CHAPTER V

The Structure of Physical Individuals

§1. Limited Independent Variety. Whether a claim is supported by data depends on what data there are and, as importantly, on what necessities there are. Where the only necessities are the logical necessities, there can be no support, however reassuring the data may appear. So, if support is to be given to claims by data, there must be necessities other than the logical ones. Up to this point, we have only hinted at how extensive such a realm of physical necessity must be. It is plain from Chapter IV that it must be extensive enough for evidence individuals relative to any hypothesis to have a chance of having the projected property corresponding to that hypothesis of necessity. But to satisfy this and other relevant demands, must one adopt the doctrine that in any individual no two properties are independent of one another? If such were the price, one might be willing to conclude that the only way to make inductive practice reasonable is to adopt an inherently unreasonable ontology. Is there a way short of this extreme necessitarianism? A modification of Keynes’ principle of limited independent variety does, I wish to show, provide a way. It will indicate the sort of mixture of necessary and contingent properties that evidence individuals must have.

The emphasis here is on the properties of individuals rather than on properties considered as entities apart from individuals. And Keynes’ strategy is precisely to concentrate on individuals. He imposes no limitation on the number of properties considered as separate entities; rather, the limitation concerns the properties a given individual can have. It is assumed that the properties in question are physical. According to Keynes, “We seem to need some such assumption as that the amount of variety in the universe is limited in such a way that there is no one object so complex that its qualities fall into an infinite number of independent groups.”¹ Thus, finite independent variety in individuals does not preclude an infinity of individuals, each with, say, a different temperature. Infinite independent variety among individuals is allowed by finite independent variety in individuals. It will not suffice here to assume merely that there is a chance an individual will have limited independent variety. For then an evidence individual might have infinite independent variety. But without limited independent variety, one will be unable to show that the individual has a chance of having the projected feature of necessity.

Keynes understood independence in a modal fashion. What he called “laws of necessary connection” were the basis for the limitation of independent variety.² Thus, if F and H are independent properties, then, for any individual x, it is not a necessity of x that if it is F then it is H, or conversely that if it is H then it is F, and it is not a necessity of x that if it is F then it is not H, or conversely that if it is not H then it is F. So independence or dependence holds between properties, not just for individuals of special natures, but for individuals generally.

The principle of limited independent variety asserts that:

(I) The number of properties in the set B_a of basic properties of an arbitrary individual a is greater than zero and less than some given finite number.

The requirement that the number of B_a be less than a given finite number rather than merely finite was noticed by Nicod rather than Keynes.³ “Basic” properties are not just underived; together they suffice for the derivation of all other relevant properties. The set of all relevant properties of a, P_a, has already been limited to physical ones, and will soon be subject to two more restrictions. So a member of B_a is underived from any other properties of P_a, and any member of P_a is derived from a subset of B_a. H “derives” from a property or set of properties F when any individual is necessarily such that if it is F or has all the properties of F then it is H. Notice that basic properties need not themselves be necessary. Indeed, the basic-derived distinction cuts across the necessarycontingent one.

Let us call the subsets of B_a "groups.״ If a has limited independent variety, then it has only finitely many groups. But even if there are only finitely many groups, there may be an infinity of different properties in a. This is possible if there are necessary connections making the groups sufficient for an infinity of properties.

Suppose properties of the set G and the property H belong to P_a. Suppose further that H is derived from G. Is the “derivational property” (G_ → H_) itself to be listed among the basic and derived properties of a? If P_a is infinite and B_a is finite, a has infinitely many such derivational properties. Only a finite number of these could be basic. It might then seem convenient to treat them all as properties derived from their own antecedents. After all, it might seem that if having G is sufficient for having H, then, just as a matter of logic, having G must be sufficient for this connection between G itself and H. But not only is this not logically true for relevant implication but also it fails to determine the status of (G_ → H _) for any individual that lacks G. Yet if the connection is a true derivation, all individuals will have (G_ → H_) whether or not they have G. It will not do to say that the derivational property is basic in individuals lacking G. For, again, there will be an infinity of such properties in any non-G. It is then imperative to exclude derivational properties from among those considered by Keynes’ principle. Thus P_a includes properties between which derivations hold, but no derivational properties.

To make things relevant to induction, imagine that we observe a to be both F and H. Let this observation support the hypothesis that all F are H. The projected feature is then (F_→ H _). There are now two questions. First, can (I) account for there being a chance that a has this projected feature of necessity? Second, can (I) help to explain how individuals other than a have a chance of having the projected feature? (I shall deal with this question in the next section.) To begin to deal with these questions, let us define any group B^F_a as follows. On the one hand, any such group is a subset of B_a from which F is derived. On the other hand, F is derived from no proper part of B^F_a. In Keynes’ terminology, F “specifies” any of the B^F_a. Since there may be several such groups even in a single individual, F may specify a “plurality of generators” in a.

Regarding the first question, it is clear that, with only finitely many groups in a, the possibility that H derives from a group in a specified by F is one among a finite number. For H, being a property of a, will derive from some group of a. And this group may be a B^F_a. Since this possibility is one among a finite number, may we say that there is a chance that F and H derive from one and the same group of a? One might object that, in Chapter IV, §5, we seemed to argue that chances are not based on possibilities. The way seems blocked.

