Philosophical Studies (2007) 132:137–159 DOI 10.1007/s11098-004-0017-y

Ó Springer 2006

NATHAN STEMMER

HUME’S SOLUTION OF THE GOODMAN PARADOX AND THE RELIABILITY RIDDLE (MILL’S PROBLEM)*

ABSTRACT. Many solutions of the Goodman paradox have been proposed but so far no agreement has been reached about which is the correct solution. However, I will not contribute here to the discussion with a new solution. Rather, I will argue that a solution has been in front of us for more than two hundred years because a careful reading of Hume’s account of inductive inferences shows that, contrary to Goodman’s opinion, it embodies a correct solution of the paradox. Moreover, the account even includes a correct answer to Mill’s question of why in some cases a single instance is sufficient for a complete induction, since Hume gives a wellsupported explanation of this reliability phenomenon. The discussion also suggests that Bayesian theory by itself cannot explain this phenomenon. Finally, we will see that Hume’s explanation of the reliability phenomenon is surprisingly similar to the explanation given lately by a number of naturalistic philosophers in their discussion of the Goodman paradox.

Goodman’s paradox, or as he calls it, the new riddle of induction, is basically the problem of specifying the universal hypotheses that are ‘‘confirmed by their positive instances’’ (Goodman, 1965, p. 81). Goodman discusses this problem in the context of an analysis of confirmation and induction, and he concentrates on Hume’s description of our inductive inferences, a description that is supposed to allow us to specify the relevant hypotheses. But Goodman thinks that Hume’s description of the inferences, and indirectly of the confirmable hypotheses, is inadequate. In particular, it apparently opens the door to the paradoxical result that a particular observation may confirm hypotheses yielding conflicting predictions. Thus, let us define ‘‘grue’’ as ‘‘green and the time is before the year 3000 A.D., or blue and the time is after this year’’. Then the observation of a grue, i.e., presently green, emerald would confirm

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not only the hypothesis ‘All emeralds are grue’ but also ‘‘All emeralds are green’’. It is now more than fifty years that Goodman’s problem has been discussed, and many different solutions have been proposed. However, as can be clearly discerned from the impressive bibliography published by Stalker in his book Grue! (1994), and from further publications on this topic, no agreement has so far been reached about which is the correct solution of the problem. But I will not contribute here to the discussion with a new solution. Rather, I will argue that a solution has been in front of us for more than two hundred years – we merely have to read carefully Hume’s description of our inductive inferences. Such a reading shows that Hume’s account of these inferences does enable us to specify the hypotheses that are confirmed by their instances, and these confirmations do not generate paradoxical results. But I will also make a further claim here, namely, that Hume solved another riddle, the reliability riddle. This riddle, which so far has received little attention, challenges us with the demand to explain the remarkable reliability of the inductive inferences made by children and naı¨ ve adults. And this reliability is even more remarkable if we consider that the inferences were often based on the observation of only a few positive instances. (We will see that John Stuart Mill did bring up the issue, but was unable to explain the phenomenon.) Now, Hume not only examines the reliability phenomenon but he also explains it, and he does it with the help of a well-supported empirical hypothesis. The discussion of Hume’s conclusions also suggests that Bayesian theory by itself does not account for this phenomenon. Moreover, we will see that Hume’s explanation of the reliability phenomenon is surprisingly similar to the explanation that has lately been given to the phenomenon by a number of naturalistic philosophers (Quine, 1969, 1974, pp. 19–20; Quine and Ullian, 1970, pp. 57–58; Stemmer, 1971, 1978). Finally, I will argue that Hume’s solution of Goodman’s problem is superior to a number of well-known alternative solutions that have been proposed.

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1. THE ACQUISITION OF EXPECTATIONS

Hume treats the inductive acquisition of expectations as a psychological process that is initiated by the observation of conjoined events. These observations generate expectations about events that are similar to the observed events. For example, Hume points out that ‘‘we always presume, when we see like sensible qualities, that they have like secret powers, and expect that effects, similar to those which we have experienced, will follow from them’’ (Hume, 1748/1975, p. 33), or that ‘‘all arguments from experience are founded on the similarity which we discover among natural objects, and by which we are induced to expect effects similar to those which we have found to follow from such objects’’ (p. 36).1 Hume also gives an example that illustrates this psychological process: ‘‘When a child has felt the sensation of pain from touching the flame of a candle, he will be careful not to put his hand near any candle; but will expect a similar effect from a cause which is similar in its sensible qualities and appearance’’ (p. 39). The example illustrates how the observation of two conjoined events – the flame of a candle and the sensation of heat – generates the expectation that similar flames will be accompanied by similar effects. Hume describes the acquired expectations as being about events that are similar to the observed events. This implies that the effectiveness of Hume’s description depends essentially on a precise definition of his similarity notion. Now, Hume does not explicitly define the notion; however, it is not difficult to supply a definition that does justice to Hume’s ideas. For Hume attributes our disposition to acquire the expectations to ‘‘custom or habit’’ (p. 43), to an ‘‘instinct or mechanical tendency’’ (p. 55). Therefore, the similarity to which he is referring is the specific similarity that governs our innately determined extrapolations, the extrapolations that directly derive from our extrapolation instinct. This similarity has been investigated in experiments on stimulus generalization with naı¨ ve organisms (see, e.g., Walker, 1987, pp. 255–260; Catania, 1998, pp. 135–139), and Quine has shown that these experiments allow us

