CHAPTER IV

Induction and the World

§1. Reasonable Practice. The inclusion of natures in the required ontology is based on the premiss that there are physical necessities. Having just elaborated on the notions of necessity and nature, we must now establish that premiss. This will be done by showing that ontologies excluding necessity come into conflict with the requirements of the practice of action based on prior sense experience.

I shall say of a practice whose requirements are satisfied by a given ontology that it is a “reasonable” practice relative to that ontology. Suppose a practice is not reasonable relative to two different ontologies. If one ontology can be extended so that the practice becomes reasonable relative to it, then it is merely insufficient relative to the practice. If the other ontology is incompatible with such an extension, then it and the practice, taken together, are “incoherent.” But to avoid incoherence, does one adjust practice to ontology or ontology to practice? The practice of action based on prior experience is fundamental in that without such a practice there would be no human practice at all. And an ontology is a theory that, like other theories, has practice as its context. Thus whenever an ontology is incoherent with the practice of action based on prior experience, one must adjust the ontology to the practice. There can be no harmonization of theory with practice that requires abandoning the practice of action on prior experience. I shall try to show that, as a consequence, any harmonization of ontology with practice requires an ontology that does not exclude necessities, as an ontology of simples does. This can be shown by demonstrating that the practice of action on prior experience is reasonable only in respect to an ontology that supports necessities.

The argument for an ontology with natures will then be a “presuppositional” argument. That is, natures are introduced as presuppositions of the practice of action based on prior experience. To reject natures means that the criteria for warranted action on prior experience cannot be fulfilled. But presuppositional arguments are often notoriously unsatisfying. For example, why not respond to this one by criticizing the practice of action on prior experience? If it has such repugnant presuppositions, perhaps it is only a bad habit. So to have force this argument must be accompanied by a “vindication” of the practice itself. It was vindicated by showing that it is an indispensable means to a clearly desirable end.¹ For unless there is a practice of action on prior experience, people will not communicate, make judgments, order their behavior, or create works of art. In short, there would be no human practice. Without vindication of this practice, its ontological presuppositions would remain hypothetical; without the presuppositions, the vindication would carry no ontological message. The two must be combined. One can then assert categorically that any satisfactory ontology contains all that the existence of necessary connections requires.

But the combination is not a “justification” of the practice. That is, it in no way shows that this practice will continue to be a reliable guide to the future on the basis of general principles about the world that are themselves justified without reliance on induetion. We do arrive, of course, at the general principle that there are necessities. By itself, though, this principle is not sufficient to guarantee continued success with the practice. If it were sufficient then a genuine justification would have been effected. For since the vindication makes the practice of action on prior experience essential to human practice generally, there is no way within a coherent whole of theory and practice to cast doubt on a principle about the world presupposed by this practice. Thus the familiar objection to justifications–that the general principles appealed to need justifying themselves–would be circumvented. Let us now consider that aspect of the practice of action on prior experience which leads in the direction of necessity.

Not every action based on prior experience is itself an act of judging some proposition true. Even if–as seems unlikely–every such action involved a judgment as an analytical component, it would not itself have to be a judgment. Let us agree to the following rather limited sense of induction. The conclusions of induetions are to be judgments made on the basis of prior sense experience. In this sense, not all actions made on the basis of prior sense experience are arrived at inductively. Nonetheless, I shall treat of inductive judgments in order to draw certain conclusions about the practice of action on prior experience as a whole. This procedure will be legitimate since to every action based on prior experience that is not a judgment one could correlate a judgment as to that action’s success. For such an action to be warranted as a step to a given goal, there must be support for the corresponding possible judgment. So whatever is needed for support of inductive judgments is needed for giving a warrant to a corresponding action based on prior experience that is not a judgment.

In the methodology of science, one cannot neglect inferences to structures that are not similar to objects of prior experiences. For example, one cannot neglect an inference from constant ratios of the weights of combining chemicals to the atomic structure of matter. Yet these inferences that C. S. Peirce called “abductive” are not, in our sense, actions on prior experiences, for actions on prior experiences concern possible features of new situations that would be similar to experienced features of earlier situations. Nonetheless, abductive inferential practice, like all human practices, exists only in a context of action on prior experience. It would then be incoherent with an ontology that rejects the presuppositions of induction. The reason for this is that abductive inferences are made only when the ground has been prepared by the acceptance of regularities on the basis of prior experience.

The abductive inference to the atomic theory, from the experience of constant combining ratios in some cases, requires the belief, which we can assume has the same empirical basis, that, in general, chemicals can be expected to combine in constant ratios. Otherwise, the acceptance of the atomic structure of unobserved chemicals would here lack an empirical basis. The atomic structure of matter is hypothesized precisely because it accounts for constant combining ratios. To hypothesize it where there is no belief in such behavior is to reason in the absence of any posited analogy with observed cases. Thus the premiss of the abductive inference would be irrelevent to it, and it would not be a legitimate inference at all.

