Forward” to Goodman (1983, p. ix), Hilary Putnam simply lays it down that “in order to ‘solve’ Goodman’s problem one has therefore to provide some principle capable of selecting among inferences that do not differ in logical form, that is, on the basis of certain predicates those inferences contain.” It is revealing that Putnam himself rejects the leading solutions along these lines, is wary of Goodman’s own, and has nothing better to suggest. Our solution, in sharp contrast, is not to look at the predicates, but at the structure of the inferences.
- 15.
Achinstein’s attempted resolution of the paradox (Achinstein 2001) is both typical and instructive. His argument proceeds in three steps:
-
1.
If D is to provide evidence for H, then the posterior probability of H given D must be high;
-
2.
If the probability of D given H is high, then D must be appropriately varied;
-
3.
In the case of the grue hypothesis, the green emerald data are not appropriately varied.
Therefore, …
…There are problems with the second premise, that high posterior probability requires appropriately varied evidence. This requirement does not follow from the rules of probability theory, nor is it easy to see how variety of evidence (as we emphasized in the preceding chapter), however laudable a methodological goal, is to be included in a formal account except in terms of something like a pair-wise comparison of hypotheses. But the third premise is for present purposes more problematic. Achinstein argues for it as follows. “Grue” is a disjunctive predicate; it applies to emeralds examined before some arbitrary date if they are green and after said date if they are blue. “Appropriately varied” data would therefore include observation of emeralds both before and after. But it is a premise of the argument that “before” data only are taken into consideration. Hence they cannot constitute “evidence” in Achinstein’s sense for the hypothesis that all emeralds are grue. The same line of argument would apply to all similarly disjunctive predicates, but not to such atomic predicates as “green,” the explanation of why the latter but not the former are entrenched in our verbal and experimental practice. Goodman would undoubtedly reply that “green” can also be construed as a disjunctive predicate, grue before t, bleen after it. If we were to protest that this way of speaking does not register the fact that a change from green to blue would have to be caused, whereas a “change” from grue to bleen would require no cause, Goodman could point to Hume’s analysis, on which “causes” are no more than habits and up-date it to linguistic habits. Moreover, we follow Rosenkrantz (see Footnote 11, Chap.8) in thinking that there are no good reasons for excluding disjunctive or “bent” predicates other than to avoid the Grue Paradox, an ad hoc and unnecessarily conservative move.
-
1.
- 16.
At least at one point in his career, and as expressed in correspondence with one of the authors, Glymour drew the conclusion, as did Goodman, that confirmation is relative to a particular interpretation of the syntax in which hypotheses and data are formalized, or, equivalently, to a particular language. On his view at one time, this conclusion can be extended to any formal confirmation theory, whether it be positive instance, Bayesian, or “bootstrap.” The moral, apparently, was that the Grue Paradox cannot be used to decide between existing confirmation theories. None of them can solve it without adding certain presuppositions about meaning, a view which echoes that of Carnap.
- 17.
See Rosenkrantz (1981), Chap. 7, Sect. 1. It has been proposed several times, for example, that Newton’s law of universal gravitation be amended; the force between two objects varies inversely as the square of the distance between them, up to some distance d. But for distances greater than d, the force varies inversely as the cube of the distance between them. Such proposals to “correct” Newton’s law were made by the Royal Astronomer G.B. Airy (1801–1892) in the attempt to explain observed irregularities in the motion of the planet Uranus, and again in the 20th century by the physicist Hugo von Seeliger (1849–1924) in the attempt to make the mean density of the university everywhere constant (the inverse square law implies a concentration of matter around centers of maximum density).
- 18.
As soon as the background information that emeralds don’t undergo sudden (and otherwise uncaused) color changes is factored in, then the posterior probability of the green emerald hypothesis is much higher. This background information is easily incorporated into the relevant priors; we accord the green emerald hypothesis a higher degree of belief on our knowledge of how the world works (i.e., acceptance of the grue emerald hypothesis would very much complicate our current scientific picture). According to Goodman, on the green emerald hypothesis, too, emeralds undergo sudden (and otherwise unexplained) color changes, from “grue” to “bleen.” So we can’t allow our views about color “changes” to figure in our confirmatory accounts. But (a) the green/grue change implicates changes in many other parts of physical theory than does the grue/bleen “change,” and (b) Goodman’s own view suggests that our relative familiarity with “green,” its greater degree of “entrenchment” in our verbal and other practices, will lead people to accord the green emerald hypothesis a higher degree of initial belief. In either case, the green emerald case is more highly confirmed. But the two hypotheses have, at least before t, the same evidence class. If one assumes that a hypothesis is confirmed to the extent that there is evidence for it, then paradox ensues. To avoid the paradox, we simply drop the assumption.
