Metaphysics and ‘Valid Inductions’

by Ann Plamondon

Ann Plamondon, formerly Ann P. Lowry, is Associate Professor of Philosophy at Loyola University, New Orleans, Louisiana.

The following article appeared in Process Studies, pp. 91-99, Vol. 3, Number 2, Summer, 1973. Process Studies is published quarterly by the Center for Process Studies, 1325 N. College Ave., Claremont, CA 91711. Used by permission. This material was prepared for Religion Online by Ted and Winnie Brock.


The author attempts to elucidate what seems to be a necessary condition for the metaphysical understanding of “valid inductive inference” patterns.

In recent discussions of induction it has been fashionable to distinguish at least two problems of induction. A distinction which seems to me to be quite valuable is one made by Professor Mary Hesse. She distinguishes (I) the "problem of validation of induction which Hume found to be insoluble" and (II) the problems of the "explication of the concept of ‘valid induction’" (2:232).

In what follows I wish to argue that this distinction is one with which Whitehead would have been in agreement and that a discussion of Whitehead’s ideas about induction in terms of this distinction is very much needed. It is needed for the following reasons. First, it is not generally recognized that Whitehead had a theory about (II). Second, it will show that Gary Gutting’s critique of metaphysical validations of induction in general and the Whiteheadian justification in particular is cogent in virtue of the fact that he requires the metaphysical doctrine of internal relations to perform tasks Whitehead never intended it to perform (PS 1:171-78). Third, the discussion will enable clarification of how the doctrine of internal relations is related to (II).


The attempt to elucidate ‘valid induction’ closest to that Whitehead suggested is that of Mary Hesse. (I am not at all suggesting, however, that Hesse owes any of her ideas to Whitehead.) Since her theory is the more fully worked out, it will be helpful to summarize certain aspects of it prior to considering Whitehead’s discussion in PR.

Hesse explains what she means by an explication of ‘valid induction:

I shall assume that the aim of an explication of ‘valid induction’ is to find a numerical or comparative function c (h,e) -- the ‘confirmation of hypothesis h given evidence e’ -- which is a syntactic function of the statements h and e, and which has a high or comparatively high value in those cases where normally accepted or intuitive inductive methods would direct us to accept hypothesis h on evidence e, at least in comparison with other hypotheses having lower c-value. (2:232f)

In a less technical sense she means the explication of "inductive inferences of a kind generally regarded as justifiable" -- inductive inferences to theories and to predictions (3:164).

Hesse’s explication of ‘valid induction’ is developed from criticisms of attempts to develop the logic of confirmation. In an early work, Carl Hempel suggested "conditions of adequacy for any definition of confirmation." These included the following:

C1: Special Consequence Condition: An observation report that confirms a theory confirms every logical consequence of the theory.

C2: Converse Consequence Condition: An observational report that is entailed by a theory confirms the theory. (See 1:30-35, 3:165f, 4: 97f.)

Hesse argues that a problem arises if both C1 and C2 are accepted. It arises for the special case when a theory t is equivalent to the conjunction of two sets of observational reports, e1 and e2. In this case t entails e1 and e2. Hence by C2, e1 confirms t; by C1, e1 confirms e2. However, the latter conclusion is unacceptable because e, may be unrelated to e1. "If t is produced just by arbitrarily conjoining any other statement e2 to e1, we should certainly not want a confirmation theory in general to allow e1 to confirm e2. Some further conditions must be imposed upon admissible e1 and e2 and either C1 or C2 must be modified" (3:166).

In view of this difficulty, Hempel first rejected C2, but later rejected C1 in favor of C2. The consequences of rejecting C1 were made clear by H. Putnam’s critique of Carnap’s theory of confirmation (see 6). If a theory t entails e1 and e2, e, is not more probable in virtue of the common deducibility of e2, and e2 from t than on the basis of e1 alone. It would seem, then, that ‘valid inductive inferences’ are from particulars to particulars; theories are "redundant" (2:234).

On the other hand, this conclusion seems to run counter to the scientist’s notion of ‘valid induction’. On the whole, e. is considered to be better confirmed when there is theoretical evidence for it than when such evidence is lacking (2:233). In 3:165 and 2:233f Hesse cites Putnam’s example of the prediction of the explosion of the first atomic bomb. From e1 (physical and chemical evidence) the prediction e2 (the explosion) was warranted to the scientist’s mind because of the mediation of nuclear theory. It would seem, then, that a possible construal of valid induction’ (inferences to predictions and theories scientists find to be good inductions) might be the following inference pattern.

