My 1983 Comments on Circumstantial Evidence and "Inference-on-Inference"

For the most part I stand by the comments I made in 1983 on circumstantial evidence and the supposed but largely-nonexistent prohibition against inference on inference:

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." ...

[fn3] ...

[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).

A fair assessment of the strengths and weaknesses of Morgan's theory of catenate inference first requires that an important modification of that theory be made. That modification (to be described below) is consistent with the general tenor of Morgan's theory; without that modification, Morgan's theory is not even plausible.

Morgan and his followers have assumed that the probability of a final inference that stands at the end of a chain of inferences (a derivative inference) is exactly equal to the multiplicative product of the existence of the immediately preceding inference (the supporting inference) and of the probability of the existence of the derivative inference on the assumption that the existence of the supporting inference is certain. (It also follows, on his view, that any supporting inference that is itself derived from an earlier supporting inference has a probability value that is obtained by the same type of computation.) As plausible as this theory sounds, however, it is almost surely wrong if one takes the view that the probative force of any item of evidence amounts to a measurement of the extent to which that item of evidence increases or decreases the probability of some fact in issue. If one takes the latter view, the key question is not how often A occurs when B occurs but rather how much more often does B occur when A occurs that it does when A does not occur. It is this comparison that permits us to determine whether the occurrence of B increases the probability of the occurrence of A. If one takes this (Bayesian) view of the manner in which the probative force of a piece of evidence is measured, however, the manner in which the probative force of any piece of evidence (whether or not it belongs to an inferential chain) is significantly altered, for on this view it is a gross fallacy to assume that the probability of A, given B, is simply the ratio of As to Bs. Instead, the comparison of the frequency of Bs, given As, and the frequency of Bs, given that As do not exist, produces a "likelihood ratio" (see Section 37 supra) that measures the extent to which our prior estimate of the odds of the existence of A are increased or decreased by the evidence in question (here, B). (As it happens, this likelihood ratio is described by the formula P(B|A) over the denominator P(B|-A); thus, precisely phrased, the comparison to be made is between (1) our estimate of the frequency of the occurrence of B when A is thought to exist and (2) the frequency of the occurrence of B when A is not thought to exist. See Section 37.6 supra.) If one accepts this basic Bayesian view that the probative force of evidence is a measure of its incremental probative force on some fact in issue, some interesting and surprising results eventuate, all of which significantly diverge from Morgan's central assumption that the probability of A, given B, is simply a matter of computing how often A occurs when we know that B occurs. Most pertinent to the present point, it turns out, as Schum has shown, that uncertainties with regard to the existence of B have an effect on the likelihood ratio with regard to the odds of the existence of A (the fact in issue) in a way that is not at all described by multiplying the frequency of As, given Bs, times the frequency of Bs, given Cs (where C is the initial evidence in the inferential chain that leads to B).

Instead, it turns out that very small uncertainties with regard to the existence of B — what Schum calls "source uncertainties" — can have a large impact on the likelihood ratio, viz., can diminish to a large extent the (incremental) probative value of any evidence whose existence is not itself certain but whose existence is itself probable and is inferred to some degree of probability from evidence found earlier in this chain of inferences. As it happens, a small measure of uncertainty about the existence of a penultimate inference — in our terms, a measure of uncertainty about the existence of an inference because of its derivation from a prior inference — has this seemingly inordinate impact on the probative force of the penultimate fact or inference, B, when B, in the absence of any such uncertainty about its existence, would produce a very substantial likelihood ratio, viz., would have great probative force. In other words, very slight doubts about the existence of what would be extremely probative evidence can diminish its value to a tremendous degree. See Schum & Du Charme, Comments on the Relationship between the Impact and the Reliability of Evidence, 6 Organizational Beh. & Hum. Performance 111 (1971). Schum shows that various types of uncertainties in what he calls the "source" may have various effects on the probative force of evidence; and, as it happens, none of these effects is accurately described by Morgan's simple multiplicative method.

