1A Wigmore on Evidence Section 41 (P. Tillers rev. 1983):
The following well-reasoned opinion [Wigmore wrote] shows a correct way to avoid the fallacy of rejecting an inference upon an inference and yet to give effect to the underlying distrust of inferences that rest upon too many intervening inferences:
New York Life Ins. Co. v. McNeely, 5 Ariz. 181, 79 P.2d 948, 953 (1938) ....
[I wrote:] A number of courts have used an approach similar to that found in _McNeely_; while allowing inferences to be based on inferences, they have said, variously, that the underlying inference must be shown as a "fact" or that the underlying inference must be the most plausible or reasonable of the inferences available.[fn5] Arizona itself, however, seems to have abandoned any requirement that the underlying inference be shown beyond a reasonable doubt.[fn6]
Wigmore's praise of the approach taken in McNeely should not be taken too literally. The briefest examination of Wigmore's Science of Judicial Proof (3d ed. 1937) shows that Wigmore did not for a moment believe that each underlying inference must be shown beyond a reasonable doubt (much less to a certainty). Wigmore's entire analysis in Science of Judicial Proof and in Sections 24 through 36 of this Treatise rests on the premise that single inferences, though weak when taken individually, may be substantial and powerful when added together (see Section 9, Section 10, Section12, Section 26, and Section 39 supra; but cf. Section 28 supra). The elaborate "chart method" used by Wigmore in Science of Judicial Proof clearly shows Wigmore's understanding that the probative strength of an underlying inference is a factor that affects the strength of the final factum probandum but that no mechanical rule can be laid down concerning how strong any underlying inference must be. The question is not whether any given inference in a chain is too weak but is always whether, in view of all patterns of corroborating and contradicting evidence at all levels of all inferential chains, the final factum probandum has been shown to the degree of likelihood required by the applicable standard of persuasion, whatever that standard may be. To make the sufficiency of any case depend on the strength of any single inference commits again the fallacy of "legal relevancy," so recently roundly condemned (see Section 37 supra) by the proponents of the now-popular and now-dominant theory of "logical relevancy." ...
[I wrote:] The supposed rule [against an inference on an inference] is incisively discussed in Bishin & Stone, Law, Language, and Ethics 289-291 (1972). See also Morgan, Basic Problems of Evidence 188 (1961) (in accord with Treatise); Cohen, The Probable and the Provable 68-73 (1977) (short chapter entitled The Difficulty about Inference upon Inference; Wigmore's views discussed; Cohen believes that the law requires that the initial factum probandum in a chain of inferences be established beyond a reasonable doubt; in fact, however, most legal authority does not expressly assert any such requirement).
Today most students of the problem of inference recognize that any single vision about the world or conclusion of fact rests on a multitude of inferences, premises, and beliefs, on a large complex of assumptions, and on a body of implicit or explicit principles by which the human organism perceives, organizes, structures, and understands experience; thus it is generally conceded that it is meaningless to denounce multistaged or cascaded inferences. See generally Section 37 supra, and see also the reviser's comments, with citations, in Section 24 supra. A belief in the ability to reach conclusions on the basis of a single inference merely reflects a lack of imagination and insight. Furthermore, the implicit character of many inferential steps does not render them invulnerable to attack. We cannot be sure that implicit inferential steps are reliable merely because they are made unself-consciously.
While it is now regarded as practically indubitable that the drawing of inferences from inferences is the natural and inevitable course of things -- without which the drawing of any inference is a practical impossibility -- it is very difficult to describe the precise kinds of processes involved in the cascading of inferences.
The usual analysis of catenate inference assumes that the top of the inferential chain, the final factum probandum, is always weaker than the bottom of the chain. See Section 37 supra. This view implicitly assumes the existence of what has been called the "transitivity of doubt"; it is assumed that the measure of doubt at the lower levels of the inferential chain is transferred to the upper levels of that chain (since it is assumed that the superstructure can never be stronger than its foundation and, indeed, must always be weaker by some measure that is directly related to the strength of the foundational inference). (This view, of course, disregards independent corroborative chains of evidence that intersect inferences at the level of the superstructure rather than at the level of the foundation.)