There are many possible inductive rules that might be used in judging propositions to be true on the basis of empirical data. Which of these rules is involved when one asks whether an ontology without a ground of necessity is coherent with inductive practice? One rule would tell us that if of the observed F are G, then if any hypothesis is to be accepted as true, the hypothesis that an F has an chance of being a G is to be accepted. But another rule, that appears counterintuitive, would favor, on the same observations, the hypothesis that an F has a chance of being a G. To accept the favored hypothesis in either case would be to judge it true on the basis of prior experience. Suppose I construct an argument for necessity on the assumption that the first rule is the one used. Let me call it the “uniformity” rule, and the second one the “counterintuitive” rule. But when I showed that the practice of acting on prior experience was a fundamental practice, there was no reference to one inductive rule rather than another. So consequences that follow from limiting the practice to one involving the uniformity rule need not be consequences that are required for any human practice. In particular, the consequence that there are necessities would become a consequence of limited significance. To preserve its general significance, one would have to show that a practice involving any rule other than the uniformity rule would be incoherent. But no one has succeeded in showing this.²

This is not a problem that faces us. I am concerned with the general fact that, whatever the rules of induction may be, the conelusions of legitimate inductions are supported, to some degree, by their premisses. There are important common features of the notion of support, features not tied to specific rules for saying what supports what and to what degree. One of these common features is that if there is support for a proposition about the unobserved, then there must be real necessities.

Let us now return to the above idea of a reasonable practice. It will be interesting to contrast it with the ideas of reasonable practice inherent in pragmatic and conventionalist treatments of induction.

For the pragmatist, induction by a certain rule is reasonable on the following condition: successful predictions can be made by that rule, if they can be made at all by any rule.³ Suppose they can be made by the counterintuitive rule. Then the uniformity rule can make at least one successful prediction. It can be used to predict– at a second level–that the counterintuitive rule makes correct predictions at the first level. But being reasonable in this pragmatic sense does not mean being reasonable relative to any ontology. Pragmatic reasonableness fails to carry over to ontological reasonableness. A pragmatically reasonable practice will, nonetheless, be incoherent with a Humean ontology, and hence not be reasonable relative to it. The practice requires necessities whereas the ontology rejects them.

Perhaps, though, pragmatic reasonableness is sufficient without bothering about reasonableness relative to an ontology. This would be true if one could infer from pragmatic reasonableness to genuine support. But in the context of an ontology that excludes necessities, there is no support, and thus the inference is blocked. By the pragmatic argument, inductions by a certain rule are reasonable when successful predictions can be made by that rule if by any rule. The pragmatist wants to be able to make the next step and say that pragmatically reasonable inductive conclusions are genuinely supported ones, without appeal to any ontological principles. But, as we shall show, support for hypotheses and an ontology that excludes necessities form an incoherent pair. Thus the step from pragmatic reasonableness to support fails. Pragmatic reasonableness is concerned exclusively with a connection among inductive rules. It implies nothing about the world, though support does. Thus, the inadequacy of a Humean ontology is only temporarily concealed from view by the statement that a certain inductive practice is pragmatically reasonable irrespective of what ontology one adopts. If one's ontology precludes necessities, one cannot coherently claim support for hypotheses, however pragmatically reasonable their induction may have been.

Goodman has given the most elegant statement of the conventionalist argument.⁴ He feels that standard inductive practice is a sufficient basis, just by its being standard, for its reasonableness. In view of this, we can “stop plaguing ourselves with certain spurious questions about induction.” But, once again, conventional reasonableness fails to carry over to ontological reasonableness. A conventionally reasonable practice that assigns support to hypotheses is still incoherent with a Humean ontology. This is not to say that the tribe whose conventions are the source of the conventional reasonableness must have ontological convictions. It is only to say that the philosopher, who wil! have ontological convictions, is not left free, despite the soothing association of convention and reasonableness, to choose a non-necessitarian ontology.

Moreover, for Goodman, a crucial factor in determining whether hypotheses are supportable by data in standard inductive practice is the “entrenchment” of their predicates in the language. In effect, hypotheses can be inductively supported if their predicates have occurred in hypotheses that people have in fact inductively inferred. This presumably takes the matter of support out of the hands of metaphysicians. But if one’s ontology fails to contain a basis for necessities, then, it turns out, the connection needed between data and an hypothesis, if the former is to support the latter, is broken. Relative to such an ontology, the conventional reasonableness of assigning support in a given way is just a tribal illusion.