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to give a precise definition, an instrumental definition, of Hume’s similarity relation (1974, pp. 16–20; Quine calls this relation perceptual similarity). For later reference, let me point out that the similarity relation defined by the generalization experiments determines extrapolations that are intuitively valid even for us, such as from a candle flame to other candle flames, from black ravens to other black ravens, or from green emeralds to other green emeralds. On the other hand, none of the generalization experiments shows counterintuitive extrapolations as from candle flames to other non-ravens, from black ravens to other non-emeralds, or from green emeralds to grue emeralds. My psychological interpretation of Hume’s similarity notion is based primarily on his conclusion that the disposition to acquire the similarity-based expectations derives from an instinct, from a mechanical tendency. This interpretation is further supported by Hume’s frequent qualifications of the notion in psychological terms. Thus when he describes the extrapolations from certain objects to other objects he describes them as ‘‘objects which are in appearance similar’’ to the original ones (p. 34, my italics). And in the example of the candle, Hume notices that the child will expect a similar effect from a cause that ‘‘is similar in its sensible qualities and appearance’’ (p. 39, my italics). It is important to realize that because of the psychological character of Hume’s similarity notion, his similarity is a restricted one. This similarity differs radically from unrestricted similarities such as the one described later by Peirce: ‘‘When we take all the characters into account, any pair of objects resemble one another in just as many particulars as any other pair’’ (1878/1957, p. 116). Although according to Peirce’s notion a candle is just as similar to another candle as to another non-raven, Hume, by using the predicate ‘‘candle’’ for describing the child’s extrapolation class rather than predicates such as ‘‘non-raven’’, makes it clear that the objects that are included in this class are determined by the restricted similarity that expresses the child’s innate extrapolation dispositions rather than by an unrestricted similarity.

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There is a feature that has to be added to Hume’s account of the process by which similarity-based expectations are acquired. Hume attributes the acquisition of the expectations to the observation of conjoined events. But as is shown by the above-mentioned generalization experiments, not every observation of conjoined events establishes an expectation; the events must be salient for the subject. This condition is probably implicitly assumed by Hume, since the stimuli that occur in his examples are salient ones, such as candle flames or bread (p. 33). In any case, it is important to mention this condition explicitly.2 The inductive inferences that agree with Hume’s similarity relation constitute a subclass of our inductive inferences, since it contains only elementary inductive inferences, i.e., the inferences that agree with our innate extrapolation dispositions. It does not contain sophisticated theoretical inferences. It does not even contain inductive inferences regarding classes such as ‘‘mammals’’, since naı¨ ve persons do not extrapolate from, say, bats to whales. It is important to realize, however, that elementary inferences play a crucial role in our life, for they provide the foundations on which all our empirical knowledge is based. Even the most illustrious scientist acquires her first beliefs about the world with the help of such inferences. And it is the task of later generations to continue Hume’s pioneering work. And indeed, Quine has contributed to this project. In his Roots of Reference, he quotes experiments that show how certain experiences enable organisms to acquire similarity sensations that transform classes such as ‘mammals’ into extrapolation classes for the organisms (1974, pp. 20–24).3 One of the consequences of the restriction of Hume’s analysis to elementary inductive inferences is that Hume’s solution of the Goodman paradox only holds for these inferences. I will return later to this issue. Hume is well aware of the fact that posterior experiences may induce a person to change a previously acquired expectation. In particular, the observation of negative instances of an acquired expectation may eventually induce the person to ‘‘transfer all the different events, in the same proportion as they have appeared in the past’’ (p. 59). But the analysis of these

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more advanced inferences lies outside the scope of the present paper.4 2. GOODMAN’S NEW RIDDLE OF INDUCTION

Goodman formulates Hume’s notion of observing conjoined events as experience of regularities, and he summarizes Hume’s account of the inductive acquisition of expectations in these terms: ‘‘Regularities in experience . . . give rise to habits of expectation’’ (Goodman, 1965, p. 82). But Goodman thinks that Hume’s description of our inductive inferences is inadequate: ‘‘The real inadequacy of Hume’s account lay not in his descriptive approach but in the imprecision of his description . . . Hume overlooks the fact that some regularities do and some do not establish such habits’’ (p. 82). For example: All [emeralds] examined before time t are green, and this leads us to expect, and confirms the prediction, that the next one will be green. But also, all those examined are grue, and this does not lead us to expect, and does not confirm the prediction, that the next one will be grue. Regularity in greenness confirms the prediction of further cases; regularity in grueness does not. To say that valid predictions are those based on past regularities, without being able to say which regularities, is thus quite pointless (p. 82).