The sort of imagery used above is very powerful. However, it is not entirely clear that the assumption of transitivity should always apply.
Whether the assumption should apply depends to some degree on our willingness to assume that we can consciously discern the foundations of our ultimate inference. If one supposes, as we do (see Section 37 supra), that the premises of our inferences are not always apparent to us and that their explicit formulation may lead to an unwarranted discounting of their force, it is of course apparent that it is not always appropriate to discount "subsidiary" facta probanda by some factor related to the degree of doubt we entertain with respect to the validity of the foundational inferences we have formulated. (In part, we mean to assert that it is questionable whether we are in fact capable of stating the foundations of our pyramided inferences.)
It is also possible that the imagery of pyramided inferences may draw a false picture of the fashion in which we shape inferences from inferences. The image of tiered inferences, each inference resting on another inference, seems to disregard the possibility that all inferences to some degree rest on holistic thinking of some sort, in the sense that no inferential chain is completely independent of any other inferential chain (just as no inference can be independent of any other inference with respect to the same factum probandum). This reciprocal relationship among chains of inferences may amount to more than just the usual notion of the convergence of corroborative chains of evidence toward a common factum probandum (whether intermediate or final); it is possible that a supposedly distinct inferential chain works backward, as it were, by making us rethink the character of the other inferential chain and the nature of the probability relations between the original evidence and the first inference or between intermediate inferences and successive inferences. (The picture painted by another inferential chain, in short, may make us redefine the very nature of a particular inferential chain.)
The theory of catenate inferences, if accepted as legitimate, poses other intricate problems within the terms of the theory itself. Even if we assume the transitivity of doubt, it is far from clear — indeed, it is almost certainly not true — that the right way to compute the degree of the uncertainty (viz., the probability) of a derivative inference is to multiply the prior probability of the derivative inference (viz., the assumed probability of the derivative inference, given the assumption that the foundational inference exists) by the probability of the foundational inference. (This sort of multiplication was apparently advocated by Morgan in his analysis of tiered inferences. See Section 37.4 supra.) In fact, it is quite possible that an intermediate foundational inference having a probability of less than .5 (but more than zero) will add nothing to the probability of the final factum probandum. See Section 14.1 infra. Conversely, the measure of uncertainty present when one link in an inferential chain has the high probability value of, for example, .95 may become greatly magnified under certain conditions, so that the decrease in the probability of the final probandum is far greater than that which would be obtained by the simple process of multiplying the probability values all along the chain of inferences. See generally, Schum, A Bayesian Account of Transitivity and Other Order-Related Effects in Chains of Inferential Reasoning, Rice University Research Report No. 79-04 (Dec. 30, 1979) (thus, for example, Schum argues that small reductions in the credibility of a witness have large negative effects on the probability of the event being testified to when the witness is testifying about the occurrence of a rare event). The formal analysis of the effects of tiered inference becomes substantially more complex when one considers the common problem of separate inferential chains that converge (to some measure) toward a common factum probandum. See generally Schum, A Problem in Cascaded Inference: Determining the Inferential Impact of Confirming and Conflicting Reports from Several Unreliable Sources, 10 Organizational Beh. & Hum. Performance 404 (1973). There are many reasons we are not always free to assume that there is an additive relationship between the probability statements for the same factum probandum established by independent inferential chains, and a variety of factors may affect how the probative worth of the two chains should be added together if there is in fact an additive relationship. For example, consider a situation in which it is not permissible to assume that if independent pieces of evidence separately make some final and common factum probandum more probable that, taken together, these separate and independent chains make the final factum probandum even more probable. The relationship of additivity may be destroyed by the logical relationships between elements of the two chains. Thus, for example, one inferential chain may have probative force only on the assumption that the accident in question occurred at one time and the other inferential chain has probative force only on the assumption that the accident occurred at a different time in a very different manner. If so, it clearly will not do to simply add these inferential chains together; instead, it seems, one is compelled to make a choice between them. Even if this peculiar relationship between elements of two inferential chains does not exist, the formal analysis of the effects of corroborating chains of inference involves a number of other complications, and these complications may have a powerful effect on the degree of the additivity of two separate chains. Thus, for example, it is of course evident that if we think there is some chance that the testimony of one witness to event A is affected in some measure by the testimony of another witness to event A and the question is how much the testimony of both witnesses should count on the question of the existence of event A, it will not do to simply add together (in some appropriate way; see this note infra) the probative force the testimony of each witness has when considered separately; one must take into account the extent to which the testimony of the one witness is conditioned by the fact of or the tenor of the testimony of the other witness.