However, two things are to be recalled. First, it was not argued that chances could not be based on possibilities. It was argued only that the possible-histories approach to necessity had no way of justifying, against an enumerativist like Ayer, its assumption that chances can be based on possibilities. Second, without assuming at the start that chances can be based on ratios of favorable to all possibilities, we arrived at the conclusion that evidence individuals must have a chance of having projected features necessarily. Immediately, the question arose as to how there can be such a chance. Several proposals were quickly rejected, and we drew the conclusion that such a chance depends on the way individuals are structured out of necessary and contingent properties. But inevitably this means that such a chance depends on ratios of possibilities within the structure. Rather than assuming, as in the possiblehistories approach, that chances are based on possibilities, our argument led to this claim through a sequence of presuppositions of induction. So, despite the criticism we made of the possiblehistories approach, we are free to claim here that if one among a finite number of alternatives is that F and H derive from the same group of a, then there is a chance that F and H derive from the same group of a.

But clearly we are in search of a stronger conclusion. To say there is a chance that a has a common source for F and H is not to say there is a chance a is necessarily such that if it is F then it is H. Of course, if H is necessarily implied by the B^F_a G and G is neces sarily implied by F, then indeed F would necessarily imply H. The difficulty is that though G necessarily implies F when F specifies G, the converse does not hold in general. For there may be another B^F_a besides G, and thus a could be F without being G.

Consider for a moment, though, the consequences of assuming that a group F specifies, say G, contains only properties that a necessarily has. I shall call such a group a “necessary group” of a. So a’s appearance as F points back not just to basic properties a happens to have but to basic properties that are essential to a. Since F specifies G and since G is essential to a, a will necessarily be such that if it is F then it is G. The idea is that, in a, F must have G as a necessary condition provided that F derives from G, but no proper subset of G, and that a could not be without G. Adding this new principle to the premisses that (1) G is a necessity of a, that (2) G is specified by F, and that (3) H derives from G we have, where G' is a proper subset of G, the following proof:

(1) □a(Ga)

(2) □a(Ga→Fa) • ~□a(G’a Fa)

(3) □  a(Ga→Ha)

(4) (□a(Ga → Fa)• ~□a(G’a→ Fa)) (□  a(Ga)

→□a(Fa → Ga))

.'.(5) □a(Fa → Ha)

This conclusion is indeed the one we were searching for.

Objections to our new principle (4) will arise only if one mistakenly reads a necessary relevant implication as a derivational claim. In (4), □  a(Fa → Ga) might be the claim that a is necessarily such that if it is water soluble then it has an ionic crystalline structure. But this does not mean that the structure is due to the solubility. Nor does it mean that the structure of the individual is the only thing that could account for its solubility. There might be multiple generators in a. Rather, this necessary implication is adequately supported by the fact that the solubility of this individual cannot help but be accounted for by, among other things, its crystalline structure.

Also, when ‘Ga’ is replaced in (4) by ‘Ga’, G'a’ and ‘Fa’ is replaced by ‘(Ǝx)Gx • G'a’, we get from □a(Fa→Ga) the claim that □  a(((Ǝx)Gx • G'a) → (Ga • G'a)). This claim seems false when we require for its truth that its consequent be derived from its antecedent. However, this necessary implication is adequately supported by the fact that a’s satisfying ((Ǝx)Gx • G’a) cannot help but be accounted for by a’s having G and G’. It cannot help but be accounted for in this way since, as the analogue to (1), one has that a is necessarily G and G׳ and, as the analogue to (2), one has that Ga • G’a, but no part of it, necessarily implies (Ǝx)Gx • G’a.

Keynes’ principle is not adequate for the claim that there is a chance that (5) is true. His principle, (I), gives premisses (2) and (3) a chance of being true. Premiss (4) was just argued for as being true. But for there to be a chance that (1) is true we also need the claim, not made by Keynes, that I shall call the principle of necessary basic properties:

(II) Any basic property of an individual has a chance of being a necessary property of that individual.

Since groups are, by (I), finite, it follows directly that any group had by an individual has a chance of being a necessary group of that individual. So there is a chance that (1) is true. (I) and (II) jointly provide that an evidence individual has a chance of having the projected feature of necessity.

Let us assume for the moment that (I) and (II) are not only adequate but also indispensable. Thus, they not only imply a finite chance for (5), but are also implied by a finite chance for (5). This will lead us to see that if evidence individuals do have a chance of having the projected properties of necessity then each evidence individual has some properties with physical necessity. (We will then have a direct proof of (3) of Chapter IV, §4, in addition to the indirect proof given there.)

This can be seen as follows. On the one hand, the principle of necessary basic properties requires for its truth that each individual have at least one property that is necessary, if as (I) guarantees, it has at least one basic property. This would not be required were it assumed that there are many individuals of which a fraction have necessary basic properties. For then, even though some may lack necessary basic properties, each would have a chance of at least one necessary basic property. Since there would be no reason to distinguish among basic properties, each basic property would, as (II) states, have a chance of being a necessary one. The difficulty with this is the assumption that there are many individuals.