But our previous analysis of Hume’s account shows that Goodman’s criticism is unjustified. Hume’s description of the expectations that derive from the observation of conjoined events – in Goodman’s terms, of the hypotheses that are confirmed by their positive instances – is not imprecise. He describes them in terms of his restricted similarity notion, a notion that can be instrumentally defined with the help of the abovementioned generalization experiments. By using this notion, we obtain the result that the observation of a grue, i.e., presently green, emerald induces us to expect ‘‘All emeralds are green’’ but not ‘‘All emeralds are grue’’. In other words, according to Hume’s account, the observation of a grue emerald confirms ‘‘All emeralds are green’’ but not ‘‘All emeralds are grue’’. And this agrees with Goodman’s observation that the observation of a green, i.e., presently grue, emerald ‘‘leads us to expect, and

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confirms the prediction, that the next one will be green’’ (Goodman, 1965, p. 82). For the next emerald will be the one that is similar to the observed emerald in its ‘‘sensible qualities and appearance’’ (p. 39). Actually, Goodman expresses the problem in different terms. He distinguishes between different regularities, and he assumes that the observation of ‘‘some regularities do and some do not establish habits’’ of expectation (1965, p. 82). If we use this description of our inductive inferences, then Hume’s reply would be that the observation of every regularity establishes an expectation.5 But the expectation is not necessarily about the elements of the classes to which we might refer when describing the observed regularity. If we are dealing with normal humans, then the expectation is about the entities that are psychologically similar (in Hume’s sense) to the observed instances. Therefore, the observation of emeralds that are grue, i.e., presently green, does establish an expectation. But the expectation is not about grue emeralds. Generalization experiments suggest that it is about green emeralds, the entities that are psychologically similar to the observed instance. But I will not adopt here Goodman’s description of our inductive inferences. Rather, I will follow Hume and describe the inferences in terms of similarities. Goodman’s point will be expressed by acknowledging that there are all kinds of similarities including unrestricted Peircean similarities and each of them may determine a subject’s extrapolation class from the observation of conjoined events. And if a subject’s inductive inferences are determined by unrestricted similarities (e.g., the subject is a Martian), then it is possible that the observation of a green emerald will indeed generate the expectation that all emeralds are grue because, paraphrasing Peirce, if we take all the characters into account, two green emeralds resemble one another in just as many particulars as a green and a grue emerald. But by restricting the similarities that stand behind our intuitive inductive inferences to Hume’s psychological similarity, this counterintuitive result is avoided. Frequently, Goodman’s new riddle is expressed in terms of the projectibility of the predicates that occur in the relevant

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hypotheses. The above conclusions can also be expressed in this terminology. Thus if the observation of the conjoined stimuli (or events) S1 and S2 generates the expectation ‘‘All F are G’’ in naı¨ ve persons – that is, if the extensions of the predicates ‘‘F’’ and ‘‘G’’ contain the stimuli that are psychologically similar to S1 and S2 – then I will say that ‘‘F’’ and ‘‘G’’ are subjectively projectible relative to S1 and S2. To be sure, the examples that are usually discussed deal with predicates of natural languages such as ‘‘raven’’ or ‘‘green’’, and it is doubtful whether the extensions of these predicates match exactly the persons’ extrapolation classes from the observation of the relevant stimuli. But since such linguistic aspects have been examined by Quine in several publications (see, e.g., 1960, 1969, 1975), I will not enter into these issues. Sometimes, it is preferable to apply projectibility directly to the extensions of predicates, i.e., to the corresponding classes. In these cases, if generalization experiments show that F is the extrapolation class from stimulus S for naive persons then I will say that, relative to stimulus S, class F is subjectively projectible. We notice that classes such as ‘‘raven’’ or ‘‘green’’ are subjectively projectible (relative to typical instances) while classes such as ‘‘grue’’ or ‘‘mammal’’ are not. In other words, the classes that are subjectively projectible (relative to typical instances), or the predicates that denote these classes, satisfy Goodman’s requirement for projectibility. In particular, they do not yield conflicting predictions. The subjectively projectible classes (or predicates) reflect the extrapolations of naı¨ ve persons; they derive from our innate similarity sensations. As mentioned earlier, people may acquire similarity sensations that transform classes such as ‘‘mammals’’ into extrapolation classes for them. Such ‘‘acquired’’ extrapolation classes could be said to be subjectively projectible for these persons. But in order to avoid too many qualifications, I have preferred to restrict the notion of subjectively projectible to the classes (or predicates) that reflect our innate extrapolation dispositions. The projectibility of a class (or predicate) is a relative notion, since a class is projectible relative to a stimulus S; it is the

