Over the years there has been much discussion about the conjunction paradox that is said to arise from the legal requirement that each essential element of a claim or charge (and, I might add, of an affirmative defense) be shown to some specified standard of persuasion such as proof beyond a reasonable doubt. It has been argued that a paradox arises if such a burden of persuasion requirement is given a mathematical interpretation that assumes that uncertainty is graded cardinally.
Suppose that the preponderance of the evidence standard requires a showing that of more than a .5 probability. On the one hand, if this mathematical requirement applies to each element of a claim and if there is more than one element to a claim, then the legally-requisite probability that the entire claim has been established -- element a, element b, etc. -- holds only if p(a) X p(b) ... p(n) is greater than .5. But if this is the case, it is argued, it follows that the probability of at least one element must be shown to be considerably more than .5. This is said to follow by application of the product rule, which in its simplest form holds that the probability of a & b together equals the probability of a times the probability of b. On the other hand, if the legal burden of persuasion is interpreted to require only that the probability of each element of a claim is more than .5, it is said that another absurd result arises, which is that a party can establish a claim (or an affirmative defense, I would add) even if the party fails to establish that the probability of all of the essential elements taken together is greater than .5.
Quite a few observers have noted that dependencies among the factual hypotheses that a party attempts to establish to establish a claim (or, I would add, an affirmative defense) reduce the numerical anomalies generated by a mathematical interpretation (based on cardinal numbers) of standards of persuasion. But unless the dependencies are complete among all essential elements, dependencies do not eliminate the paradox.
1. Has anyone noticed the similarity between the conjunction paradox as formulated above and the paradox that arises if it is believed that several testimonial qualities (e.g., veracity vel non, objectivity vel non, ability to perceive vel non, etc.) are essential to an assessment of the credibility of a witness and if one accepts the suggestion (made by at least one probabilist and implicitly also by Edmund Morgan's treatment of chains of inference) that the probabilities of such credibility attributes must be multiplied with each other to arrive at a judgment about the aggregate probability that a witness is credible?
2. Has anyone in the literature argued or suggested that a mathematical interpretation of burdens of persuasion requires that the prior probabilities of the various elements of a claim, charge, or defense be considered and that sometimes or often it it might be appropriate to assume that the prior probability of a fact instantiating an essential element is, for example, more than .5 or .95 or even some higher probability?
3. Has anyone considered what the application of the conjunction paradox to affirmative defenses does to the application of the conjunction paradox to the essential elements of a claim or charge and has anyone argued that the application of the product rule to affirmative defenses makes the conjunction paradox paradoxical?
It's here: the law of evidence on Spindle Law. See also this post and this post.