§2. Inductive Support. It is important to distinguish acceptance from support. If from a given set of data I infer inductively a given hypothesis, then I accept that hypothesis as true on the basis of those data. But the acceptance will not be warranted unless the hypothesis is supported by the data. The support of the hypothesis is not an act, though acceptance is. Moreover, the degree to which data may support an hypothesis may be too small to warrant its acceptance.

In this discussion, I am beginning with the practice of induetive acceptance of propositions and moving toward the notion of support. That is, I shall try to claim that warranted acceptance on the basis of certain data implies support from those data. I am not, conversely, starting from support and moving toward induetive practice. That is, I shall not try to claim that acceptance is automatically warranted when a certain high degree of support is reached.

Fortunately, there is no need to make this claim, for it has paradoxical results. In a large, fair lottery there is little support for the proposition about any given ticket that it will win. Conversely, there is a high degree of support for the proposition about any given ticket that it will not win. So, on the view that a high degree of support suffices for acceptance, one would accept the proposition about any given ticket that it will not win, assuming the lottery is large enough. One would likewise accept a similar proposition about every other ticket. So one ends up accepting the proposition that no ticket will win, which is absurd. A high degree of support then, is not a sufficient condition for acceptance.

I shall make the assumption, simply for purposes of exposition, that support can be treated as a numerical function. Having made this assumption, it is natural to assume further that it is a function with the formal features of a probability function. Here, p(h/e • ƒ) = n shall mean that, relative to the data recorded by the proposition e and the background features recorded by f, the support for the hypothesis h has the degree n, where o ≤ n ≤ 1. However, to say that, given f, h is supported by e to degree n or that, given f,e gives h the support n is to say that p(h/e • ƒ) = á while implying that pfi/e • ƒ) > p(h/f). In other words, though ‘The support for h relative to (e • ƒ) is n’ does not imply e is “relevant” to h, ‘Given ƒ, h is supported by e to degree n’ does imply the relevance of e to h. Our moderate empiricism commits us to the view that some hypotheses are acceptable only if they are given support by empirical data and are not just supported relative to such data (Chapter I, §3). We are then concerned with the consequences of the fact that hypotheses are given support by empirical data.

Whatever the full requirements of acceptability, they will inelude the minimal requirement of support. But here also there is room to distinguish h’s being acceptable given (e • ƒ) from e’s making h acceptable given ƒ. Thus h might be acceptable given (e • ƒ) simply because it is acceptable given ƒ. But if e makes h acceptable given f, then not only is h acceptable given (e • ƒ), but also h is not acceptable given merely ƒ. The requirement of support that is of interest here follows from the claim that some hypotheses are made acceptable by empirical data. When e makes h acceptable given f, then, as I shall show, it is required that h be supported by e given not just that h have support relative to (e • ƒ). If this were not the case, then moderate empiricism would not require giving support, but only that some acceptable hypotheses be made acceptable by empirical data.

There have recently been various proposals for conditions for acceptability of h given (e • ƒ) that avoid the lottery paradox. In general, these conditions include the requirement that there must be support for h relative to (e • ƒ). For example, the proposal made by Levi involves a comparison of support for any hypothesis with the disutility assigned to its acceptance should it be false.⁵ Here disutility is of an epistemic variety. Specifically, the disutility of accepting a false hypothesis that is rich in content is less than that of accepting a false hypothesis that is trivial, for hypotheses that are rich in content have a greater potential for relieving doubt. The proposal is that one is to reject h when the support for h relative to (e • ƒ) is less than the disutility of accepting h given (e • ƒ) if h is false.* It turns out, on this proposal, that for h actually to be accepted given (e • ƒ) and its competitors to be rejected, h will have to have support relative to (e • ƒ).

But now consider the case in which e makes h acceptable given f, rather than that in which h is merely acceptable given (e • f). Where e makes h acceptable given h is not acceptable given ƒ. Thus relative to ƒ alone, it either is rejected or, though not rejected, is a disjunct in an acceptable hypothesis. In either case, for h to be acceptable given (e • ƒ), some competitor to h must be rejected because of e. This means that in respect to e the disutility of believing the competitor when it is false should be larger than the support for the competitor.

On the one hand, this might occur by decreasing support for the competitor. But since the competitor will imply ~h support for hwill increase as a result of e. On the other hand, this may occur by increasing the disutility of accepting the competitor should it be false. That is, e may decrease the content of the competitor, and hence of ~h. This would mean an increase in content for h in view of e. For example, e might reduce the number of possible situations satisfying h from two to one by increasing the number of possible situations satisfying the competitor from eight to nine; h would then have more content since it would be satisfied in a smaller proportion of cases.