The view of evidence individuals developed here needs to be independent of whether there are many evidence individuals and of whether there are unobserved individuals. That is why we must rely solely on the structure of any given evidence individual. We cannot assume a multiplicity of individuals in trying to justify the claim that an evidence individual has a chance of having a feature necessarily. For the claim that such a feature might be projected if there were other individuals to project it on is a claim that is true or false of the individual in question, and it does not change its truth value if other individuals cease to exist.

On the other hand, the principle of limited independent variety requires for its truth that each individual have its derivational properties of necessity. Suppose these derivational properties all hold of logical necessity. But we are interested solely in the real physical necessities. The possibility of making inductions to physical hypotheses requires that there be a chance that two logically independent properties are connected by necessity. Such a connection will not then be logical, and hence (I) is irrelevant unless it is taken to assume that there are physical derivational necessities.

Thus individuals understood in the light of (I) and (II) are seen to have a mixture of necessary and contingent properties. The mixture contains a finite ratio of necessary basic properties to all basic properties. It also contains some derived properties that are connected to basic properties by physically necessary connections. The necessary basic properties are physically necessary since they are physical properties.

§2. Natural Kinds and the Multiplicity of Generators. The second question takes us beyond evidence individuals to unobserved ones. Is there a chance that in addition to a all other individual F’s have (F_ → H_) of necessity? If so, the datum that a is F and H would support the claim that all F are H. (When I speak of two F’s, I am speaking of two individuals with two similar properties, not with one shared property.) But the chance that a has (F_ →H_) of necessity depends on the chance that H derives from one of the groups, say G> that F specifies in a and on the chance that this group is necessary to a. So the most one can say is that there is a chance that every other F for which F specifies G has (F_ → H_) of necessity. And so the given data support only the qualified claim that all F for which F specifies G are H.

The reason for this qualification is recognized by Keynes as the possibility of a plurality of generators as between individuals.⁴ In another F, G may not be among the groups specified by F. If this individual is b, then it may be that b is F but not H. For, though H was derived from G in a, it may not be derived from any group in b that F specifies.*

Suppose the evidence individual a─which is F and H─has a chance of having the projected feature (F_→H_) of necessity. But suppose further there is no chance that all other F’s have this feature. Then clearly there would be no chance that (e > h), that is, that a’s having F and H factively implies that all F are H. It is for this reason that a chance must be secured for all other F’s having the projected feature. The reason is not the one Keynes adduced. He wanted a prior probability for the hypothesis so that, by Bayes’ Theorem, evidence could increase its probability. However, my reasoning here does not rely on Bayes’ Theorem and, to that extent, is not Bayesian. Rather, I am insisting that there be a chance that all other F’s have the projected feature (.F_→H_) for the reason that otherwise there would be no chance that (e > h). There would be no chance that (e > h) since, with the evidence in, e would be true even though there would be no chance that h is true. The need for h to have a chance to be true is based on the need for (e > h) to have such a chance and not on the need for a factor greater than zero in the numerator of Bayes’ Theorem.

As Keynes himself points out, an additional principle is needed. It can take either of two forms. Suppose it has the form:

(III) An arbitrary property F has a finite chance of specifying a common group wherever it occurs.

Call this the principle of the common generator. From (III) and (II) it follows that an arbitrary property has a chance of specifying a common necessary group wherever it occurs. If there is support for the qualified universal claim that all F for which F specifies G are H then, since there is a chance F specifies G everywhere, there is support for the unqualified claim that all F are H.

Suppose, however, the principle has the form:

(IV) For an arbitrary property F there is a chance that there is a finite set of groups such that F specifies some member or other of this set wherever it occurs.

Call this the principle of limited sufficient generators. It is compatible with F specifying an infinity of groups. It requires only a chance of a finite core of groups specified by F, where some member or other of the core shows up in every F. Under (IV) one abandons the possibility of supporting unqualified universal claims. For the chance that the F’s in larger and larger classes of F’s have the same generators tends to zero. The possibility of supporting instantial claims is retained. For, if F specifies G in a, there is still a chance it specifies G in another F, say b, since there is a chance that the G in a belongs to a finite core of groups.

Keynes’ problem of the multiplicity of generators is related to the old question as to whether there are natural kinds. It turns out that the principle of the common generator, (III), is equivalent to the claim that there are natural kinds, in the context of (I) and (II). This establishes a link between Keynes’ (III) and Broad's claim that induction is warranted only where it has to do with individuals identified as members of natural kinds.⁵

In the strong sense, a natural kind was defined as a class in which all members are physical individuals of exactly similar natures. In the weak sense, a natural kind was defined schematically as a class of physical individuals, whose natures support certain exactly similar necessities. Precisely what necessities are to be similar here? Now the collection of necessities associated with a group G of a divides into two parts. First, there are the necessities a is under to have certain basic properties of G and to have all the properties derived directly from these basic ones. This division may be empty. Second, there are the necessities a is under to have all the derivational properties (F_ → H_), where F is derived from G. I call these two the divisions of basic and derivational necessities of G.

I now make the notion of a natural kind definite by requiring that every member of a “natural kind” have one set of properties, comprised of necessary basic properties, that is exactly similar to some set in every other individual of the natural kind such that the properties of this set are still necessary basic properties of its individual. In Platonic terms, members of a natural kind have a common necessary group. Common necessary derived properties do not suffice; they may be derived from dissimilar groups in distinct individuals. Both Jones and his computer may be necessarily rational, but if the bases for rationality in the two are dissimilar groups, it is not established that they belong to the same natural kind.