Professor Morgan's theory of "catenate inference" (Wigmore's phrase) or "cascaded inference" (Schum's phrase) is by and large still the most common theory espoused by judges and by writers of treatises and text-books. Closer inspection of that theory is therefore warranted. A theory of catenate inference affects a wide variety of problems, including matters such as conditional relevancy (see Section 14.1 supra), the appropriate treatment of real evidence, and the methods to be employed for determining the probative force of evidence (a determination required, for example, to decide whether evidence is unduly prejudicial; see Section 10a supra).
Nonetheless, Morgan's thesis, even when taken in its most general sense, has other difficulties. While it is hardly doubtful that any assessment of evidence and its probative force must take into account in some way whatever uncertainty happens to pertain to the assumptions, inferences, beliefs, and so on that seem to be involved in the assessment of the value of any given item of evidence, it is not equally clear that the image of a chain of inferences, with each inference depending in some way on the strength of a prior inference, is an image that accurately portrays the manner in which the inferential process works. Although it would be rash to say that no series of inferences ever takes the form of a chain, it may be that the chain metaphor is applicable far less often than is usually supposed, and it also may be that efforts to portray the inferential process in this way are almost invariably intrinsically bound to produce serious distortions in our understanding of the inferential process and thus perhaps may impair our ability to assess the probative force of evidence in a proper fashion. The basic reason we suspect that the metaphor of a chain is misleading is that it tends to neglect what we believe to be almost universal relations of interdependence that exist between any so-called independent inferential chain and various ... things apart from that chain that necessarily have a bearing on the probative force of the chain. We believe that these relations of interdependence are so pervasive that it is misleading to describe any series of inferences as an independent chain of inferences that may be considered in isolation.
To the extent that it is presented above, the Bayesian modification of Professor Morgan's analysis of an inferential chain does not challenge his central thesis that there is a continuous relationship (if not a one-to-one correspondence) between the strength of any evidence standing at the top of a chain and the strength of inferences falling below the apex, supporting it. Even if that relationship is not linear, it is continuous and the probative force of any evidence derived from prior inferences may be mathematically conceived as a function of the strength of those prior inferences. In some cases, however, we may discover, perhaps counterintuitively, that the probative force of a derivative inference is not continuously related to the probative force of its supporting inferences, even though the probative force of the ultimate derivative inference is related in some fashion to its supporting inferences and even though, in a real sense, the exact probative force of the final inference is genuinely dependent on the strength of preceding inferences in the chain. For the sake of convenience, let us call the penultimate inference in the chain FP-2. Let us say that the fact in issue, toward which the chain is eventually directed and of which FP-2 is evidence, is called A. Let us also say that the proposition or inference that gives rise to FP-2 is called FP-1. And let us say that the evidence giving rise to FP-1, the supporting inference, is called F. Using this terminology, we say that it may happen that as we increase our estimate of the probability of FP-1, the probability of FP-2 will not always continuously increase (or decrease) as FP-1 increases, even though the probability of FP-2 is a derivative of the probability value attached to FP-1.