But when we speak of action on prior experience, it will be true that, since experience does not limit possibilities for unexamined cases of unrestricted hypotheses, it in no way alters the content of the hypothesis corresponding to the action. To observe that water boils at 100₀ C on many occasions does not reduce the content of the hypothesis that the next pan of water will boil at 100₀ C. So increasing the disutility of a competitor when it is false is not open to us as a means of rejecting competitors. One must accept the conclusion that if e makes h acceptable given f, then e gives support to h in respect to ƒ. It seems likely that this would hold, not just for Levi's proposal, but for any proposal adequate for inductive acceptance.

In what follows, I shall have in mind three kinds of inductive claim: universal generalizations, statistical generalizations, and qualified instantial claims. I shall suppose the generalizations are cast in the form of relevant conditionals. Since the universal generalization ‘Any x is such that if it is F then it is G’ is represented as ‘(x)(Fx→ Gx)’, the statistical generalization ‘Any x is such that if it is fthen it has an chance of being G’shall be represented as ‘’.I shall also suppose that ‘All F are G’ can be represented in the first way and that ‘All F have an chance of being G’ can be represented in the second way, even though in each case this involves the false assumption that one is talking about the x’s rather than merely about the F’s. Now it might be highly unlikely that there are no exceptions to a universal conditional. But still it might be likely that any new instance taken by itself that satisfies the antecedent of the conditional will satisfy the consequent. There is support, not for the universal claim, but for instances of it qualified as satisfying its antecedent. Thus though p((x)(Fx→ Gx)/e) is negligible, for any x, p(Gx/Fx • e) is not negligible.

We want our conclusions to be valid for the practice of action on prior experience generally. Thus we want our conclusions to hold at least for induction generally. We cannot, then, limit ourselves to considering, say, only inductions to universal claims. Consequences derived from the fact that there is support given by evidence to inductively acceptable universal claims need not be part of the required ontology. For the practice of action based on prior experience is possible without the practice of inductions to universal claims. The same is true also of statistical claims and of qualified instantial claims. We must then look for consequences of support that do not depend on the kind of claim that is supported. We shall, then, be looking for consequences that are common to support for all three types of claim.

§3. Conditional Support and Support for Conditionals. How, then, does necessity become a requirement for support? Necessity is involved, it turns out, in several ways. But one among them is fundamental in that only by noticing it does one see how the others are required. The key idea in regard to this fundamental way is this. There is to be a chance that the individuals from which the data for an hypothesis are gathered have necessarily, or by their natures, the feature that the hypothesis projects onto other individuals. For if it is certain that the individuals in the data have the projected feature only contingently, their having it depends on their circumstances. Thus their having that feature would be irrelevant to the question of whether other entities, which may be in quite different circumstances, stand a chance of having it.

This requirement of necessity is not very useful by itself. It is useful only if unobserved individuals covered by an hypothesis have a chance of belonging to a natural kind to which the individuals in the data belong. Through natural kinds necessity enters in a second time, as will be discussed in the next chapter.

The first step in my argument involves a seemingly innocent transformation from one expression for support to another. It seems undeniable that an hypothesis can be supported by data only if there is some chance that if the data exist, then the hypothesis is true. If there were no chance that the data implied the hypothesis, then the data simply fail to support the hypothesis. My claim is for the case in which data increase support. When there is merely support relative to the data, there need be no chance that the data imply the hypothesis, since the data may have nothing to do with the hypothesis. So my claim is made only when e is positively relevant to h, that is, when p(h/e • ƒ) > p(h/f).⁶ The claim is then that:

(1) If not only p(h/e • f) > o but also p(h/e • ƒ) > p(h/f), then p(e>h/f)> o.

Here ‘if e then h’ is symbolized as (‘e > h’, which as we shall see, is not a relevant conditional.

It might seem reasonable to claim also that the converse of (1) is true. Indeed, it might even seem reasonable to make the stronger claim that the values of the two support functions are equal, that is:

(1') p(h/e • ƒ) = p(e > h/f), where p(h/e • f) > p(h/f).

This is not to be faulted by claiming that the left side of the equation presupposes the truth of e whereas the right side does not. For one can speak of the support an hypothesis has relative to certain evidence both when the evidence has actually been observed and when it is merely assumed. The meaning of the support claim is the same in each case.

Moreover, it is not to be faulted by considering the case in which an assumed, but actually false, e is incompatible with ƒ. In such a case it might be argued that since e and ƒ are incompatible, they together validly imply anything, and in particular h. Holding this implication to be not just true but valid, one reaches the undesirable result that, since the value of the left side of equation (1') is unity, the value of each side of equation (1') must be unity.⁷ However, if the expression “validly implies” means the same as “validly relevantly implies,” it is surely false that e and ƒ validly imply h simply on the basis of their incompatibility. Their incompatibility alone does not guarantee that they relevantly imply h. Still, however solid (1') might be, I shall need only the weaker claim (1).