Further, a common contingent group would suffice for common derivational necessities. But derivational necessities hold for individuals generally, not just for those with a given group in common. So a group cannot be the basis for a natural kind due to its division of derivational necessities. Nor can the divisions of dérivational necessities associated with different groups define natural kinds. For the members of each such division apply to all individuals. We are led, rather to associate natural kinds only with those groups whose divisions of basic necessities include all their properties.

Keynes’ (III) was needed for inductions to universal claims. I shall now show that, in the context of (I) and (II), it is equivalent to the claim that an arbitrary property has a chance of being coextensive with a natural kind. Even induction to universal claims does not, then, require appeal to natural kinds in the strong sense of individuals with similar natures.

Suppose there is a chance that F specifies the same necessary group wherever it occurs. There is, then, a chance that there is a necessary group common to all the F’s. So there is a chance that the F’s make a natural kind.

Conversely, suppose there is a chance that the F’s make a natural kind. There is, then, a chance that (i) there is a necessary group common to all the F’s. Also, there is a chance that (ii), when some group is a necessary group common to all the F’s, F specifies that group in all the F’s. Since there is a chance that (i) and a chance that (ii), it follows that there is a chance that F specifies a common necessary group in all the F’s. Thus (III) is satisfied.

How though are we sure there is a chance that (ii)? Here we rely on (I) and the fact that derivational properties hold universally. In view of the finitude of groups, there is a chance that, when some group belongs to an F, the property F specifies that group. If the group that F specifies in this individual is common to all the F’s, then, in view of the universality of derivational properties, F specifies that group in all the F’s. So, there is a chance that, when some group is a necessary group common to all the F’s, F specifies that group in all the F’s.

Keynes’ principle of limited sufficient generators (IV) suffices to provide support for instantial claims. Natural kinds are then superfluous. Still, in the context of (I) and (II), Keynes’ (IV) is equivalent to the assumption that there is a chance that any two members of the set of F’s share a group whose properties are basic and necessary for each. Any two members may, of course, share a necessary group when there is no necessary group common to all. The group that a and b share may not be the one b and c share. If any two members of a class share a necessary group, then I shall call the class a “family kind.” Human family resemblances do not require a single obvious common and distinguishing feature in all members of the family, but generally any two members have some distinguishing point in common─the shape of the nose, the texture of the skin, or the shape of the mouth. In a family kind there is not only no single nature shared by all, but there is also no need for a common necessary group.

Assume there is a chance that there is a finite set of necessary groups such that, among the groups specified by F in any individual, there is one from among this finite set of necessary groups. This is just (IV) combined with (II). There will then be a chance that, between any two individuals that are F, the property F specifies the same necessary group. Thus there is a chance that F is coextensive with a family kind.

Conversely, assume there is a chance that the F’s make a family kind. If there is such a kind, let a be a member of it. By (I), a has only the finite number of groups G₁, G₂, . . . , G_n. By definition, a has a group in common with any other member of the family kind coextensive with F. Indeed, it has a common necessary group. All individuals in this kind can be put in the finite number of possibly non-disjoint classes {G₁}, {G₂}, . . . , {G_n}, where {G_i} is the class of all individuals in the kind sharing G_i when G_i is a common necessary group. Since a given G_i need not be a common necessary group, some of these classes might be empty.

By (I), there is a finite chance that F specifies any G_i in a. Since there is a chance that G_i is one of the necessary groups of the family kind, there is a chance that individuals with G_i form {G_i}. Moreover, if F specifies G_i and the individuals with G_i form {G_i}, then F specifies a necessary group G_i throughout {G_i}, since derivation holds universally. Therefore, there is a finite chance that F specifies a necessary group G_i in every individual in {G_i}. But then there is a finite chance that F specifies either G₁, G₂ . . . , or G_n in any individual F and that each of these is a necessary group. Hence there is a chance that there is a finite set of necessary groups such that among the groups specified by F in any individual there is one from among this set.

Earlier, in Chapter III, §3, I noted that the practice of action on prior experience does not depend on assuming the existence of classes whose members have the same nature. I noted that at most the assumption of natural kinds in the weak sense was needed. Now it appears that the assumption of natural kinds, even in the weak sense, is needed only when we assume that induction to universals is an essential part of inductive practice. If, however, we content ourselves with induction to instantial claims, we see that the assumption of natural kinds in the weak sense is unnecessary. All that is needed is family kinds. The assumption that there is a chance that any property is coextensive with a family kind would seem to be the weakest assumption that would still allow inductive practice.* It cannot then be claimed that natural kinds are part of the required ontology.