Hesse argues, however, that inference pattern (1) is inadequate because it suggests that e2 is more probable in virtue of its relation to e1 via t than on the basis of e1 alone. Yet this is not the case. "No probabilistic confirmation theory of any type yet developed will allow us to infer with greater than prior confirmation from e1 to e2 merely in virtue of the fact that both are deductive consequences of some theory" (3:165). Hesse’s point is that t does not add to the probability of e2 because the probability of t is derived from e1 and e2. "Theories cannot be pulled up by their own boot straps, but only by support from external models" (3:175). In general, the inference from e1 to e2 is not justified unless there is a ‘probabilistic dependence’ between e1 and e2 ‘independent’ of their deductibility from t (3:166).

The probabilistic relation which suffices to justify the inference from e1 to e2 which Hesse takes to be the clue to an adequate inference pattern is that of analogy. Her schematic inference pattern is the following (3:174):


Here the inductive inference involved is an analogical one from t_ to t based upon an analogy between e_ and e1; t deductively entails e2. There is no questionable generalization from evidence to theory. The inference, rather, is from particulars to particulars (2:244). But theories are not redundant. On Hesse’s view theories are not to be conceived as arranged in layers such that higher levels are generalizations from lower ones and allow predictions on the lower levels to be deduced. However, theories do have a role to play. Her "tentative suggestion" is that "the function of the theory is the indication and systematic extraction (or abstraction) of analogies between a number of empirical systems" (2:243). Hence her view is able to explicate inferences of a kind scientists find justifiable. (In 2:242f she discusses the prediction of the explosion of the first atomic bomb in terms of her theory.)

These notions constitute an elaboration of Hesse’s view that theoretical inference always takes place by way of a model or analogy for which laws are already known (4:115).


I wish to suggest that Whitehead held a theory about the "explication of the concept of ‘valid induction’" quite similar to (2). The language is strikingly different because he was writing about generic ‘valid inductions’ rather than what has come to be a technical problem in the logic of confirmation.

Whitehead clearly maintained the basis of ‘valid inductions’ to be analogy.

Thus, according to the philosophy of organism, inductive reasoning gains its validity by reason of a suppressed premise. This tacit presupposition is that the particular future which is the logical subject of the judgment, inductively justified, shall include actualities which have a close analogy to some contemporary subject enjoying assigned experience.... It is also presumed that this future is derived from the present by a continuity of inheritance in which this condition is maintained. There is thus the presupposition of the maintenance of the general social environment. (PR 310. my italics)

An inductive argument always includes an hypothesis, namely that the environment which is the subject matter considered contains a society of actual occasions analogous to the society in the present. But analogous societies require analogous data for their several occasions; and analogous data can be provided only by the objectifications provided by analogous environments. But the laws of nature are derived from the characters of the societies dominating the environment. Thus the laws of nature dominating the environment in question have some analogy to the laws of nature dominating the immediate environment. (PR 312)

Thus the basis of all probability and induction is the fact of analogy between an environment presupposed and an environment directly experienced. (PR 314)

Whitehead includes in this discussion a passage from SMW which he calls a ‘summary’ of the first passage quoted above.

You will observe that I do not hold induction to be in its essence the derivation of general laws. It is the derivation of some characteristics of a particular future from the known characteristics of a particular past. The wider assumption of general laws holding for all cognizable occasions appears a very unsafe addendum to attach to this limited knowledge. (PR 310)

These passages suffice to show that Whitehead did not accept inference pattern (1); clearly, ‘valid inductions’ do not essentially involve inferences of increasing generality.

What kind of inference can be abstracted from these passages? I take the following to be a close approximation to Whitehead’s notion of valid induction.

There is an analogical inference from environment E_ to environment E based on an analogical relation between the actualities of evidence ae1_ and ae2. Since laws of nature are abstractions from an environmental order, one can deduce predictions as to behavior and properties of the actualities of ae2; the actualities of ae2 will conform to the laws expressing the dominant order of environment E.

