
Legal Application: Conjunction Paradox
The payoff of the fuzzy logic approach emerges as one realizes how it affects the view of the proof process. Consider the bestknown statement of the infamous conjunction paradox:
We purport to decide civil cases according to a moreprobablethannot standard of proof. We would expect this standard to take into account the rule of conjunction, which states that the probability of two independent events occurring together is the product of the probability of each event occurring separately. The rule of conjunction dictates that in a case comprised of two independent elements the plaintiff must prove each element to a much greater degree than 50%: only then will the plaintiff have shown that the probability that the two elements occurred together exceeds 50%. Suppose, for example, that a plaintiff must prove both causation and fault and that these two elements are independent. If the plaintiff shows that causation is 60% probable and fault is 60% probable, then he apparently would have failed to satisfy the civil standard of proof because the probability that the defendant both acted negligently and caused injury is only 36%. In our legal system, however, jurors do not consider whether it is more probable than not that all elements occurred in conjunction. Judges instruct jurors to decide civil cases element by element, with each element decided on a moreprobablethannot basis. Once jurors have decided that an element is probable, they are to consider the element established, repress any remaining doubts about it, and proceed to consider the next element. If the plaintiff proves each element by a preponderance of the evidence, the jury will find in his favor.... Thus, jurors may find a defendant liable even if it is highly unlikely that he acted negligently, that is, the conjoined probability of the elements is much less than 50%. In such cases, the verdict fails to reflect a probable account of what happened and thus fails to minimize the cost of judicial errors.... .... ... Although courts direct juries to consider and decide each element seriatim, juries do not consider each item of evidence seriatim when deciding whether a given element is proved. The jury must decide each element by looking at all of the evidence bearing on proof of that element. Thus, although the jury does not assess the conjunction of the elements of a case, it does decide each element by assessing the conjunction of the evidence for it. (113) The implications are profound but boggling. Allowing recovery on a 36% showing of causation and fault is not only unfair but inefficient. How embarrassing for the law!
For another boggle, ponder the apparent criticality of how exactly the ancients (and moderns) divided our causes of action and defenses into elements: the more subdivisions, the lower the conjunctive probability that would produce victory. (114) And yet:
Anyone who has ever litigated a real case knows the exact opposite of the conjunction paradox is true: the more disputed elements the plaintiff has to prove, the less likely the plaintiff is to prevail .... [A]lthough it is possible that a particular plaintiff could obtain an unjust verdict in a case with several disputed elements, [there is an increased] probability that the jury will find at least one element to be less likely than not. (115) Admittedly, the conjunction paradox turns out to be not such a serious problem in practice. Only one element might be in dispute, or the disputed elements might not be really independent. The judge might not clearly state, or the jury might not fully understand, the proper elementbyelement approach to the standard of proof.
Or, because humans might tend to construct a story for the whole case rather than proceeding elementbyelement, the factfinder might end up applying the standard of proof to the conjoined elements. In fact, many psychologists agree that the factfinder naturally constructs such stories, although perhaps not in a very systematic manner. (116) The broadly accepted story model of evidence processing holds that the factfinder, over the trial process's course, constructs from the evidence the story that makes maximal sense; and the factfinder then chooses, among the available decisions, the one that fits best with the constructed story:
Several authors have recently proposed a model for juror decisionmaking based on the concept of a story as an organizing and interpreting schema. The story model attempts to explain how jurors organize and interpret the vast amount of information they encounter at trial and apply the appropriate decision criteria.... ... The jurors construct a story adequately describing what happened. At the conclusion of the trial, they construct the verdict categories based on the instructions given by the judge. The individual juror arrives at his decision by determining the best match between his story and the available verdict categories. The task of the jury in deliberations then becomes one of selecting a story from among those offered by the jurors and fitting it to the available verdict options. (117) If the jurors construct a story (or stories (118)) for the whole case, or otherwise cognitively process the entirety while the trial progresses, and then the judge instructs on standard of proof, it might be that the jurors actually apply the standard to the whole claim or defense. It might also be that, being human, a judge when acting as factfinder proceeds in essentially the same manner, testing whether the already conjoined elements are more likely than not.
Indeed, by providing obscure instructions only at the end of oral trials, the law seems determined to encourage overall consideration and to discourage applying the standard of proof elementbyelement. Although the judge does instruct literally in elementbyelement terms, (119) this may work only to encourage the jurors' detailed evaluation of the evidence and to stress the requirement that any story must contain all of a series of elementsjust as many evidence rules may work to brake any undesirable tendency of the factfinder to rush toward creating a story. (120)
So, the conjunction paradox may not inflict great practical effects. Nonetheless, the big theoretical problem of the conjunction paradox will unavoidably pose at least some practical difficulties. The law sometimes enforces its elementbyelement theory and thereby impedes the holistic practice. An obvious example would be when the judge requires a special verdict that asks the jury to find each element by a preponderance. (121) The conjunction paradox therefore remains troubling, and theorists twist themselves into pretzels trying to explain it away.
It would be troubling, however, only if theory really calls for the product rule. But theory does not. Instead, it invokes the MIN rule. The truth of the conjunction equals the minimum of the truths of the elements. If each element is more likely than not, then the truth of the conjunction is more likely than not. To use the above example, if the plaintiff shows that fault is .60 true and that causation is .60 true, then he has shown to .60 that the defendant both acted negligently and caused the injury.
Thus, there is no conjunction paradox. It implodes under the force of fuzzy logic. The MIN operator provides that belief in the conjunction will match the least likely element, which has already passed the standard of proof. The MAX operator meanwhile indicates that belief in the negative of the conjunction, that is, in the disjunction of each element's negation, will never reach equipoise. The story of liability will not only be the most believable story, but will be more believable than all the stories of nonliability combined.
Comfortingly, under the MIN rule, applying the standard of proof elementbyelement works out to be equivalent to applying it to the whole conjoined story. So, if the factfinder actually does follow the story model, that practice would not directly endanger the standard of proof. The apparent criticality of the number of elements melts away too. Because the MIN rule applies to each set of evidence to be conjoined, it does not matter where the law draws formal lines between elements, or whether the elements are independent or interdependent. Nor does it matter if I sloppily labeled identity as an "element" in my examples above. (122)
Moreover, the proof process within elements is not dissimilar to the proof process between elements. Within elements, the factfinder uses intuitive techniques in a nonquantitative and approximate fashion. Between elements, and for separate facts within elements, the factfinder uses the fuzzy operator for conjunction that works in a similar style.
The law does seem to know what it is doing, then. Whenever it phrases its instruction to require applying the standard of proof elementbyelement, it is instructing to apply the MIN operator. But do actual factfinders apply the MIN operator as they should and as the law tells them to do? We do not know. Some experimental evidence arguably suggests that the lay person tends to apply the product rule rather than the MIN operator. (123) Nevertheless, no sign exists that factfinders in the legal system are using the product rule. After all, a concern that they were ignoring the product rule generated the unfounded fear of the conjunction paradox in the first place.
Theorists also claim there is a converse paradox, involving multiple theories. These observers lament that the law denies relief to a supposedly deserving plaintiff (or to a defendant with multiple defenses almost proved):
Consider a case involving three different legal theories and three different factual foundations. Plaintiffs deserve to win if one of the stories embodying one legal theory is true; defendants deserve to win only if all of their competing stories are true (for if this is false, one of the...
Death of paradox: the killer logic beneath the standards of proof.
Author:  Clermont, Kevin M. 
Position:  II. Conjoining Assessments B. Legal Application: Conjunction Paradox through Conclusion, with footnotes, p. 11061138 
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