If universal hypotheses are formulated explicitly in terms of natural kinds, then not only is Keynes’ assumption (III) about multiple generators superfluous but so is my assumption (II) about necessary basic properties. This is obvious since F’s being coextensive with a natural kind was just seen to imply F’s specifying a common necessary group. Suppose the hypothesis is that (x)(Nx → (Fx → Hx)). Recalled that ‘N’, as a natural-kind term, signifies a property, the kind property, that belongs to each N necessarily. Though (I) is needed if there is to be support for this hypothesis, (II) and (III) are not. This may be shown as follows. The evidence individual a is, we suppose, an N and has (F_→ H _). Now let the kind property signified by ‘N’ be J. There is a chance, in view of (I), that (F_ → H_) derives from a group, G, specified by J. Since a has limited groups, G has a chance of being the necessary group common throughout N. So there is a chance that □a(Ga) holds. In view of (1)-(5) of §1, there is a chance that □ a(Na → (Fa→ Ha)).

What is the prospect for generalizing this? Instead of assuming (III) at this point, one merely notes that, by definition, members of a natural kind have a necessary group in common. In view of limited independent variety in a, there is a chance that G, the necessary group specified by the kind property J in a, is the common necessary group. Thus not only is there a chance that G is necessary in every N, but also there is a chance that (F_ → H_) is derived, in any N, from a group that the kind property of N specifies in that member of N. That is, there is a chance that (x) □  x(Nx → (Fx → Hx)), which by the Principle of Essentialism is interderivable from (x)(Nx → □  x(Fx → Hx)). This last we called a kind-specific necessity. Universal inductive hypotheses characteristically have this form, or at least the implied non-modal form. So without kind-specific hypotheses, or ones implied by them, induction requires the strong principles (II) and (III). The Principle of Essentialism transforms a kind-specific necessity into the full necessity that is given support by (I) alone in the above argument.

§3. Relations and Limited Variety. Among the properties of an individual are its relational ones of being spatially apart from each of the other individuals in the world. In addition, there are its relational properties of being darker than, cooler than, larger than, slower than, and heavier than various individuals in the world. Now how is it possible to contend that any individual has a limited number of independent properties in view of these facts? Being distant from Sirius and being distant from Spica would seem to be independent properties of an earthling. Being darker than this flower and darker than that leaf would seem to be independent properties of a color sample. So it would seem that either there is no limited variety in individuals or there is only limited variety for non-relational properties. On the first alternative, no claims are supportable that go beyond the observed. On the second, only those claims are supportable that do not involve relations between individuals.⁶

Suppose the color sample is darker than the flower. There will then be the two relational properties, darker than the flower and lighter than the color sample. As I shall show in Chapter VII, there cannot be relational properties without certain foundations. In other words, for the color sample to have the relational property darker than the flower, it is requisite that two conditions be satisfied. First, the color sample must have a certain shade of color; second, the flower must have a certain shade of color. So there are two distinct property foundations for any one relational property. The situation for a purely spatial or temporal relational property is more subtle. Action, which is also a component of individuals, is the foundation for both temporal and spatial separation, as I shall argue farther on.

Many relational properties of a single individual have the same foundation in that individual. Yet they will have distinct foundations in other individuals. The shade of the color sample is the foundation for its being darker than any number of other individuals. So with only a finite number of independent properties other than relational ones, an individual may still have an infinity of independent relational properties. But the important thing is that those finite independent non-relational properties can include directly or by derivation all the foundations in that individual for the infinity of its independent relational properties. This is possible since each of those relational properties also has a foundational property outside the individual.

In view of this, it becomes clear how the principle of limited independent variety can be restricted to non-relational properties of an individual without losing its applicability to all hypotheses, including relational ones. The set P_a of properties of a that are relevant to limited variety are then physical, non-derivational, and non-relational. Consider the relational claim that electrons are lighter than protons. The claim is true if any electrön has a foundational property from a certain range, a range of masses, and protons have masses from another range. So there is support for the relational claim if there is support for the two non-relational claims about foundational properties. The principle of limited independent variety is applied to protons and electrons separately. One is not forced to apply it to artificial entities, each containing an electron and a proton. It still remains to deal with the more complex problem of allowing for inductions to claims asserting temporal posteriority within the context of limited variety. I shall return to this problem in Chapter VIII, §1.

§4. Levels of Limited Variety. Keynes’ principle of limited independent variety, (I), is not required by inductive practice. A more general principle─the principle of levels of limited variety─can do the job that needs to be done. A view of individuals that incorporates it still requires real physical necessities.

Formally speaking, a level is merely a class of properties of an individual. Levels of the same individual are disjoint. There is no limit to the number of levels an individual may have. The formal notion of level gains some content only under the restrictions of the principle of levels. The first restriction is that each level in an individual must be characterized by limited independent variety relative to the properties at that level, and that some finite number is larger than the amount of independent variety at any level. A property that is basic relative to a level is simply one at this level that is not derived entirely from properties at this level. It may have a derivation that involves a property of a different level and thus not be basic, absolutely speaking. Suppose F and H belong to one level and J belongs to another. Conceivably, H could be derived from F with an assist from J. Still H would be basic at its level if it were not derived entirely from any other properties at its level. The second restriction is that between any two levels there must be a connection in the sense that having the properties of some group at the one level necessarily implies having the properties of some group at the other level. Here, of course, a group is relativized to a level and is thus defined as a set of properties that are basic relative to a given level.

The principle of levels of limited independent variety says:

(I*) There is a set of levels of the properties of an arbitrary individual such that there is limited independent variety at each level, and there are connections between any level and each of the others.