Clearly inference pattern (2.1) bears a resemblance to (2). The closeness of the resemblance, however, can only be decided after a clarification of the Whiteheadian notion of ‘environment’. Such a clarification will show that (2.1) as it stands is only a close approximation to a ‘valid inductive inference’ pattern for Whitehead.


The term ‘environment’ refers to the order obtaining in a finite spatiotemporal region. The region is necessarily finite because, for Whitehead, ‘order’ is conceived as a correlative of ‘disorder’; order is never complete, merely dominant in some region (PR 127f). An environmental order is necessary for the existence of any particular entity or structure of order for which it is the environment (SMW 155f; PR 138f). The entities of ae2º cannot exist without the environmental order provided by E_ and those of ae1 cannot exist without that provided by E. It is because of this condition for the existence of entities that analogy between the entities of the two environments warrants an inference to be made as to the analogical relationship between the two environments. And, since laws of nature are abstractions from environmental order, the inference provides a context for inferring predictions about entities in the environment E as well.

We can now see how the doctrine of internal relations is involved for Whitehead, in a ‘valid inductive inference’. The relationship between any entity and its environment is an internal one, and the inductive inference is ‘valid’ ultimately in virtue of this internal relation between entity and environment:1

Survival requires order, and to propose survival, apart from the type of order which that type of survival requires is a contradiction. It is at this point that the organic philosophy differs from any form of Cartesian "substance-philosophy." For if a substance requires nothing but itself in order to exist, its survival can tell no tale as to the survival of order in its environment. Thus no conclusion can be drawn respecting the external relationships of the surviving substance to its future environment. For the organic philosophy, anticipations as to the future of a piece of rock presuppose an environment with the type of order which that piece of rock requires. Thus the completely unknown environment never enters into an inductive judgment. (PR 311f)

In brief, the internal relationship between an entity and its environment is a necessary condition for a ‘valid inductive inference’. It is not, however, a sufficient condition. But I do not think it can be argued that Whitehead meant the doctrine of internal relations to be a sufficient condition for the validation of inductive inferences. In PS 1/3 (Fall, 1971), 174, Gutting has asked this metaphysical doctrine to provide a measure of similarity, or positive analogy. Of course it cannot. Clearly Whitehead is suggesting this when he brings up the point that analogy leaves a "margin of uncertainty" (PR 312).

The ‘valid inductive inference’ pattern (2.1) -- abstracted from the passages prior to the passage mentioning the uncertainty of analogy -- suggests that Whitehead’s meaning was simply that if there is analogy, a further condition for making the inference is still required, viz., the internal relationships between entities and environments. But this is not sufficient. It still remains to determine whether or not there is sufficient positive analogy. At this point Whitehead goes on to add the further assumptions which Gutting has shown to amount to the Keynesian principle of the limitation of independent variety (PS 1:177). These additional assumptions are necessary for determining whether or not there is sufficient similarity to warrant the inference from one environment to another. But the inference from one environment to another is also grounded in the metaphysical doctrine of internal relations, and this grounding is the prior one. The completed Whiteheadian ‘valid inductive inference’ pattern thus requires that we specify an internal relatedness between ae1 º and Eº, and between ae1 and E, in our diagram (2.1).

The denial of the necessity of the doctrine of internal relations leaves one with Hume’s problem, which is indeed insoluble: "The question, as to what will happen to an unspecified entity in an unspecified environment, has no answer" (PR 312). The presupposition of internal relations is not sufficient to specify the entity. But in virtue of the doctrine of internal relations, if the entity is specified, so are some aspects of its environment.

Let us now return to assess the resemblance of the inference patterns (2) and (2.1) as specified in terms of internal relatedness. In both patterns, the induction is essentially "from particular to particular" and is founded on analogy -- the analogy between the entities constituting the evidence and that between systematizations of order relevant to these entities (‘theories or environments’). Both are claiming that an analogy between two sets of evidence warrants an analogy between the two systems of order bound up with the respective evidence; and the S stem of order to which the inference is made deductively entails certain predictions. That is, an analogy between e1º (ae1º) and e1 (ae1), and an analogical inference from tº (Eº ) to t (E), and the fact that t ------> e2 (E------> ae2 ) warrants the conclusion e2 (ae2) with higher than prior confirmation. Both insist that theoretical inference takes place by way of a model whose order is already known (‘theory’ or ‘environment’); t (Eº) is a model for the elucidation of t (E).