In taking this first step, one is already moving away from the kind of thinking that treats the world as a multiplicity of disjointed entities. One is already thinking in terms of connections. Data fail to support hypotheses where there are no chances of connections, where there are merely associations. Suppose the conditional (e > h) is no more than the material conditional (e כ h), for which, as we saw, both (h כ (e כ h)) and (~e כ (e כ h)) are valid. Then, the consequent of (1) would automatically be satisfied provided either p(h/f) > o or p(~e/f) > o. But I shall show that (e > h) is true, unlike (e כ h), only when e makes h true.

If e supports h, or makes h likely, in respect to f, then there will be a chance, in respect to f, that, when true, e makes h true. To make something likely is to stand a chance of making it true. Suppose the individuals that e is about are both F and G and that e says of them that they are both F and G. Hypothesis h projects the conditional property, G if F, to other individuals. Could we still say that e supports h if it were certain that e does not make h true?

If it were certain that e does not make h true, then it would also be certain that the having by individuals that e is about the having of properties similar to those of other individuals, where the similar properties are those signified by 'F', would not influence the latter individuals to have G. For if the similarity in regard to F were to influence the unobserved individuals favorably for having G, then there would be a chance of circumstances in which this influence would make these individuals have G. But assume that the similarity in regard to F is a basis for a chance that the unobserved individuals will be made to have G. Then, that the evidence individuals are both F and G stands a chance of making other individuals G if they are F. So if the similarity in regard to F is a basis for a chance of making individuals have G, e has a chance of making h true. But if the similarity in regard to F in no way influenced unobserved individuals to have G and hence gave no basis for a chance of making individuals have G, then e would certainly not make h likely. In short, if e did not have a chance of making h true, it would not make h likely, which it does merely if it supports h. Clearly, then, (e > h) is more than a material conditional since it holds only if, when true, the antecedent makes the consequent true. It can naturally be called a “factive” conditional.

Two things are important to note about the factive conditional. First, it is not necessarily a causal conditional, even though it has been described in terms of the idea of making something true. Suppose what a grain of salt does when it is put in water happens just because it is a grain of salt. Then a true proposition about what it does can be said to make true a proposition about what another grain of salt would do if immersed. The truth-making lies in the fact that both propositions are about grains that behave the same way in water because they are salt. It does not lie in the fact that one grain acts causally on the other. Second, e may make h true, and thus factively imply h, even though under different circumstances it might not make h true, and thus not factively imply h. In general, factive implication will be context dependent. It may depend for its truth on matters beyond its antecedent and consequent. Thus one can agree with C. I. Lewis when he said that “real connections” need not be necessities.⁸ When e is true, it may make h true only because certain favorable circumstances happen to be true. When e is false, and we are thus dealing with a counterfactual conditional, e makes h true only because in a selected possible situation in which e is true there are circumstances that allow e to make h true.⁹ On the other hand, our relevant conditional differs from the factive conditional in not being context dependent.

§4. Necessities for Evidence Individuals. The second step in the argument characterizes the connection that holds when a factive conditional is true as one with a modal condition. The connection holds only if some necessity holds. Nonetheless, the connection itself is not a necessity.

Among the propositions comprising the evidence for an hypothesis there will be singular propositions such as φa If φa can increase support for h, then a is an “evidence individual” in respect to the evidence e and hypothesis h.

It would be convenient to have a unified conception of the features projected by inductions, even when the inductions are to quite different kinds of proposition. An induction to a universal claim of the standard sort indicated in §2 projects onto any entity the feature(F_ → G_). That projected by a statistical claim is the feature . Inductions to instantial claims can be V á ) brought into line with these cases by using the idea behind (1). That is, we shall simply assume that an induction to Gb from Fb and e could be replaced by an induction to (Fb > Gb) from e. One then projects onto b the feature (F_ > G_). It will be true in all three cases that if the inductive conclusion is true then each of the evidence individuals will have a chance of having the projected feature. This might not seem to be the case when one infers Gb from Fa and e, for there may be no chance that a have G. But if such an inference is replaced by one to (Fa > Gb) from e, then the projected feature is (F_ > Gb), which the evidence individual a has a chance of manifesting.

Not only will the evidence individuals have a chance of having the projected feature, but also they will have a chance of having it necessarily. It is by consideration of the factive conditional that this additional requirement emerges. It is because there is a chance of (e > h) that each evidence individual will have a chance of having the projected feature of necessity. So:

(2) If p(e > h/f) > o then, where x is any evidence individual in respect to e and h and H is the projected feature corresponding to h, p(□x(Hx)/f)>o.

This is the second step of my argument. It can be justified if it can be shown that to have the sort of connection introduced in the first step between evidence and hypothesis, each evidence individual will have, of necessity, the feature projected onto other individuals by the hypothesis.