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extrapolation class from the stimulus (see also Stemmer, 1975). It seems that the relative status of projectibility is generally ignored because it is tacitly assumed that the stimulus that generates an extrapolation class is a typical element of the class described by the predicate. Thus, when Goodman claims that all emeralds ‘‘examined before time t are green, and this leads us to expect . . . that the next one will be green’’ (1965, p. 82), he is probably assuming that the examined emeralds are typically green. And the word ‘‘green’’ is supposed to apply to all the emeralds that are involved in the extrapolation process. But for the sake of simplicity, I will generally ignore here the relative character of projectibility. Notice, however, that the relative status of projectibility is automatically taken care of by Hume since he expresses our extrapolations in terms of similarities, and the notion of similarity is a relative notion. An extrapolation class contains the entities that are similar to the original entities; it is an extrapolation class relative to these entities. (Extrapolation classes have a further feature that introduces complications; they have vague boundaries. But for the sake of simplicity, I will ignore here this feature. It is discussed in Stemmer, 1981.) We have seen that Hume’s reply to Goodman’s request is descriptive; he describes the inductive inferences that are intuitively valid. The reply has no normative component; it does not justify these inferences. And the recent naturalistic solutions to Goodman’s problem that were mentioned above also have this feature; they merely describe the intuitively valid inferences. But from a naturalistic point of view, the lack of a normative component is no shortcoming and Goodman himself seems to agree. Goodman thinks that an inductive inference is justified if it conforms ‘‘to accepted inductive practice’’ (1965, p. 64). He therefore looks only for an adequate description of our inductive inferences, and he criticizes Hume not because his account is descriptive but because he thinks that the description is inadequate: ‘‘The real inadequacy of Hume’s account lay not in his descriptive approach but in the imprecision of his description’’ (1965, p. 82). This suggests that Goodman also agrees that an adequate

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description of our intuitive inductive inferences does not have to include a normative component.6 And as I have tried to show so far, Hume indeed gives us an adequate description of these inferences. 3. THE RELIABILITY OF OUR INDUCTIVE INFERENCES AND MILL’S PROBLEM

But Hume not only gives us an adequate description of our elementary inductive inferences; he takes a further step. He examines a surprising feature of these inferences, namely, their remarkable reliability. For the inferences are not only intuitively valid but many of them have also been highly reliable. This has been the case even if, as in the case of the candle flame, the expectations were acquired by young children, i.e., by people who did not consult background knowledge for forming the expectations, and even if they were based on the observation of very few positive instances. Moreover, Hume not only calls our attention to the reliability of the inferences; he also gives a scientifically valid explanation of the phenomenon. To this effect, he first notices, as we have seen already, that the extrapolations derive from an instinct, an instinct that can even be detected in animals, as when a ‘‘creature expects from the present object the same consequences which it has always found in its observation to result from similar objects’’ (p. 106). In a second step, Hume notices the general usefulness of instinctive behavior, since it enables us, and other organisms, to survive. He remarks, for example, that it is an instinct ‘‘which teaches a bird, with such exactness, the art of incubation and the whole economy and order of its nursery’’ (p. 108). Finally, Hume conjectures that just as other innate dispositions, the extrapolation instinct, too, has ‘survival value’; it generates inferences that are [almost] ‘‘infallible’’ (p. 55). More specifically, Hume proposes an empirical hypothesis which accounts for the reliability of the inferences that agree with our similarity sensations: As nature has taught us the use of our limbs without giving us the knowledge of the muscles and nerves by which they are actuated, so has she implanted in

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us an instinct which carries forward the thought [the operation of the mind by which we infer like effects from like causes] in a correspondent course to that which she has established among external objects (p. 55).