However, in inference pattern (2) no explicit claim is made about metaphysical doctrines of any kind. Indeed Hesse eschews metaphysical justifications of induction (4:105). But in what sense can this be done? Is inference pattern (2) a ‘valid inductive inference’ if the doctrine of internal relations is not an implicit necessary condition? I do not think that it is. The Whiteheadian argument seems quite general; it is about generic valid inductions. Let us reconsider this argument in face of the claim that such postulates as the Keynesian principle of limitation of independent variety constitute adequate grounding for ‘valid inductive inference’.


The Whiteheadian argument, as we have seen, is that ‘valid inductive inference’ requires that the existence of any entity (and thereby the behavior and properties of that entity) be bound up with an environmental order. This is an assertion of internal relatedness. The entities of an environment contribute to the environmental order and, at the same time, the entities cannot exist without that environmental order. The alternative to recognizing the internal relatedness of entity and environment is to assert that an entity does not require a particular environmental order to exist. In this case, it seems quite clear that inference pattern (2) is not a ‘valid inductive inference’ because the analogies between entities cannot be a ground for analogies between systems of order. Entities could exist under any alternative orders and inductive inference, it seems, could never be justified.

The Keynesian argument, however, seems to be at odds with the Whiteheadian one. Keynes maintains that a ‘valid inductive inference’ requires that the "universe of phenomena . . . present those peculiar characteristics of atomism and limited variety which appear more and more clearly as the ultimate result to which material science is tending" (5:427). These two assumptions are not "formally equivalent" but are tantamount to the same thing (5:260). Atomic uniformity is a thesis about the relationship of parts to complexes into which they enter (or organize) (5:249). The limitation of independent variety is an assumption to insure that the number of properties relevant to the behavior of any object is not infinite (5:258). This assumption guarantees that the probability of finding a particular set of properties in unobserved cases when they have already been observed together will be increased (see 5:260, 4:105).

Keynes’s argument as to why these two assumptions are required for ‘valid inductive inference’ is crucial for resolving the conflict about the role of the doctrine of internal relations. Consider the following passages.

Yet there might well be quite different laws for wholes of different degrees of complexity, and laws of connection between complexes which could not be stated in terms of laws connecting individual parts. In this case natural law would be organic, and not, as it is generally supposed, atomic. If every configuration of the universe were subject to a separate and independent law, or if very small differences between bodies -- in their shape and size, for instance, -- led to their obeying quite different laws, prediction would be fin-possible and the inductive method useless. Yet nature might still be uniform, causation sovereign, and laws timeless and absolute.

The scientist wishes, in fact, to assnme that the occurrence of a phenomenon which has appeared as part of a more complex phenomenon, may be some reason for expecting it to be associated on another occasion with part of the same complex. Yet if different wholes were subject to different laws qua. wholes and not simply on account of and in proportion to the differences of their parts, knowledge of a part could not lead, it would seem, even to presumptive or probable knowledge as to its association with other parts. Given, on the other hand, a number of legally atomic events and the laws connecting them, it would be possible to deduce their effects pro tanto without an exhaustive knowledge of all the coexisting circumstances. (5;249f)

If the fundamental laws of connection changed altogether with variations, for instance, in the shape or size of bodies, or if the laws governing the behavior of a complex had no relation whatever to the laws governing the behavior of its parts when belonging to other complexes, there could hardly be a limitation of independent variety in the sense in which this has been defined. And, on the other hand, a limitation of independent variety seems necessarily to carry with it some degree of atomic uniformity. The underlying conception as to the character of the System of Nature is in each case the same. (5:261)

In these passages Keynes is maintaining that one must make an assumption to guarantee the limitation of independent variety. If laws for complexes were not expressible in terms of laws governing the parts of complexes, then different complexes into which the same parts enter could be governed by quite different laws (indeed, incompatible ones). That is, knowledge of the behavior of parts in one complex could never warrant knowledge of the behavior of the same parts in another complex. Hence predictions would not be possible, and no explication could be given of ‘valid inductive inference’. What is required, then, is that the laws governing the behavior of parts in one complex be related to laws governing those parts in other complexes.