If we adopt the Bayesian modification of Morgan's analysis of inference upon inference (as described above in this note), it is easy enough to visualize examples of the sort of inferential chain Morgan had in mind. Thus, for example, suppose that we believe that when A thinks it will rain he takes his umbrella to work 80 percent of the time and that he takes his umbrella to work 40 percent of the time when he believes that it more likely than not will not rain. If the fact is shown that A thought it will rain, the evidence of this fact — viz., the fact itself — clearly has probative value for the question of whether A took his umbrella to work. Morgan's claim, however, is that in many cases we cannot know whether the actor (for example) thought it would rain and that in many cases we are uncertain whether this is so. In these cases, then, we must discount the probative force of the evidence by taking into account that the evidence only shows that it is likely that the actor had such a belief and by taking into account that the evidence does not show for certain that the actor had such a belief. How much do we discount the evidence because of this sequence of inferences (evidence belief it will rain taking of umbrella) from the evidence? Morgan's answer, as modified here, is that there is a direct and continuous relationship between the probability that A took his umbrella and the ratio of the relative frequency with which a belief of the probability of rain is present when certain evidence is present and the frequency with which a belief of the probability of rain is present when such evidence is not present. Thus, for example, if the evidence shows that the actor told his wife, "It will rain" and we believe that such statements are made in 40 percent of the cases in which the actor does in fact believe it will rain and that the actor makes such statements in only 1 percent of the cases in which in fact he does not believe it will rain, the evidence of the statement clearly has probative value for the question of whether the actor took his umbrella with him. Morgan's thesis, as modified here, is that the likelihood of the taking of the umbrella is directly and continuously related to this ratio. For example, the probative force of the statement "It will rain" with respect to FP-2 becomes exactly half of what it was before if we revise our estimate of the ratio with which the actor makes such statements, so that we now assert that he makes such statements in only 20 percent (and not 40 percent) of the cases in which he in fact believes it will rain. In short, as we revise our generalizations in a way that tends toward the conclusion that the actor says "It will rain" with equal frequency when he believes it will rain and when he believes it will not rain, the evidence of the making of the statement "It will rain" loses probative force for the question of his belief and, thus, revision of generalizations in this direction has a direct and continuous impact on the likelihood that the actor took his umbrella with him.
This example of evidence of rain and the assessment of its impact on the question of the taking of an umbrella seems to offer persuasive support for Morgan's general thesis. In fact, however, the example only illustrates that in some cases a Morgan-like analysis may be right; it falls far short of showing that a Morgan-like analysis is always right.
Suppose that the matter in issue, the final matter, is whether X will take his gun to work. The final factum probandum, therefore, is "X will take his gun to work." Call this FP-2. We wish to know the probability of FP-2. What is the likelihood X will take his gun to work? The hypothetical we pose, however, is curious because, as we imagine the situation, our estimate of the likelihood that X will take the gun to work will be affected by our estimate of the ways in which X usually acts when we make another estimate (as well as the way he acts when we do not make that other estimate). In short, we have created a situation in which that person's behavior (we believe, to a certain probability) is altered by what we, the factfinder, estimate about something else, viz., the probability estimate we make of something else. In doing so, we have established a relation of dependence between one estimate (our estimate of the probability that X will take the gun to work) and another estimate (of the likelihood of something else, that "something else" to be described momentarily). By hypothesis, then, our estimate of the probability of FP-2 is "derivative" from the probability of FP-1. However, as we choose to construct our hypothetical, there is no direct or continuous relationship between the likelihood of FP-2 and the likelihood of FP-1. This we do simply by saying that the actor's behavior (most frequently) is altered when and only when the odds of FP-1 reach a certain value, say fifty-fifty, and that otherwise (we estimate, by some generalization) the actor (probably) does not alter his behavior, viz., he does not permit our estimate of FP-1 to alter (we think) the frequency with which he takes the gun to work under various conditions. Therefore (we conclude, because of what we believe his behavior is likely to