Hume attributes here to us a disposition, implanted in us by nature, which enables us to extrapolate in such a manner that it corresponds to what occurs in the external world or, as he states a couple of lines earlier, he conjectures that ‘‘the ordinary wisdom of nature’’ has created ‘‘a kind of pre-established harmony between the course of nature and the succession of our ideas . . . the operation of the mind by which we infer like effects from like causes’’ (pp. 54–55). And this hypothesis is supported not only by the fact that many of our elementary inductive inferences have indeed been highly reliable but also, and independently, by the general usefulness of innate dispositions as manifested by ‘‘the [successful] use of our limbs’’ (p. 55). Hume’s explanation of the reliability of our elementary inductive inferences has been largely ignored. But lately the above mentioned naturalistic philosophers have given a similar explanation of this reliability; however, since they were already acquainted with Darwin’s theory, they attributed the reliability to natural selection. This evolutionary process has favored extrapolation instincts that were useful for us, that agreed with the course of nature (see, e.g., Quine, 1969, 1974, pp. 19–20; Quine and Ullian, 1970, p. 58; Stemmer, 1971, 1978). But Hume’s explanation was probably the best that could have been formulated in his time. It is interesting to note that in one of his latest publications, Quine, too, appeals to a pre-established harmony in order to explain the reliability of elementary inductive inferences: ‘‘What we have is . . . a pre-established harmony between standards of perceptual similarity and the environment’’ (Quine 1996, pp. 160–161). This harmony is accounted for by natural selection because: Successful expectation has always had survival value, notably in the elusion of predators and the capture of prey. Natural selection has accordingly favored innate standards of perceptual similarity which have tended to harmonize with trends in the environment (p. 161).

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Since Quine’s ‘trends in the environment’ expresses the same as Hume’s ‘course of nature’, Quine’s harmony hypothesis differs very little from Hume’s harmony hypothesis according to which the wisdom of nature has created a kind of pre-established harmony between the course of nature and the succession of our ideas.7 The reliability phenomenon also called Mill’s attention, and in a well-known passage he asks: Why is a single instance, in some cases, sufficient for a complete induction, while in others, myriads of concurring instances, without a single exception known or presumed, go such a very little way towards establishing a universal proposition? (1895, Book III, Chapter III, Sec. 3)

Mill did not find an answer to his question and he continues: ‘‘Whoever can answer this question . . . has solved the problem of induction.’’ Now, Hume did not solve the problem of induction: the justification of induction. However, we have just now seen that Hume did solve another problem, perhaps not as important as the traditional problem, but still important, namely, the reliability problem or, to coin a new term, Mill’s problem. He gives a well-supported answer to Mill’s question. By acknowledging the usefulness of our instincts in general, and of the extrapolation instinct in particular, Hume is able to explain the high degree of reliability of the hypotheses that agreed with our extrapolation instinct even if they were based on the observation of just a few instances. Let me close this section with a brief comment on the role of background knowledge in inductive inferences. The role of this knowledge has often been stressed in connection with these inferences. For example, in an analysis of the intuitive validity of inductive inferences, Hempel speaks of the ‘‘additional knowledge that we happen to have at our disposal’’ which may affect our judgment of the relation between a given evidence and a corresponding hypothesis (1965, p. 19). And background knowledge has recently been adduced by Okasha in a criticism of Hume’s account of induction. Okasha thinks that ‘‘Hume’s description of our inductive practices was badly deficient’’ because it ignores the fact that ‘‘expectation-formation takes

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place in the light of a vast store of background information’’ (2001, p. 309). Now, background beliefs have indeed an important function in sophisticated inductive inferences. But there is no doubt that they play only a minor role in the acquisition of the elementary expectations of children. And Okasha’s description of a Bayesian agent who ‘‘begins with an allocation of subjective probabilities to a set of propositions’’ (p. 316) on the basis of background beliefs clearly does not apply to the child who has acquired the expectation ‘‘All candle flames are hot’’. Psychological theory does not support the assumption that a child’s early acquisition of expectations is preceded by an allocation of subjective probabilities to a set of propositions on the basis of background information. (Notice that usually there are an infinite number of Goodmanian descriptions of a person’s past experiences, each of which may generate a different background proposition. Thus in Hume’s example, the child’s background propositions may be based on experiences such as ‘‘The milk I drank yesterday was hocold’’, where ‘‘hocold’’ means hot today or cold next week, or cold next year, or cold in 3000 A.D., etc.) Moreover, and this is crucial, and is the reason for introducing the topic here, not even the best specialist in Bayesian theory is able to assign on the basis of this theory itself a degree of probability to the hypothesis ‘‘All candle flames are hot’’ that does justice to its objective probability, a probability judgment that the child obtains by merely ‘‘consulting’’ her well-adapted extrapolation dispositions. To be sure, Bayesians can assign a high prior to the hypothesis ‘‘All candle flames are hot’’. But then they can also assign a high prior to the hypothesis ‘‘All candle flames are hocold’’. Bayesians need an additional instrument, an instrument that enables them to assign high priors to certain propositions rather than to others. And Hume’s psychological theory together with his harmony hypothesis may give them such an instrument. 4. THE JUSTIFICATION OF INDUCTIVE INFERENCES

Hume’s account of our inductive inferences in terms of his similarity notion enables us to specify a basic set of classes that