But before justifying (2), let me contrast the view about the world implied by (1) and (2) with a familiar view about order in the world. This is the view that for hypotheses to be supportable by data it suffices for the world to be orderly. That is, it suffices for there to be a great number of simple regularities, universal and statistical, though all of these are matters of the sheerest contingency. Given a great number of universal regularities, observed exceptionless regularities will support universal hypotheses. Given a great number of statistical regularities, observed statistical regularities will support corresponding statistical generalities. Now why will such a cosmology of contingent orderliness not do? It would do if (e > h) were true simply by h being true. For to assume order is only to assume that a great number of simple hypotheses are true. Thus if the assumption of order is to suffice, it must suffice to make (e > h) true in a great number of cases. It will do this only if (e > h) is merely a material conditional. But if e is to give support to h, a connection stronger than material implication is required. In view of (2), I am also claiming that this connection holds only when certain entities have certain features by their natures. Mere order has to be supplemented by necessities, and hence by natures, to arrive at an adequate cosmology for induction.

How is this claim to be defended? Assume the evidence individuals relative to e and h have the projected feature corresponding to h in only a contingent fashion. Could (e > h) be true? There will be types of circumstances in which the evidence individuals would not have the projected feature. This leads us to consider two cases. First, there is the case in which the evidence individuals are such that the context for which (e > h) is to hold is incompatible with circumstances in which an evidence individual would lack the projected feature. Second, the evidence individuals allow that this context is compatible with circumstances in which an evidence individual wouïd lack the projected feature. In each case, (e > h) must be rejected unless a requirement of necessity is recognized.

In the first case, though the evidence individuals have the projected feature contingently, they have of necessity the conditional property that if they are in the context supposed by (e > h) then they have the projected feature. For they are such that this context excludes circumstances in which an evidence individual might lack the projected feature.

In the second case, the evidence individuals allow that the context supposed for (e > h) is compatible with circumstances in which an evidence individual would fail to have the projected feature. But then we are faced with a grave difficulty. An evidence individual does not have the projected feature of itself, since it has it only contingently. So it will not be because of what the evidence individuals are of themselves that e will make h true. Will, then, e make h true because of what the evidence individuals are in the supposed context? Not at all; the context is compatible with circumstances in which the evidence individuals do not even have the projected feature. Nothing is fixed about the unobserved individuals with which the hypothesis is concerned since they may be in precisely those circumstances that would exclude the projected feature. In this case, then, there is no obvious way that, when true, e could make h true. In this case, (e > h) holds only if the evidence individuals have the projected feature of necessity.

The conclusion to be drawn from taking these cases together involves a qualification on my original (incautious) way of putting the second step. One should not say simply that, when (e > h), the evidence individuals will have the projected feature of necessity. Rather, one must say that, when (e > h) holds for a certain supposed context, the evidence individuals will have of necessity either (1) the conditional property that if they are in this context they will have the projected feature or (2) the property that is the projected feature itself. In either case, there is a necessity, and we have thus uncovered at least one way that action based on prior experience involves necessity. Nothing essential will be changed if, now that we have reached a requirement of necessity, we ignore the qualification introduced here and return to our incautious way of putting the second step. This means ignoring the context dependent character of the factive conditional.

A third step is still needed. All that is required so far is that there be a chance that any evidence individual have the projected feature of necessity. Does this mean that there will be any necessities? And if so, will they be necessities of the evidence individuals? My answer is affirmative to both questions. Thus, where ƒ is in fact true:

(3) If, where x is any evidence individual in respect to e and h, and H is the projected feature corresponding to h, p(□x(Hx)/ƒ) > o, then each evidence individual will have some properties of necessity.

The net effect of (1)-(3) is vastly different from saying that an induction presupposes that there be some necessities or some laws. It is, rather, the more specific claim that an induction presupposes that the evidence individuals have some properties of necessity. The properties that they in fact have of necessity need not be the projected features. Moreover, on this account, determinism is not required for induction. It is not required that each evidence individual have each property either of necessity or because of some cause. As far as (3) is concerned, the evidence individuals can have many contingent properties. Moreover, the idea of causation is not even introduced by (3).

Let me consider two objections to (3). First, there are many combinations of properties an individual might have. Each combination is a possible over-all state of the individual. The property red might belong to some of the combinations for the individual a. If, all told, there are a finite number of combinations for a, then it will stand a chance of being red. Likewise, if a is to have a chance of being necessarily red, then the property necessarily-red ──if there could be such a property─will have to belong to one or more of the combinations. But the combinations of properties to which it belongs may all be different from the combination that a actually has. And, indeed, there need be no property involving necessity that a actually has. So to stand a chance of having a certain property of necessity does not─as (3) claims it does─imply having any property necessarily.