- 19.
See Goodman (1983), p. 74: “Then at time t we have, for each evidence statement asserting that a given emerald is green a parallel evidence statement asserting that the emerald is grue” (our italics).
- 20.
It might be objected that our way with the paradox is ad hominem. That is, although Goodman conflates confirmation and evidence, we could restate the paradox in terms of confirmation alone: incompatible hypotheses are confirmed by the same data, observations of green emeralds, whether or not these data are said to constitute “evidence” for either one. This objection presupposes, what both we and Goodman deny, that the two hypotheses have the same priors; indeed, it goes through only if we assume that the “positive instance” account is correct. At the same time, it is hardly surprising, much less paradoxical, that incompatible hypotheses might be confirmed by the same data. This fact has long been known and gives rise to the very difficult “curve-fitting problem” at the foundations of statistical inference. The real bite comes when we begin, as with Goodman, to speak of “evidence,” for this suggests a way of distinguishing between other indistinguishable hypotheses.
- 21.
Thus green emeralds constitute evidence for the hypothesis that all emeralds are green vis-à-vis the hypothesis that all emeralds are bleen, but do not constitute evidence for the green hypothesis vis-à-vis the hypothesis that all emeralds are grue. The likelihood of green emeralds (examined before some arbitrary time t) on the green hypothesis is vastly greater than the likelihood of green emeralds on the bleen hypothesis, whereas it is the same on both green and grue hypotheses.
- 22.
That the intuition involved in our account, data count as evidence just in case they serve to distinguish hypotheses, has a long history is indicated by William Caxton, in his Deser Eng. of 1480: “He maketh no evidence for in neyther side he telleth what moveth him for to saye.”
- 23.
Although some prominent Bayesians, particularly those of an “objectivist” orientation, maintain that the air of paradox is illusory. See, for example, Roger Rosenkrantz, “Why Glymour is a Bayesian,” in Earman (1983), especially pp. 85-6. In the same volume, Daniel Garber, “Old Evidence and New,” essays a “subjectivist” attempt to disarm the problem. See Bandyopadhyay (2002) for reasons why the Bayesian account of confirmation cannot, on either of its standard variants, solve the old evidence problem.
- 24.
See Glymour (1980), Chap. III.
- 25.
The argument can be re-cast in such a way that it does not depend on Pr(D) = 1.
- 26.
Ibid., p. 86.
- 27.
Summarized in Chap. 6.
- 28.
See Brush, “Prediction and Theory Evaluation: the Case of Light Bending,” Science, 246 (1989), pp. 1124-129; Earman and Janssen, “Einstein’s Explanation of the Motion of Mercury’s Perihelion,” in Earman, et al., eds., The Attraction of Gravitation (Cambridge, MA: MIT Press 1993); Roseveare, Mercury’s Perihelion from LeVerrier to Einstein (Oxford: Oxford University Press 1982). We are especially indebted to Earman’s account of the tests of the GTR.
- 29.
For our purposes it is not necessary to decide any of the historically delicate questions concerning what Einstein knew and when he knew it; what he knew or didn’t know at the time of his discovery of GTR has nothing to do, as against Glymour’s paradox, with the evidential significance of M. In trying to determine what constitutes “new” evidence, Imre Lakatos and Eli Zahar make a rather desperate appeal to what the scientist “consciously” knew when he came up with GTR. See their (1975).
- 30.
See Taylor and Karlin (1998, p. 7).
- 31.
See Pawitan (2001, p. 427).
- 32.
We owe this formulation of the background assumptions for both GTR and Newton’s theory to John Earman in an email communication.
- 33.
See also Lange (1999) for a different approach to the “old evidence” problem..
- 34.
In the limit case, adding as many parameters as there are data points.
- 35.
Careful statisticians do post hoc data analysis all the time, but they label it as such and consider their results more as hypothesis-generating than as evidence-supporting.
- 36.
Gould and Lewontin (1979).
- 37.
This was not the end of the matter. Among the most interesting of the many papers critical of the Gould-Lewontin thesis is Dennett (1983).
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Bandyopadhyay, P.S., Brittan, G., Taper, M.L. (2016). The Paradoxes of Confirmation. In: Belief, Evidence, and Uncertainty. SpringerBriefs in Philosophy(). Springer, Cham. https://doi.org/10.1007/978-3-319-27772-1_9
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