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are subjectively projectible. And we have just now seen that by acknowledging the usefulness of instincts in general and of our extrapolation instinct in particular, Hume is able to explain why these classes also had a high degree of objective projectibility; they enabled us to acquire expectations that were frequently highly reliable in the real world. It is important to realize, however, that the usefulness of instincts can only be asserted for the past. Instincts have indeed been useful, but we do not know whether they will continue to be so, in particular because, as Hume remarks, ‘‘the course of nature may change’’ (p. 35). And this is the reason why Hume’s reliability conclusion must be expressed in the past tense. The harmony hypothesis only implies that the classes that are subjectively projectible had a high degree of objective projectibility. The restriction to the past of Hume’s reliability conclusions has a curious effect on predicate pairs such as ‘‘green’’ and ‘‘grue’’. Since ‘‘green’’ is subjectively projectible, Hume’s harmony hypothesis suggests that the predicate has been objectively projectible. But the extension of ‘‘green’’ has so far been identical with the extension of ‘‘grue’’. It follows that although with respect to subjective projectibility there is a clear difference between these predicates, they do not differ with respect to past objective projectibility. But this result does not affect Hume’s solution of the Goodman paradox. The subjective aspect of projectibility continues to be covered by Hume’s restricted similarity notion, which determines the nature of our inductive inferences about similar events, including events that will occur in the future. On the other hand, Hume’s conclusion that the course of nature may change suggests that he is not committed to the future validity of his harmony hypothesis, and this suggests that he is not committed to the objective validity of predictions about future events including of conflicting predictions about the nature of emeralds. Consequently, the fact that the extension of ‘‘green’’ has so far been identical with the extension of ‘‘grue’’ does not commit Hume to conflicting predictions. The limitation to the past of Hume’s reliability conclusions also bears on the justification of the inductive inferences of

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hypotheses of the form ‘‘All F are G’’. Since such hypotheses are usually supposed to also hold for the future, Hume’s harmony hypothesis cannot tell us anything about their reliability even if we have observed positive instances of the hypotheses and even if the predicates ‘‘F’’ and ‘‘G’’ are subjectively projectible. In other words, Hume’s harmony hypothesis does not justify these inferences. If we wish to justify them we must introduce an appropriate futurity assumption – for example, the assumption that the course of nature will continue to agree with our extrapolation instincts. But although such an assumption may reflect our intuitions regarding the future, it is not supported by empirical evidence. (On this subject, see also Stemmer, 1978; Konyndyk, 1980.) Clearly, the evolutionary conclusions regarding the reliability of our innately based inductive inferences are also restricted to the past. Although evolutionary theory tells us that instinctive behavior usually had survival value, we do not know whether it will continue to have such value. Consequently, the above conclusions about the status of ‘‘grue’’ and about the reliability of hypotheses of the form ‘‘All F are G’’ also hold for the evolutionary solutions of the Goodman paradox. They are limited to the past. This result is expressed in Goodman’s criticism of the evolutionary solution of the paradox given by Quine and Ullian (1970): . . .no appeal to survival of the fittest will explain why H [‘All emeralds are green’] but not K [‘All emeralds are grue’] is projected or projectible. They have been, and until 2000 AD will be, equally useful for survival (Goodman, 1972, p. 358).

But this criticism affects only the objective interpretation of projectibility and then only the interpretation that applies the notion to the future. It affects neither the subjective interpretation of the notion nor the objective interpretation that is limited to the past. To be sure, Quine and Ullian do not discuss explicitly the three different aspects of projectibility, and this may have prompted Goodman’s claim. But the distinction is implied by their analysis. Still, I will not deal here with this analysis. What is important for us is that Hume clearly separates

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the three aspects. Subjective projectibility is expressed by our similarity-based expectations about objects and events including about their future nature. Objective projectibility can be asserted for the past, because the general usefulness of our instincts has determined ‘‘a kind of pre-established harmony between the course of nature and . . . the operation of the mind by which we infer like effects from like causes’’ (pp. 54–55). As to future objective projectibility, nothing can be said even if the expectations are derived from useful extrapolation instincts because ‘‘the course of nature may change’’ (p. 35; see also Stemmer, 1988). 5. OTHER SOLUTIONS TO THE GOODMAN PARADOX

Goodman’s new riddle is thus the problem of describing the hypotheses that are confirmed by their positive instances or, as is usually expressed, the problem of characterizing the predicates or classes that are projectible. We have seen that Hume’s account of our inductive inferences in terms of his similarity notion enables us to specify a basic set of classes that are subjectively projectible (relative to certain entities). Moreover, by acknowledging the usefulness of instincts in general and of our extrapolation instinct in particular, Hume is able to explain why these classes also had a high degree of objective projectibility. Several other criteria have been proposed for selecting projectible predicates (or classes). For example, some authors think that projectible predicates (or their extensions) are sortals (Ackerman, 1969), qualitative (Carnap, 1947), simple (Harman, 1994), express subjective experiencing (Hetherington, 2001), non-temporal (Lange, 1994), or non-disjunctive (Sanford, 1994). But these proposals and several others have at least one of the following shortcomings: (a) It is very difficult, often even impossible, to give a precise characterization of the criteria. Goodman, for example, states, ‘‘I simply do not know how to tell whether a predicate is qualitative. . .except perhaps by completely begging the question at issue’’ (1965, p. 79).