Whether or not one accepts the Bayesian modification of Morgan's analysis of inference upon inference, it is at least evident that Morgan's views about how probabilities along a chain of inferences are to be combined are problematic. However, even if one accepts the need for a Bayesian modification of Morgan's thesis (which seems to presuppose some sort of a frequentist theory of probability; see Section 37 supra), the heart of Morgan's theory remains intact since under either approach there is agreement about the need for some type of discounting of the probative force of evidence by the use of some formula that takes into account the uncertainties that are found to exist in the chain of inferences that ultimately leads to the evidence bearing on the fact in issue.

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.

Before we turn to a brief examination of our general thesis concerning the problematic character of the chain metaphor, we shall first attempt to substantiate our thesis by indirection, by showing that there may be relations of interdependence within a chain, not at first apparent, that may make the chain metaphor inappropriate. The illustration we offer is meant to suggest the general lesson that what appears to be a chain will often, on closer inspection, turn out to be something quite different. After discussion of this specific illustration, we will turn to more general considerations that suggest why the metaphor of a chain, and the methods of analysis and computation that that metaphor suggests, are ordinarily, if not always, misleading and dangerous.

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.

This problem is peculiar, to be sure, but perhaps not as peculiar as it seems, since, in fact it may and often does happen that persons about whom predictions are made will alter their behavior in some way that is related to the kinds of predictions made; and in a society governed by various truth-constraints, it may be that in a certain percentage of cases a person's predictions about the behavior of others will reflect the true beliefs of the person making the predictions, which are based on the available evidence. (It may be noted that examples of the sort of perverse behavior shown by the actor in the example are not unique.

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.

The example of the perversely behaving gunslinger is useful because it suggests, quite rightly, that Morgan-like analysis of inferential chains applies only if inferential chains can somehow be held separate from lateral relations of interdependence, as well as from (in our hypothetical) vertical relations of interdependence, that may exist along (so to spjeak) the length of the chain. The Morgan-like analysis holds only if we can prevent the links of a chain from interacting with one another. Similarly, that analysis holds only if separate inferential chains can be kept apart. If that cannot be done, the Morgan-like analysis fails, or at least becomes so attenuated and complex that there is no reason to suppose either that any factfinder could actually use Morgan-like reasoning in any conscious fashion (see Section 37 supra) or that any factfinder somehow uses such a complex process implicitly, without knowing that he is doing so (see again Section 37 supra).

Returning for a moment to the more technical aspects of Morgan's analysis, it is worth remarking that a Bayesian modification of Morgan's analysis inexorably leads to the conclusion that the existence of corroborating inferential chains with regard to a fact in issue entirely eliminates the propriety of supposing that every increase in the probability of FP-1 has anything like a direct impact on the probability of FP-2. In this situation, involving corroborative chains of evidence, odd-shaped curves and graphs may have to be constructed to depict the eventual impact of FP-1 on FP-2. See 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). The complexities of analysis grow apace if one considers in addition to the problem of corroboration, the problem, already discussed, of measuring the impact of "source uncertainties" on the probability of FP-2. The complexities of analysis further increase (as shown in Section 37.6 supra) when it is supposed that one piece of evidence bearing on a fact in issue to some degree duplicates information by another piece of evidence bearing on the same fact in issue. (In the parlance of probability theory, this is the problem of dependent conditional probability.) When this happens, the force of one piece of evidence must be discounted to some degree to account for the extent of the duplication involved. We can only imagine (faintly) the further difficulties of analysis and description that may arise if one also supposes that the evidence presented in one inferential chain simultaneously happens to affect the force of generalizations that are pertinent to the assessment of the force of an inference in one link of another inferential chain. Suppose, for example, in a negligence action the issue is whether the floor in a supermarket was wet. The first piece of evidence offered is that it was quite warm in the supermarket. The factfinder believes that this tends to show that the floor was wet because he believes that floors are generally more likely to be wet when the air is warm than when it is dry. The plaintiff, however, offers evidence that shows that ten other people fell at the spot in question in the supermarket within a ten day period; the theory on which such evidence is offered is that falls in such frequency are unlikely to occur in the absence of some negligence or malfeasance on the part of the supermarket. The proponent asks the factfinder to also infer that water was probably the cause of the accident on which the suit is based since he offers evidence that in those prior accidents there was water on the floor. As it happens, however, those prior falls took place on warm days. The factfinder therefore discounts his earlier hypothesis about the relationship between warm weather and wet floors, and thus the force of the earlier evidence is diminished. In short, the probative force of separate inferential chains may depend on logical relations and implications of the theories that support the separate inferential chains and on the way those implications relate to other generalizations and theories involved in the analysis of other inferential chains.