be in response to our estimates) many variations in the likelihood of FP-1 have no effect on the likelihood of FP-2. To make this example concrete, suppose that we suppose that X (we believe), will not take his gun to work in 90 percent of the cases in which we inform him of our previously private estimate that he will take the gun to work. (X's aim you see, may be to avoid the possibility of being caught.) We assume, for the sake of simplicity of description (though, for present purposes, it makes no difference in principle), that we, the factfinder, feel bound (for ethical reasons, let us say) to disclose to X our original private estimate of whether he will or will not take the gun to work. This estimate (by the reviser's stipulation) does not concern (what we believe) are X's probable responses to our original estimate but relates solely to what we think he probably would do if we made no disclosure of our private estimate of his probable behavior. Now to complete our hypothetical and our demonstration, all we have to do is to suppose that X, to a certain likelihood, probably will not vary his behavior in response to our original estimates in any way unless and until our estimate of probability crosses the fifty-fifty threshold, viz., X will act as we describe above only when we make the statement that it is more probable than not that he will take his gun to work. In addition, assume the factfinder estimates that X will not further alter his behavior if the fact-finder states (for example) that it is very likely that X will take his gun to work.
In the above example we maintain that the probability of FP-2 is in a very real sense dependent on the probability we attach to FP-1 (a person's taking his gun to work when certain evidence is present and when the person is not informed of FP-1), and yet it is evident that the likelihood of FP-2 is not continuously related to changes in FP-1. Hence, we have given the lie to any Morgan-like thesis about the nature of the relationships between the probabilities of facta probanda in an inferential chain. It is of course true that one may avoid this result by using the semantic trick of saying that our example is not a "true" example of an inferential chain since — so the argument goes — the only evidence of any pertinence is evidence showing whether the factfinder said he probably would take the gun to work and by saying that in this situation any chain would arise only insofar as there is a measure of uncertainty as to the making of any such statement, in which event the FP-2 should be discounted by the usual Morgan-like rules described above. But, we maintain, to take such a course is nothing more than a semantic trick, for in fact a descriptive account of how any real factfinder would in the end assess the probability of FP-2 would have to say (under the constraints we have established on the factfinder's behavior) that the factfinder does think that the likelihood of FP-2 is affected by (though not determined in all instances by) the likelihood of FP-1 (because the fact-finder, by our stipulation, feels bound to express that estimate, in every case, as he believes it). Thus, in this factfinder's mind, in this situation, the likelihood of FP-2 really is dependent (in his mind) on his estimate of the likelihood of FP-1 (in his mind), and it is therefore appropriate to speak of an inferential "chain." And yet, of course, it is clear that not every increase or decrease in the likelihood of FP-1 is continuously or directly related to increases or decreases in the likelihood of FP-2.
Consider the field of intelligence work, in which opposing sides attempt to deceive each other and in which each side attempts to determine the rules of deception the other side presumably follows. This sort of problem is the subject of "signal theory.") However, whether or not the problem is peculiar, it suffices to lend credence to our general thesis, which is that close inspection of an inferential chain may show that there are relations of interdependence among the links of the chain, so that it cannot be said that a final inference is continuously related (whether positively or negatively) to increases in the probability of a supporting inference. Thus we believe the example lends some credence to the even more general hypothesis that in many situations involving putative inferential chains there may be relations of interdependence which either make the nature of the chain far more complex than it may appear at first glance or which, at certain extremes — if only because of the complexity of calculation required to maintain the image of the chain — may render the chain metaphor almost entirely worthless. Worst of all, the complexity of the relations of interdependence are so great that the effort to portray a series of inferences as a chain will lead to serious distortions in understanding and evaluation. It may also be the case that the very complexity of the description required to maintain the chain metaphor and to make it tenable, either descriptively or prescriptively, suggests that, practically speaking, there is no good reason to hang on to the metaphysical or epistemological assumptions that lend force to the chain metaphor.