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(b) There is no independent explanation of why many of the hypotheses in which the relevant predicates appeared, and that were based on the observation of only a few positive instances, were often highly successful in our world. Why should the use of, say, qualitative predicates have been more appropriate for arriving at successful inductive inferences than the use of nonqualitative predicates? Hume’s naturalistic solution of Goodman’s problem includes both aspects; a clear characterization, that can be improved with the help of generalization experiments, of a set of classes that are subjectively projectible (relative to typical instances), and a plausible explanation, that can be improved by adding evolutionary conclusions, of the past objective projectibility of these classes. It therefore seems reasonable to conclude that Hume’s solution is better than the above alternatives. Goodman selects projectible classes on the basis of the history of successful projections of the corresponding predicates. This proposal seems to account for the objective success of the projections, although in a somewhat circular way. There is no independent explanation of the success as in Hume’s solution, which is independently supported by the general usefulness of innate dispositions. But, more critically, Goodman does not explain the high degree of success of the elementary inductive inferences made by children and naı¨ ve adults. We can hardly attribute this success to a record carried out by these persons of past successful projections, which would ‘entrench’ the corresponding predicates. In particular, since Goodman thinks that ‘‘entrenchment derives from the use of language’’ (1965, p. 95), he cannot account for the success of the inductive inferences of preverbal children. It appears, therefore, that Hume’s solution is also better than Goodman’s linguistic entrenchment solution. I noticed above that Hume’s harmony hypothesis accounts only for the past objective projectibility of the extrapolation classes that are subjectively projectible. It is possible that some of the scholars who have proposed the alternative solutions intended the selected predicates or their extensions to be objectively projectible also in the future. But this does not point to an advantage of these solutions since there is no doubt that

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any claim about future objective projectibility requires an appropriate futurity assumption. For example, even if one could show that the classes denoted by non-disjunctive or by well-entrenched predicates have been objectively projectible, we have no reason to conclude that the classes will continue to be objectively projectible unless we introduce an appropriate futurity assumption. But if we already introduce a futurity assumption, then Hume’s solution also gives us future objective projectibility. Hume’s solution of the Goodman paradox has other limitations, since it only specifies a limited set of classes (or predicates) that are subjectively and objectively projectible, namely, those that agree with our innate similarity sensations. With respect to subjective projectibility, further research may allow us to add additional classes. And as mentioned above, Quine has described the experiences that transform classes such as ‘‘mammals’’ into extrapolation classes for the relevant persons. But this is basically a psychological enterprise, although it is doubtful whether there are general psychological laws that account for the subjective projectibility of theoretical predicates such as ‘‘gravitational force’’ or ‘‘phlogiston’’. Moreover, it is unlikely that the alternative solutions to the Goodman paradox – e.g., qualitative, non-disjunctive, entrenched, etc. – can give us a method for determining whether or not such theoretical predicates are subjectively projectible. Note in particular the linguistic entrenchment of ‘‘phlogiston’’. This suggests that neither Hume’s solution nor the alternative solutions can give us a general method for selecting the theoretical terms that are subjectively projectible, and this implies that with respect to these terms the alternative solutions have no advantage over Hume’s solution. As to objective projectibility, several scholars have identified factors that have lead to highly reliable theories, such as varied evidence, coherence with other theories, simplicity, or refutability (see, e.g., Quine and Ullian, 1970, pp. 42–53). And further studies in the Philosophy of Science may allow us to increase the precision of these factors and perhaps add others. But the study of these factors has no direct connection with the

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Goodman paradox. Moreover, the reliability conveyed to the theories by the factors is not independently supported as in the case of the elementary inductive inferences, where the usefulness of innate dispositions gives such an explanation. I will therefore not pursue this topic here. There is, however, one aspect of theoretical reasoning that is related to Hume’s conclusions because, at least in certain initial stages of theoretical reasoning, the usefulness of our innate extrapolation dispositions continues to play a crucial role in ensuring the reliability of the inferred hypotheses. Consider classes such as ‘‘mammals’’, ‘‘vertebrates’’, or ‘‘metals’’. These classes are not innate extrapolation classes, i.e., classes that directly derive from our extrapolation instinct. However, as said earlier, certain experiences may transform such classes into acquired extrapolation classes for the subjects who undergo the experiences. But although the psychological processes that stand behind these transformations derive from innate learning mechanisms, the end results depend on a person’s idiosyncratic experiences. It is therefore unlikely that we can attribute the reliability of hypotheses about such classes – e.g., the reliability of the hypothesis ‘‘All mammals have a heart’’ which we have derived from the observation of a limited number of positive instances – to the general usefulness of innate dispositions. Nevertheless, an important part of the reliability of such hypotheses can indeed be attributed to this usefulness. For many of these acquired extrapolation classes are unions of classes that are identical with, or very close to, innate extrapolation classes (relative to certain typical entities). For example, the class of mammals is the union of the innate extrapolation classes of giraffes, whales, bats, etc. Now, according to Hume’s harmony conjecture (or for us, according to evolutionary theory), innate extrapolation classes have been objectively projectible. Therefore, hypotheses about these classes – e.g., ‘‘All giraffes have a heart’’, ‘‘All whales have a heart’’, or ‘‘All bats have a heart’’ – were frequently highly reliable even if they were based on the observation of a limited number of positive instances. And it is likely that an important part of the reliability of ‘‘All mammals have a heart’’ derives from the