This objection is incompatible with the notion of necessity developed in Chapter II, §4. And that notion was precisely the notion of real necessity required in a discussion of induction. If, in one of the combinations of properties, we find necessarily-red, then a is possibly necessarily red. But then a could have a nature that supports red. If it could have such a nature, it actually has it. For, otherwise, if its actual nature were not the same as the nature it could have, we would no longer be speaking of the same individual. So it would be b not a, that is possibly necessarily red. But we had assumed that the combinations of properties were possibilities for the same individual. Only if the evidence individual has the projected feature necessarily will there be a chance that it has it necessarily. But this is clearly too strong, so some other method of accounting for the chance of necessity should be sought.

Second, an individual has a certain chance of having a specified property when considered as a member of a reference class in which the proportion of members with that property is equal to that chance. On this premiss the objection will be raised that the chance of an evidence individual having the projected feature of necessity does not imply that this individual has a necessity but only that some proportion of the individuals in the appropriate reference class has the projected feature of necessity. But what reference class? All individuals? Certainly not, for only a vanishing proportion of them will have the projected feature of necessity. Then perhaps all evidence individuals? Suppose, though, there is only one evidence individual. On this account it will be a certainty it has the projected feature of necessity. But this cannot be, since there are cases in which single instances provide support, even though it becomes clear later that they manifest the projected feature only contingently if at all. In the absence of other likely alternatives for a reference class, I conclude that an entirely new picture of the matter is called for.

The idea of possible combinations of properties was unsatisfactory, and so is that of a reference class of actual individuals. Neither accounts for the chance of having the projected feature of necessity. In their place I propose the following. An individual has certain properties, and among these are some it has of necessity. How can there be a chance that it has one of its properties with necessity? This is possible if within the individual there is an appropriate mixture of necessary and contingent properties. The details of this proposal will be worked out in the following chapter. All that is important now is that for there to be such a mixture the individual must have some properties of necessity.

§5. The Possible-Histories Approach. The above argument for necessity as a presupposition of induction is, in one crucial respect, less vulnerable than the more familiar one that sets out from the notion of possible histories. The latter argument can be summarized as follows.

Suppose Hume is right in adopting an ontology without anything that would support a real physical necessity. What are the implications of this for inductions to, say, universal claims, such as that all F are G? The observed regularity that is the evidence for this claim is compatible with an infinity of courses of subsequent events. With logical consistency as the only constraint, the number of possible histories that contain the observed regularity and that do not violate the universal claim is insignificant by comparison with the total number of possible histories that contain at least the observed regularity.

If the possible histories containing the observed regularity are treated as equiprobable, then the observed regularity would not support the universal claim, for there are an overwhelming number of possible histories with which the latter is incompatible. On the one hand, it might be said that the possible histories are not themselves equiprobable. It is rather, say, types of possible histories that are equiprobable; a history has an equal chance of being of that type in which one-half of the F’s are G’s as it does of being of that type in which three-fourths of the F’s are G’s, even though there may be more histories of the former type than the latter.

Thus Joseph─an advocate of the possible-histories approach to necessity─says, “But if, as the empiricist insists, all things are antecedently equally possible, then all proportions of regularity to irregularity in the world are equally possible antecedently”¹⁰ However, Keynes─an advocate of necessity, though not of the possible-histories approach─holds that the individual histories are equiprobable.¹¹

Keynes’ point is that equiprobable hypotheses ought to be of the same logical complexity. Yet the hypothesis that one-half of the four balls in an urn are black and the rest white is equivalent to a disjunction of six compositions for the urn, whereas the hypothesis that one-fourth are black is equivalent to a disjunction of only four compositions. Being disjunctions of six and four basic possibilities, respectively, the hypotheses are of different logical complexity and hence are not equiprobable.¹²

But even if types of possible histories, rather than individual possible histories, are assigned equiprobability, the observed regularity would not support our universal claim. For there are innumerable types of possible histories that are unfavorable, and only one type that is favorable to the universal claim. So the above objection can be ignored here.

On the other hand, it will be objected that it is illegitimate to introduce probabilities at all when we have considered only possibilities. “How probable it is that these logical possibilities are realized in a balanced or unbalanced way can be estimated,” says Ayer, “only in the light of experience.”¹³ For Ayer there is no connection between few favorable possibilities and improbability, or between many favorable possibilities and likelihood. It is not counting possibilities but counting observed frequencies that is relevant to estimating probability. In particular, it is the observed regularity and not the overwhelming number of unfavorable possible histories that is relevant to the question of whether there is support for our universal hypothesis. But if counting possibilities is irrelevant, how is counting frequencies relevant? If counting frequencies is to be relevant to judging the support for an hypothesis, must it not already have been agreed that there is a significant proportion of favorable possibilities?