I shall now speak of a set of levels satisfying both restrictions of (I*) simply as a set of levels.

The properties of an individual divide up in various ways. Thus to say they can belong to the levels of a specified set is not to say they cannot belong to another set of levels. Whether a person’s weight, blood type, and intelligence quotient all belong to the same level or not depends on factors beyond the principle of levels. If there are several sets of levels, all of these properties may belong to one level in one set, and each may belong to a distinct level in another set. And even if there should be a unique set of levels in an individual, (I*) does not specify which properties belong to which levels.

Now an individual with a set of levels may have an infinite number of absolutely basic properties. Suppose the individual has infinitely many levels, and that two properties are basic in respect to each level. If one of these properties at each level is not connected with any property at any other level, then it is basic in an absolute sense. A universe containing some such individuals would violate Keynes’ principle, (I), but would still satisfy the levels principle, (I*). Such individuals might seem strange since one property at each level is shielded from all other levels. They fit neither a reductionist nor an emergentist world view. But this is no objection here since the practice of action on prior experience does not require either a reductionist or an emergentist ontology. Typically, the reductionist would think in terms of a single level. The emergentist would think of all properties of at least some levels as derived from, though not reduced to, those of other levels. This case of an infinity of absolutely basic properties is not the only one that conflicts with Keynes’ (I). For (I*) does not require that there be any absolutely basic properties at all.

Does the principle of levels provide a basis for saying that an evidence individual has a chance of having the projected feature of necessity? Let the evidence individual, a, be F and H and the projected feature be (F_→H_). Now a may have several sets of levels. But nothing in our argument will depend on the peculiarities of any one set. In any set of levels, F will specify a group at some level, say i, and H will belong either to i or to some other level j. On the one hand, if it belongs to i, the possibility that H derives from a group specified by F is one among a finite number of possibilities, there being only finitely many groups at i. On the other hand, let j be the level at which H is derived. Since levels are connected, one group at i necessarily implies some group at j, or conversely. If i has m and j has n groups, the possibilities that a group specified by F and one specified by H are connected is one out of─at most─m•n possibilities. So the possibility H is derived, through inter-level connection, from a group specified by F is one out of─at most─2 \³□(m•n possibilities. Even with an infinity of levels, the connections among levels secures a chance that F and H have a common source.

This is not yet what we want. Is there a chance that, for a, F necessarily implies H? For this to be the case we need to assume:

(II*) Any individual has some level at which any property that is basic at that level has a chance of being a necessary property of that individual.

But since there are connections among levels and since the connections stand an equal chance of running in either direction, any group at any level has a chance of being necessary. Suppose, then, F specifies the group G_i at level i and that H derives from G_j. By the interconnection of levels, G_i may necessarily imply G_j. So by (II*) there is a chance F and H derive from the same necessary group G_i And by the argument (1)-(5) of Section 1, there is a chance that, for a, F necessarily implies H.

Of course, to project (F_ → H_) in a universal claim, there must be either an assumption regarding the plurality of F’s generators or a use of ‘F’ as a natural-kind term. The assumption would be:

(III*) There is a set of properties G such that for any individual that is an F there is a finite chance that G is a group relative to some level in that individual and that F specifies G.

We cannot say simply that there must be a chance F specifies the same group at the same level. For we have given no sense to the idea of sameness of level in individuals that differ in some properties. As before, (III*) in the context of (I*) and (II*) is equivalent to the claim that there is a chance the F’s constitute a natural kind. A natural kind is now a class in respect to which there is a set of properties G such that, in any individual in the class, G is a group relative to some level of that individual, and the properties of G are necessary for that individual.

There is an analogy between what I have called levels of limited independent variety and levels of scientific inquiry, such as physics, chemistry, biology, psychology, and the various levels within each of these. The levels of limited independent variety may be infinite in number, and Böhm has suggested that the levels of scientific inquiry are infinite.⁷ The important thing about an infinity of levels of independent variety is that an infinity of levels allows that properties may be basic only relative to given levels, and not absolutely so. Keynes’ principle, however, requires that all individuals have absolutely basic properties. Böhm and others argue that it is out of keeping with the scientific spirit to posit, as Keynes does, an ultimate level, that is, a level of properties that are absolutely basic.

I am not concerned here with this aspect of the scientific spirit, but only with the question of whether induction requires an ultimate level. I am saying that a universe of the sort Böhm thinks the scientific spirit posits─a universe in which individuals have no ultimate level─can contain the proper allotment of necessity to make support for hypotheses possible. Induction does not require absolutely basic properties, as Keynes thought it did. Bohm denounces as “mechanistic” any natural philosophy involving “the assumption that the possible variety in the basic properties and qualities existing in nature is limited.”⁸ However, my objection to such a natural philosophy is only that its assumption is not the most general one that allows induction to be reasonable. Induction allows for─though does not require─infinite levels.

Levels of scientific inquiry are generally assumed to be serially ordered, but no requirement of order is imposed by the principle of levels. Bohm’s levels, for example, are ordered by the fact that each one is only approximate as judged by reference to the ones below it, the latter taking into account factors the former leaves out. In addition, levels of scientific inquiry are distinguished by distinctive types of properties such as physical properties and biological properties. Mass and distance belong to the physical level since they are properties of the physical type. Properties of the same level in individuals with different properties are simply properties of the same type. But the levels of (I*) are distinguished by the individual properties they contain, not by the types of these properties.

