I wrote this -- substantially this -- to a good friend today:
Assume that legal liability depends on elements a, b, c & d. Assume that these four elements (or, in any event, 3 of these 4 elements) can be classified either as elements of a claim [assume a civil case] or as elements of an affirmative defense. (In theory affirmative defenses could be done away with altogether -- by making all elements elements of claims.)
Assume that claimant has the burden of persuasion (assume this is defined in terms of probabilities) of slightly more than p(.5) on each element of each claim and that respondent has the burden of persuasion of slightly more than p(.5) on the negation of each element of each affirmative defense.
If the product rule (not modified for dependencies) applies to legal claims (which are assumed to consist of elements) and also to affirmative defenses (which are also assumed to consist of elements) -- and why should not the product rule apply to affirmative defenses if it applies to claims?--, does it follow that if elements a, b & c rather than just elements a & b are elements of the claim, that the claimant's burden of persuasion for the entire claim (with three elements) is roughly .125 rather than .25 (with two elements) and does it also follow that the respondent's burden (on the entire affirmative defense) rises to roughly .5 (from .25)? Can this (logically) be?
Look at the problem this way. Assume the claim has two elements a & b. Assume the affirmative defense has elements c & d. If the claimant's burden of persuasion in this situation is _roughly_ .5 on each of the four elements, does it not follow that if an affirmative defense is "in play" (because properly raised by the pleadings or otherwise), the claimant's burden on all four elements -- the conjunction of those four elements -- is roughly .0625 whereas the respondent's burden on the affirmative defense (the conjunction of not-c and not-d) is roughly .25? Is there a paradox here, a paradox about the assumption that the product rule applies here?
Put it this (equivalent?) way: If the conjunction rule applies and all four elements are elements of the claim (and there is no affirmative defense), the respondent wins if it establishes that the probability of the negation of any one element is roughly (slightly more than) .5. However, if two of the four elements are part of an affirmative defense (and if, therefore, they are not elements of the claim), respondent wins by relying only on a showing of "high" probabilities (a probability slightly greater than .5) of the negation of each of the two elements -- i.e., by showing that the probability of the negation of each element of two elements is slightly more than .5. If so, how can it be (on one horn of the conjunction paradox) that claimant's responsibility is to show (at least) that p(c & d) is roughly .25 whereas respondent must show that the probability of p(~c & ~d) is roughly .25? Doesn't the rule against self-contradiction, or the inverse relationship between the probability of some thing X and the probability of some thing ~X and the convention that probability of some thing X and the probability of the complement of X must sum to one (1) mean that the probability of the subset (c & d) and the probability of the complementary subset (~c & ~d) must equal one? If so, is it logically incoherent, self-contradictory, to assume that the product rule applies to the problem of the burden of persuasion on a claim? (My basic argument remains unchanged if you assume that the burden of persuasion on a claim requires that probability of the entire claim be slightly more than .5 -- and that this means [the other horn of the conjunction paradox] that the probabilities of some elements of a claim must be more than .5, often [depending on the number of elements] much, much more than .5. But I think the counter-riddle I have tried to describe also arises here, though in a somewhat different guise. But don't ask me to explain!)
N.B. The conjunction paradox does not go away just because alternative factual scenarios (which have facts that instantiate or satisfy an element of a legal claim) can establish a legal claim: The paradox remains. (I once worked this out to my satisfaction. But, again, please don't ask me to do so here.) But my counter-paradox (meta-paradox?) also remains, I think.
But perhaps my argument (which definitely falls short of a proof!) -- perhaps my argument about the existence of meta-paradox has a logical flaw? It's entirely possible that it does!
Postscript: As the comments to this post show, my musings (see above) did not lead anywhere. What makes the conjunction paradox so agonizingly difficult is that one has the sense -- or, in any event, I do -- that the proper mathematical interpretation of the burden of persuasion on a claim (or an affirmative defense) is that the jury must be instructed (or the trial court must understand) that it can find for the claimant (or for the respondent on the basis of an affirmative defense) only if it finds (or the judge concludes) that the claimant (or respondent) has established the probability of the claim as a whole (or, in an appropriate situation) the affirmative defense) meets or exceeds the probability required by the applicable burden of persuasion -- and not just that each element of the claim (or affirmative defense) meets or exceeds the requisite probability. The difficulty with this interpretation is twofold: (i) it seems to be inconsistent with the way that the law usually expresses the burden of persuasion and (ii) if each element must be shown to the probability required by the legal burden of persuasion, and if the claim or affirmative defense has more than one element, the application of the product rule says (even when there is some dependence, but less than complete dependence, between the probabilities of the two elements) that the overall claim or affirmative defense must be established to a probability that exceeds the legally-stated burden of persuasion for the claim as a whole -- and, moreover, unless there is a very high degree of dependence among the probabilities of the elements of a claim or affirmative defense, if there are many elements in a claim or affirmative defense, the requisite level of probability for the claim or affirmative defense as a whole is much higher than the legally-stated burden of persuasion (much higher than, e.g., .5 in a civil case, if a probability of slightly more than .5 is the level of probability that must be shown for each element in the usual civil case).
If that's the case -- if, for example, in the usual civil case the trier of fact should uphold a claim or affirmative defense with numerous elements only if it finds that the probability of the claim or affirmative defense very substantially exceeds .5 -- does it follow that the jury or trial court should almost always find that the claim or an affirmative defense has not been sufficiently established? No, I think that does not follow at all -- because if the claimant or respondent has oodles of powerful evidence to support each element, the jury or the trial court should, of course, find that the claim as a whole [or the affirmative defense] has been sufficiently established, and it probably will do so. The high probability seemingly required for the claim [or affirmative defense] as a whole speaks only [if it does so] to the question of whether an investigation will uncover evidence that sufficiently establishes a claim or an affirmative defense. Moreover, whether an investigation will uncover sufficient evidence of numerous elements of a claim or an affirmative defense depends in part on the characteristics of the part of the world that is being investigated -- and this means it may or may not be very difficult to uncover evidence that will establish all of the various elements to a very high probability. Yes? No?
It's here: the law of evidence on Spindle Law. See also this post and this post.