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reliability of the underlying hypotheses. After all, we have not examined all individual giraffes, whales, and bats regarding their being chordate. It is in this sense that the usefulness of certain low-level theoretical terms, and perhaps even of some higher-level terms, is partially determined by the usefulness of our innate extrapolation dispositions.

6. CONCLUSIONS

Hume’s account of our inductive inferences shows his extraordinary genius. It is a sophisticated, insightful, and, what is most important, a scientifically correct account. He treats the process by which we inductively infer universal hypotheses as a psychological process that is initiated by the observation of (salient) conjoined events, and these observations then generate expectations about entities that are psychologically similar to the observed events. By describing the inductive inferences in terms of his similarity relation, a relation that can be instrumentally defined, Hume’s description of the inferences satisfies Goodman’s demand of a clear description of the hypotheses that are confirmed by their positive instances. Hume’s most famous conclusion about inductive inferences is of course the conclusion that the inferences cannot be justified. But since this paper is not concerned with this aspect, I have not delved into it. However, there is one aspect that is related to this topic, namely, the reliability phenomenon. Even though inductive inferences cannot be justified, the elementary inferences, those that directly derive from our innate extrapolation dispositions, were frequently highly reliable even if they were based on the observation of only a few instances. And Hume explains this reliability with the help of his independently supported harmony hypothesis. In other words, Hume does not solve the problem of induction, but he does answer Mill’s question. Reasons have been given that suggest that Hume’s solution to the Goodman paradox is superior to those given by a number of alternative solutions – in particular, because the

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description given by the alternative solutions to the projectible predicates is frequently imprecise or because they do not explain why the hypotheses in which the predicates occur have been highly reliable even if they were based on the observation of a few positive instances and even if they were inferred by people who did not consult relevant background information. A central feature of Hume’s approach is the distinction between the subjective and the objective aspects of induction. We have seen that these are different issues, and they require different treatments. It is likely that the reason why so far no agreed upon solution to the Goodman paradox has been found is that, except for the naturalistic solutions mentioned above, the other solutions have not systematically distinguished between these two aspects. NOTES * I am indebted to Max Hocutt, to Joseph Ullian, and especially to an anonymous referee of this Journal for many helpful comments on earlier versions of this paper. Correspondence about this paper should be sent to [email protected]. 1 All page references not otherwise attributed are to Hume (1748/1975). 2 On the role of salience in the acquisition of similarity-based expectations and on the dimensions of our innate standards of salience, see Quine (1974, pp. 24–27). 3 It is with the help of these acquired similarities that children learn the correct use of words such as ‘‘toy’’ or ‘‘furniture’’ which refer to objects that are not innately similar. 4 Still, let me call attention to the results of experiments on discrimination which show that if a person observes a negative instance of the expectation ‘‘All F are G’’ which she previously acquired upon observing an F that is G, and if the person noticed a discriminative feature D that was possessed by the positive instance but is absent from the negative instance, then the observation of the negative instance will frequently induce the person to ‘‘replace’’ the original expectation by a restricted expectation ‘‘All FD are G’’ where FD contains those elements of F that also have feature D (see, e.g., Walker, 1987, pp. 252–301; Catania, 1998, pp. 128–147; see also the section on restricted predicates in Stemmer, 1971, 1981). 5 Of course, certain additional conditions have to be fulfilled such as the above mentioned conditions of salience. But this is not Goodman’s point. 6 See also Quine’s account of naturalistic positions:

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I see philosophy not an as an a priori propaedeutic or groundwork with science, but as continuous with science . . .There is no external vantage point, no first philosophy. All scientific findings, all scientific conjectures that are at present plausible, are therefore as welcome for use in philosophy as elsewhere (Quine, 1969, pp. 126–127). 7 The relevance of Hume’s harmony hypothesis for solving the reliability riddle is discussed in Stemmer (1983, p. 64).

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