The levels principle provides an interesting perspective from which to view Goodman’s “new riddle of induction.”⁹ We used the predicate ‘grue’ to apply “to all things examined before t just in case they are green but to other things just in case they are blue.” Now the property signified by this curious predicate─if indeed there is such a property─is not derived from the property green, and conversely green is not derived from it. For there are green things that are not grue (green things not examined before t are not grue) and there are grue things that are not green (blue things not examined before t are not green). So green and grue do not specify the same group.

Now ‘grue’ is only one of the predicates we can think up alongside ‘green’. There is also ‘gred’─the corresponding blend of green and red─and a host of others. Assuming they signify properties, observations before t on emeralds that indicate they are green are also observations that indicate they are grue and gred. Thus the conflicting hypotheses that all emeralds are green, that all emeralds are grue, and that all emeralds are gred have the same observed positive instances. Yet one would instinctively suppose that not all these hypotheses are equally supported by these observations. The problem is, why not?

The principle of levels allows for many sets of levels in the same individual. But in empirical practice some set of levels comes to be thought of as the set of levels. Thus the physical, chemical, and biological levels are thought to be in the set of levels of living things. Similarly, if one admits grue-type properties, one might segregate them into a level other than that at which properties like emerald and green are located. This would mean that one could speak of an emerald-green level for common-sense properties and of an emerose-grue level for any grue-type properties. (‘Emerose’ applies to all things examined before t that are emeralds and to all other things that are roses.) I wish to suggest that it is in terms of a segregation into these levels that our feeling that hypotheses crossing these levels have less support is to be accounted for.

At a given level we look for the chance that two properties are derived from the same group. Between levels we look for the chance that the groups specified by the properties of the two levels are connected. I shall try to show that, in general, the intra-level chances outweigh the inter-level chances. In particular, this will be the case when some of the groups at one level are not connected with those at the other level.

Is there a group at the emerald-green level that is not connected with a group at the emerose-grue level? A group specified by green does not necessarily imply a group specified by grue. Conversely, a group specified by grue does not necessarily imply the group specified by green. Assuming that the property of being examined before t belongs to the emerald-green level, then not only will green not be connected with grue, but it will not be connected with any group at the emerose-grue level. Perhaps this proves too much; perhaps it shows that there are no connections at all between groups at the two levels. We would not then have levels in the sense of (I*). But clearly there are some connections. A group specified by green-and-examined-before-t necessarily implies one specified by grue in any individual, as does a group specified by blue-and-not-examined-before-t.

Suppose then that there are m groups at the emerald-green level, k of which are connected with some of the n groups at the emerosegrue level. On the one hand, at the emerald-green level there is a chance that a group specified by emerald gives rise to green. At the emerose-grue level there is a chance that emerose is similarly related to grue.

On the other hand, the inter-level situation is more complex. What is the chance that a group specified by emerald necessarily implies a group specified by grue? Any group emerald specifies has a chance of being necessarily connected to some group at the emerose-grue level, and hence a chance of necessarily implying some such group. But any group grue specifies has a chance of being a group necessarily implied by a group that emeraldspecifies. So there is a chance that a group specified by emerald also generates grue. It was shown that k by indicating that there were inter-level lacks of connection. Now á is at least as great as m because there are multiple counterparts at the emerosegrue level for most properties at the emerald-green level. Therefore, support for the hypothesis that emeralds are grue is significantly less than that for the hypotheses that emeralds are green and that emeroses are grue.

Introducing grue-type properties does not destroy limited independent variety on the levels model, though it would if there were no levels. If there is limited variety at the emerald-green level, there will be limited variety at the emerose-grue level. For the properties at the emerose-grue level will then correspond to combinations of properties based on finite variety. The major difficulty is that we can use different critical times to determine the corresponding properties. Given an infinity of times, the emerosegrue level becomes a level of infinite independent variety. There would then be a grue for which the critical time is t₀, one for which the critical time is t₁ and so on. To counter this difficulty, each critical time must be associated with a different level. There would then be an emerose-grue-t₀ level, an emerose-grue-t₁ level, and so on.

§5. The Statistical Case and Limited Variety. Limited variety is also important in regard to the support of statistical claims by observed frequencies. The statistical hypothesis that all F have an chance of being H is supported by some ratio─not necessarily the projected ratio of of H’s to F’s in a sample. Assume the properties of the individuals in the sample are segregated into levels of limited variety in the sense of (I*). In each sample individual, the property F will specify a group at some level. If a sample individual has the property of having an likelihood of being H─ that is, the property H─then either this property is at the level of F or at another level.

In asserting the statistical hypothesis on the basis of the sample, it is already believed─though, as I shall claim, not on the basis of a prior induction─that each sample individual has H. For there to be support, each sample individual must have a chance of having the projected feature necessarily. Thus, in any individual in the sample, either the property H follows from a group at the level of F or from a group at another level. With finite variety at each level and with connections among levels, there is a significant chance a group specified by F in each sample individual generates H in that individual. Given (II*), there is a significant chance that each sample individual has the projected feature of necessity. Where ‘F’ is not a natural-kind term, induction beyond the sample requires a limitation on multiple generators for F. (Ill*) suffices for this.