It is not useful to multiply examples of the complexities that are occasioned by efforts to employ a Morgan-like analysis of inferential chains. To do so might give the impression that we believe that something like a Bayesian perspective on inference and probability assessments is the correct one. We do not take such a view of either Bayesian analysis or of Morgan-like analysis, at least not insofar as either of those methods purports to constitute a description of the general character of all inferential processes. We have seen that the use of such methods of analysis leads to enormous complexities, which prevent any facile conclusions about the appropriate methods to be used for combining inferences and probability assessments from various types of evidence in various circumstances. The moral we draw from our story in this note is not that we as yet have an inadequate understanding of analytical techniques of a Bayesian sort; rather, we draw a different moral, a moral that explains why we have discussed Morgan and Bayesian analysis in one breath. Our moral is that any method of analysis that pretends to yield a prescriptive account of inferential processes will fail entirely if that analysis demands that all conditions of dependence among various items of evidence and between evidence and other things (such as beliefs) be exhaustively, systematically, and precisely described. There are simply too many potential relations of interdependence to allow for the possibility of any such comprehensive account. Thus, for example, as already argued in Sections 37.6 and 37.7 supra, it is highly doubtful that it is ever possible, in principle, to state fully or exactly all the generalizations and "background information" (including general and specific hypotheses about how the world works) that are pertinent to any particular inference. If we are right, the view that a probability assessment at any step in the inferential chain is a direct function of a stated generalization necessarily misstates (to some necessarily indeterminate degree) the force of the inference that is warranted at any particular stage of the inferential chain; and, of course, if we are right, it is patently wrong to quantify the level of uncertainty that is generated by the stated generalization alone and to combine it in some computational fashion with similarly infirm (and distorted) probability assessments found to hold (as a result of some stated generalization) at some later point in the inferential chain.