How might our anti-possibilist respond? There seems to be no way of showing, by relying solely on the probability calculus, that observed instances support hypotheses. It has long been recognized that Laplace failed to show this by his inversion of Bernoulli’s theorem to get the so-called inverse law of maximum probability.¹⁴ If such purely formal attempts are unavailing, what recourse is there? There remain the pragmatic and conventionalist arguments referred to in §1. Based on considerations internal to inductive practice, these arguments attempt to show that frequencies, but not possibilities, are relevant to support. The pragmatic argument leads to the reasonableness of using the uniformity rule to assign support. For even if some other rule successfully predicts events in the world, the uniformity rule would be successful at least in predicting the success of this other rule. Since, in general, our practice is to project the frequencies that we have actually observed, the conventionalist argument also leads to the reasonableness of the uniformity rule to assign support.

One is tempted to reply that a healthy realism seems to make it obvious that possibilities at least partially determine support. So the pragmatist and the conventionalist must already be assuming that the possibilities have been reduced in number from the Humean logical possibilities to the necessitarian’s physical possibilities. Now, though it is obvious to Kneale and Carnap that possibilities are to be considered in determining support,¹⁵ this obviousness to them is no answer to the anti-possibilist. At this point, the possibilist might resort to pointing out some of the difficulties of the pragmatic and conventionalist arguments. Is there only one rule such that, if any rule is successful, this one rule will be?¹⁶ Does our common inductive practice fit only one induetive rule, or are there many that can be used to represent it? But the possibilist has no way of showing that these difficulties are insuperable. He cannot deny that reasons could be given from inside inductive practice for using a rule that accords support without regard to possibilities. So there appears to be a stalemate.

Nonetheless, the possibilist wants to retort that whatever the internal reasons might be for according support without reference to possibilities there are, in a Humean world, simply too many equiprobable alternatives. So in such a world, the uniformity rule cannot be held to be successful, for it would be unreasonable to hold that any rule would be successful. But how can the possibilist make such a retort here? The context of the debate is that the anti-possibilist challenged his right to say a priori that the alternatives have probabilities, whether equal or unequal. Yet here he has supposed they are equiprobable precisely in order to attack the anti-possibilist’s assertion that frequencies determine support through a rule that makes support insensitive to possibilities.¹⁷ In short, the possibilist is unsuccessful in showing that pragmatic and conventionalist arguments for the reasonableness of assigning support by certain inductive rules fail in a context where the logical possibilities are not cut down by physical necessities. It does not help the possibilist to allow him to distribute the probabilities unequally among possibilities, for the question is not whether the probabilities are equal or unequal but whether possibilities have any probabilities a priori.

The argument for necessity in §3 and §4 avoids the stalemate facing the possible-histories approach. The former does not depend on the association of probabilities with possible histories. Rather, by following up the consequences of the correlation between conditional support and support for a conditional, we were forced to grant that the evidence individuals had some properties of necessity. This of itself does not limit possible histories containing the evidence, since it concerns only the evidence individuals. It does limit these histories when it is assumed that there are future individuals with the same necessities.

Whether this or a related assumption about the future is needed will be the subject of part of the next chapter. But even at this point, it is established that any purely enumerativist approach to induction is a mistake. That is, in addition to the data enumerated one needs the assumption that the individuals involved in the data have some physical properties of necessity. Otherwise, the data cannot support an hypothesis. But this conclusion has not been reached by an argument from the multiplicity of possible histories.

It remains to be pointed out that the argument for necessity in Sections 3 and 4 is not one that ties us down to any particular inductive rules. In particular, it does not tie us down to the uniformity rule.

(a) Suppose on the evidence that F are G there is support for the hypothesis that (x)(Fx → Gx). Even though the evidence itself fails to show a ratio of G’s to F’s equal to the projected chance, it may nonetheless be the case that each evidence individual has a chance of being G if it is F. Indeed, given that the evidence supports the above hypothesis, each evidence individual has some likelihood of necessarily having the property of having a chance of being a G if it is an F.

(b) Suppose on the evidence this emerald is green there is support for the hypothesis that anything is either green if it is an emerald observed before 1975 or blue if it is an emerald not observed before 1975. In short, it supports the hypothesis that all emeralds are “grue” (that is, green and observed before 1975 or blue and observed after 1975). Again, in order for there to be such support, this emerald, which I observe in 1971 to be green, will have to have a chance of being necessarily grue, assuming it is an evidence individual.

There might well be reasons for rejecting such odd inductive rules as the ones assumed in (a) and (b). But I am not concerned here with their acceptability. What is important is that the requirement for necessity depends on the notion of support and not on the kind of rule from which support is inferred. This is important because the notion of action on prior experience does not limit one to rules of certain kinds. So one can now say that the practice of acting on the basis of prior sense experience would be incoherent with a denial of real physical necessity.

* One need not reject h when it has zero support, for there might be zero disutility in believing it when it is false.