Two consequences of this treatment of statistical induction may seem to weigh against it. First, when one considers whether a group that F specifies is a group that generates H in the sample individuals, it is assumed the sample individuals have H. But the limited-variety model provides no account of how the fact that, say, ‘F’s in the sample are H’s supports the claim that each sample individual has an chance of being H. In effect, then, my account treats H as a directly ascribed, rather than an inductively inferred, property of the F’s in the sample where F’s are H.¹⁰

Some will find this strange indeed. They will reason that, since one can be in error about the chance of being H while correctly noting the proportion of H’s in the sample, the chance is induetively inferred from that proportion. But by the same logic, they should object to treating the non-statistical properties emerald, green, and spherical as directly ascribed rather than inductively inferred. For, when one has limited acquaintance with a physical thing, one can be in error in ascribing to it one of these physical properties even while correctly describing one’s sensations of it. Again, on the limited-variety model, there is no account of how sensations provide support for a claim that a thing has a certain physical property.

In general, the limited variety model is applicable only when evidence individuals can be supposed to have the properties mentioned in the corresponding hypothesis. But this is no objection if a distinction between conceptual interpretation and inference is granted. One conceptualizes a sample of a certain composition as made up of individuals with a definite chance of having a certain property. One conceptualizes an object of sensation as one with certain physical properties. Our mental pathways are such that, after having certain kinds of sensations, we say a thing is an emerald. Likewise, on observing a sample in which F’s are H, we ascribe H to each of those F’s, where normally the two ratios are equal. Described in this way, the matter does not require that our observations be converted into premisses for inductive inference. Since this description is available, it is not objectionable that the limited-variety approach to statistical induction treats the chances sample individuals have as non-inductively ascribed.

Second, in our account statistical properties are grounded in non-statistical properties. Thus H is grounded in, say, the group the reference property F specifies. But in familiar cases, it is not a group of properties specified by the reference property but chance variations in certain properties among individuals that are the source of a statistical property. The chance of one-half that 3 coin has to land on its head is not derived from a group specified by the property of being a tossed coin.¹¹ If it were, it would be impossible to devise a physical tossing device that controlled the initial conditions in such a way that the coin had to land heads up. Rather, the chance of one-half for heads depends on variations of the initial conditions from toss to toss. It depends, in effect, on the distribution of possible initial conditions, that is, on the chance that each one of various sets of initial conditions has of prevailing. On the other hand, the situation is quite different for an orthodox quantum entity. Such an entity does not depend for all of its statistical properties on chance variations in experimental set-ups. The chance that a radioactive nucleus will decay in a certain period is grounded in its internal structure, not in chance variations in circumstances.

Statistical induction, properly speaking, is, then, severely limited in extent. Suppose a sample does support an hypothesis. The hypothesis projects a statistical property, H, over the reference class coextensive with F. It would then be incoherent to hold that this statistical property has no chance of being derived from a group specified by the reference property in a sample individual. For if there were no chance of this, there would be no chance that the individual has of necessity. So if the statistical property is grounded only in the chance variations of circumstances into which individuals with the reference property enter, the sample does not support the hypothesis. It follows then that there can be statistical induction for uranium decay but not for coin tosses.

For consolation, one can complicate statistical hypotheses by adding antecedents to them that posit distributions of initial conditions. From a sample one can infer the hypothesis that any tossed coin has a one-half chance of landing heads up provided there is a uniform distribution of initial conditions. However, this is possible only if the sample individuals can be assumed to have the chance properties assigned by such a distribution. In many interesting cases, the distribution assumed on the basis of prior observation is doubtless false. The distribution assumed for a statistical political claim would, if correct, lock people into an eternal pattern of response to changing circumstances that is in fact only a reflection of the way current society has conditioned them to respond.¹² This is to assume a passive populace, whereas some “political action aims precisely at raising the multitudes out of their passivity.”

Even so, only those statistical hypotheses with assumptions about chance variations of conditions built into them can be inductively inferred in a world in which the only properties derived from nonstatistical ones are other non-statistical properties. Such a world need not be deterministic, for to say that being H is not generated by being F is not to say that being H is generated by being F. On the other hand, simple statistical hypotheses, which do not have assumptions about chance variations built into them, can be inductively inferred only in a world in which some nonstatistical properties generate statistical ones. Such a world will indeed be non-deterministic. Since inductions to simple statistical hypotheses are not needed for the practice of action on prior experience, the fact that induction to simple statistical hypotheses requires a non-deterministic world does not imply that the required ontology is non-deterministic.

Statistical properties derived from non-statistical ones will, in Chapter X, be interpreted as dispositions for frequencies. Statistical properties depending on chance variations in conditions lack at least one feature of most genuine properties: they cannot be inductively projected. Thus claims apparently involving them are best viewed not as ascribing genuine statistical properties to individuals, but as saying what proportion of individuals in the reference class have a certain property.

* It is to be emphasized that here we have had to treat the universal claim that all F are H as a claim about F’s rather than about any individuals whatsoever.

* This anticipates the fact that statistical induction requires no weaker assumption.