It seems to me that the Keynesian necessary condition for the limitation of independent variety is an affirmation of the Whiteheadian claim that "survival requires order."

It may seem as if Keynes is maintaining a "reductionist" conception of the relationship of parts and complexes which contradicts the organic conception. However, the two views are not incompatible: in order for the reductionist conception to make sense at all, the "laws governing parts" must include implicit reference to the behavior of these parts with respect to any complex into which the parts may enter. This is in agreement with the organic view of the relation of parts and complexes, as I have shown elsewhere.’

Both Whitehead and Keynes are maintaining the following: If [a1, . . . , a (n)] organize complex C° and [a~ a (r)] organize complex C, then the laws governing a~ must include a description of the behavior of a+ in both C_ and C -- behavior which we empirically know is rarely the same. But this amounts to saying that what a1 is depends upon the orders which it forms; a complete description of a1 cannot be given without reference to its potential to behave differently in different complexes. Clearly this constitutes a rejection of what Whitehead terms the "Cartesian ‘substance-philosophy’." Such a rejection is necessary if predictions are to be made and ‘valid inductions’ explicated. To repeat: "For if a substance requires nothing but itself in order to exist, its survival can tell no tale as to the survival of order in its environment. Thus no conclusion can be drawn respecting the external relationships of the surviving substance to its future environment" (PR 311).

Thus it seems that any explication of ‘valid inductive inference’ requires as a necessary condition the metaphysical doctrine of internal relations. This doctrine is a necessary condition for the limitation of independent variety, which is, in turn, a necessary condition for ‘valid inductive inference’ to predictions and to theories.

The intent of this paper has not been to provide a metaphysical justification of induction but, rather, to attempt the very limited task of elucidating what seems to be a necessary condition of ‘valid inductive inference’ patterns and to insist that metaphysical doctrines are not irrelevant to the understanding of ‘valid inductive inference’. Certain necessary conditions for ‘valid inductive inference’ (the limitation of independent variety) are grounded in further necessary conditions (internal relations) which constitute metaphysical presuppositions.



1. C. Hempel. "Studies in the Logic of Confirmation," reprinted in Aspects of Scientific Explanation. New York: The Free Press, 1965. Pp. 3-46.

2. Mary Hesse. "Consilience of Inductions." The Problem of Inductive Logic. Edited by I. Lakatos. Amsterdam: North-Holland, 1968. Pp. 232-57.

3. Mary Hesse. "An Inductive Logic of Theories." Minnesota Studies in the Philosophy of Science, IV. Edited by Radner and Winokur. Minneapolis: University of Minnesota Press, 1970. Pp. 164-80.

4. Mary Hesse. "Positivism and the Logic of Scientific Theories." The Legacy of Logical Positivism. Edited by Achinstein and Barker. Baltimore: The Johns Hopkins Press, 1969. Pp. 85-114.

5. John Maynard Keynes. A Treatise on Probability. London: Macmillan and Co., Ltd., 1948.

6. Hilary Putnam. "‘Degree of Confirmation’ and Inductive Logic." The Philosophy of Rudolf Carnap. Edited by P. A. Schilpp. La Salle, Ill.: The Open Court Publishing Co., 1963. Pp. 761-83.



1 The important internal relationship in the valid inductive inference pattern is not the relationship of prehension as Gutting suggests (PS 1:174); rather, the relevant internal relations axe those between entity and environmental order. The entities of an environment contribute to the order and yet cannot exist without that order. The clearest summary of this is, I think, to be found in SMW 215; here, Whitehead says anything is "otherwise than what it would have been if placed elsewhere," i.e., if in a different environment. It would be otherwise because the relations between entities and their environments are internal ones. I do not mean to suggest that environmental order does not depend on prehension for Whitehead. I merely wish to point out that the internal relationship of entity and environment which is required for any ‘valid induction’ may be a doctrine of a metaphysical theory which does not include the doctrine of prehensions. See, e.g., D. Bohm, "Some Remarks on the Notion of Order" and "Further Remarks on Order," in C. H. Waddington, ed., Towards a Theoretical Biology. 2, Sketches (Edinburgh: Edinburgh University Press, 1969), pp. 18-40 and pp. 41-60, respectively.

2A. Lowry, "Emergence," forthcoming in Mind.