In response to what we say about Morgan-like analysis of evidence — a mode of analysis that relies heavily on the dissection of evidence into pieces in an effort to discern interrelationships — it might be said, by way of confession and avoidance, that there are, to be sure, limitations on the capacity of analysis, but it is still worthwhile to be as systematic and as precise as we can be. And there is no doubt, of course, that this is a powerful rejoinder. But whether it is an entirely adequate rejoinder seems doubtful to us, for if the limitations on our analytical capacities mentioned above are assumed, it seems to us that neither the Morgan-like nor the Bayesian analytical technique can offer a basis for asserting that the method by which it organizes data and evidence is superior to any other systematic method that similarly requires that we pay attention to detail. (One may examine evidence systematically by attentively reading, in order, all the testimony given at a trial.) Our ruminations on the complexities generated by Morgan-like and Bayesian analysis suggest that, in some cases, care should be taken to examine evidence in ways that keep in the forefront, rather than in the background, the question of the manner in which the various pieces of evidence relate to each other and the kinds of general visions or beliefs that shape and should shape our interpretations of a mass of evidence. (The danger, to revise James's dictum, is that we will overstate or overestimate the degree to which our minute analytical analyses support the inferences we reach.) The importance of looking at the parts and the generalizations that relate to each of those parts must not overshadow the importance of looking at the whole and at the kind of general notions that we may use to tie the parts together in a meaningful way. Various parts of a whole can be put together in quite different ways to tell quite different stories. Consider, for example, a case of libel in which the evidence largely centers on the actions of the defendant and the motives and purposes he sought to achieve by uttering the various things he is said to have uttered and the things he might have uttered. At question in this trial is why the defendant uttered what he did and the question of what he did utter and just how he uttered whatever he did utter. In this sort of trial, the various bits of evidence presented will in subtle ways serve to establish a certain kind of general image of the alleged malfeasor, and we will use this image to evaluate much of the evidence submitted and its force. As the evidence is submitted, piecemeal, we may find that we, the factfinders, are constantly revising our image of the defendant and, as we do so, are constantly returning to the evidence earlier presented in order to see, first, whether it fits the image we have and, second, to see whether we have properly interpreted just what that man "did do" (did he sneer or smile?). This examination, in turn, will make us review the general image we have of that man and will make us revise that image, so that it somehow better fits the image we now (tentatively) have of the type of man involved. In this kind of trial, the interpenetration of theory and fact is almost complete and the separateness of different inferential chains, as well as the composition of the links in the inferential chains, is as much a product of our theorizing as it is of our examination of what we (initially) take to be separate pieces of evidence. In this kind of process, it is most difficult to give overweening significance to separate pieces of fact, and there is no reason we should do so. In reality, it seems that the facts somehow manage to generate in us certain general images and theories and that these general images and theories are then used to review the facts, in order to reconstruct them and interpret them in some sensible fashion. We may suppose that this sort of interaction between fact and theory, in this sort of trial, continues until, at some indefinable point, the factfinder feels that he has reached the limits of his imagination and understanding. For our own part, we have no reason to suppose that this sort of process is not well designed to produce reasonable estimates about what probably really happened, and we are inclined to think that this process is a good one. Hence, for all the reasons discussed here and in Sections 37.6 and 37.7 supra, we think it is important to discard all analytical efforts that somehow serve to deflect people from this most natural, and possibly most rational, method of getting at the truth about those sometimes-elusive "facts." We wonder why it was ever thought that factual questions have "hard edges"; the imagery quite belies the reality.

The result of this rather extensive discussion is a relatively modest conclusion. Perhaps it is useful in some way to see inferences as belonging in chains and to think of them as chains and to analyze them in that fashion, but it we do use some sort of a "chain" analysis to think about some problems of evidence, we should of course recall that the reasoning we use depends, in great measure, on oversimplification and that when we see a chain before us we are presupposing an enormous number of matters and disregarding the possibility of a great many interrelationships with other matters, both evidentiary and conceptual. Accordingly, we should keep in mind that any analysis using the chain metaphor is highly provisional and tentative; perhaps the chain metaphor, at best, should constitute a kind of visual or mental aid, and little more. Furthermore, we think it equally important to remember that from time to time it is wise to sit back and take a "holistic" or impressionistic approach to the evidence in an attempt to see how the various pieces fit together given the kind of general theory or story that those various pieces, as we see them, seem to presuppose. Then, through a series of reciprocal interactions, we can do what we usually do: We try to somehow make the various pieces of evidence and the general stories we have devised fit together to make a plausible and consistent whole.

In connection with this discussion of the problem of catenate inference, we note that a prominent logician has taken the view that, at least in some contexts, it is rational — and perhaps desirable, — to require that foundational inferences (inferences supporting later inferences in the chain) be established as practically certain and that only when this is done is it legitimate to rest subsequent inferences upon the prior inference. Compare discussion in the text infra. While that author draws support for this view from what he views as conventional legal treatment of catenate inference and from a reading of this section in the third edition of the Treatise, his approach does not entirely depend on the law; he also offers independent grounds (which he describes as being founded in "Baconian" logic) for his point of view. See generally Cohen, The Probable and the Provable (1977) (especially § 22, § 23, and § 72).

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The dynamic evidence page

It's here: the law of evidence on Spindle Law. See also this post and this post.