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.
8 comments:
It would be great if all our errors could be called paradoxes!
Your questions presuppose that there is a conjunction paradox in the way legal decisions on liability are made. But there isn’t really (although I know it’s well recognised in the literature). Consider how the fact finder will address the liability question: if there are three elements to be proved, and if each (as you want to avoid discussing conditional probabilities) is independent of the others, then the fact finder will consider the evidence relevant to each in turn and will evaluate whether, in a civil case, each is proved on the balance of probabilities. If that is proved for each element, then liability is established. There is no need for the fact finder to ask, “Gosh, what were the chances of the elements occurring together?” because they did.
There is nothing paradoxical about the probability that n independent events will occur together being less than the probability that (n – 1) of them will occur together if none of them are certain to occur. It accords with common sense.
As to your questions:
1. The fact finder will assess the credibility of a witness by taking into account veracity, objectivity and perception in a way that is more like moving a point back and forth on a scale of credibility, than like multiplying probabilities.
2. I don’t know about the literature – just about everything has been said by someone sometime, surely (yes yes, “don’t call me Shirley”).
3. The elements of affirmative defences should be found by asking whether the evidence for them reaches the necessary standard of proof, again without any paradox arising.
Where the elements are independent, legal fact finders go straight to the evidence on each element and assess its strength without reference to the probability that that evidence would exist jointly with the evidence for some other element.
I think that before a huge argument starts about how to deal with the alleged paradox, it is appropriate to sort out whether there really is a paradox arising from the way facts are determined in law.
Don, interesting comment. N.B. I don't endorse the conjunction paradox. I state it. There is something wrong with it. I want to defeat. To that end: your following comment is interesting: "If that [proof of each element of a civil claim to a balance of the probabilities] is proved for each element, then liability is established. There is no need for the fact finder to ask, “Gosh, what were the chances of the elements occurring together?” because they did."
The proposition "If that [proof of each element of a civil claim to a balance of the probabilities] is proved for each element, then liability is established. There is no need for the fact finder to ask, 'Gosh, what were the chances of the elements occurring together? because they did." does not respond to the question of the trial lawyer, "Gosh, what are the chances I can prove this claim?" but, then, neither does the application of the product rule as described by L.J. Cohen -- in part because the chances the lawyer's chance of persuading the jury to the requisite degree of probability that each element of the claim is valid depends in part on the evidence available to support (and oppose) the claim, and not on some chance taken out of the air -- say .5 or any other number -- that each element will be adjudged true or valid. Am I missing something? But the product rule (modified for dependencies) may apply - yes? - to the lawyer's calculation of the lawyer's chances of convincing the jury (by a preponderance, in a civil case) of the validity or truth of the various elements of the claim. Am I missing something? Of course factors other than evidence -- e.g., the pre-existing attitudes of the jurors, the rhetorical skills of the opposing lawyer, etc. -- will affect that assessment. In any event, L.J. Cohen's conjunction paradox turns out to be a parlor room trick.
Progress(?) (at last!) toward a satisfying solution of L.J. Cohen's conjunction paradox:
Step 1: The elements of a claim, charge, or affirmative defense are not events that have some probability of happening -- such as probability of .95 [for example] of "death" in murder cases or probability > .5 of "foreseeability" in negligence cases. Essential elements (to some legally-required level of persuasion) are not events or possible events at all. They are legal requirements for the validity of a claim, charge, or affirmative defense.
Step 2: If the product rule is to be applied at all, it must be applied to the possible facts that a proponent seeks to establish in order to establish some legal claim, charge, or defense. But it is meaningless to ask, "What is the inherent probability of some element such as 'death.'" (It is no good, in short, to pretend that such possible facts are possible facts like the event "face 1 of an evenly-balanced six-sided die turns up".)
Step 3: If we wish to assess the chances of the combination of F-1 that instantiates E-1 (Fact 1 instantiating Element 1), F-2 that instantiates E-2, ... F-n instantiating E-n, and if we think of "establishing F-1 ... F-n" as a series of possible events [e.g., chances of getting the jury in a civil action to declare that all F-1 ... F-2 are more probably than not true], we must ask, not what is the pre-existing probability of F-1, of F-2, etc., but, for example, what is the probability of F-1 [e.g., "jury decision that James is dead"] given the evidence available to the proponent (and given the evidence available to the opponent, given the rhetorical skills of the proponent and opponent, and a wide array of factors and circumstance).
Step 4 (conclusion): If Step 3 is correct, a product rule of some kind may apply to the "probability of a claim," but the product rule that is applied won't look much like the one L.J. Cohen describes. For example, the probabilities of proving the individual instantiating facts -- F-1, etc. -- will vary enormously. And isn't that as it should be? The conjunction paradox is a parlor trick.
N.B. Nothing important in the above argument changes if we focus on the chances of the underlying instantiating facts F-1...F-n. For example, the probability of "death of James" depends (in part) on the available evidence (which may, of course, be sparse initially; but that is neither here nor there).
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These thoughts of mine were precipitated by the above exchange with Dan Mathias.
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Is the solution this simple? Am I missing something?
Progress(?) (at last!) toward a satisfying solution of L.J. Cohen's conjunction paradox:
Step 1: The elements of a claim, charge, or affirmative defense are not events that have some probability of happening -- such as probability of .95 [for example] of "death" in murder cases or probability > .5 of "foreseeability" in negligence cases. Essential elements (to some legally-required level of persuasion) are not events or possible events at all. They are legal requirements for the validity of a claim, charge, or affirmative defense.
Step 2: If the product rule is to be applied at all, it must be applied to the possible facts that a proponent seeks to establish in order to establish some legal claim, charge, or defense. But it is meaningless to ask, "What is the inherent probability of some element such as 'death.'" (It is no good, in short, to pretend that such possible facts are possible facts like the event "face 1 of an evenly-balanced six-sided die turns up".)
[go to next comment, below]
Step 3: If we wish to assess the chances of the combination of F-1 that instantiates E-1 (Fact 1 instantiating Element 1), F-2 that instantiates E-2, ... F-n instantiating E-n, and if we think of "establishing F-1 ... F-n" as a series of possible events [e.g., chances of getting the jury in a civil action to declare that all F-1 ... F-2 are more probably than not true], we must ask, not what is the pre-existing probability of F-1, of F-2, etc., but, for example, what is the probability of F-1 [e.g., "jury decision that James is dead"] given the evidence available to the proponent (and given the evidence available to the opponent, given the rhetorical skills of the proponent and opponent, and a wide array of factors and circumstance).
Step 4 (conclusion): If Step 3 is correct, a product rule of some kind may apply to the "probability of a claim," but the product rule that is applied won't look much like the one L.J. Cohen describes. For example, the probabilities of proving the individual instantiating facts -- F-1, etc. -- will vary enormously. And isn't that as it should be? The conjunction paradox is a parlor trick.
N.B. Nothing important in the above argument changes if we focus on the chances of the underlying instantiating facts F-1...F-n. For example, the probability of "death of James" depends (in part) on the available evidence (which may, of course, be sparse initially; but that is neither here nor there).
&&&&
These thoughts of mine were precipitated by the above exchange with Dan Mathias.
&&&&
Is the solution this simple? Am I missing something?
Hello again. I see you have been busy during what we here in NZ have been calling "night time".
I think your analysis is very helpful. Yes, the product rule applies to facts needed to establish an element. And yes, the chances of establishing all the elements can be no greater than the chance of establishing the least likely of them - in fact it should be the same as the least likely element.
So I think I agree with all your steps. Didn't mean to suggest you agreed with the "paradox".
I notice my name has become "Dan", no doubt because that's how you pronounce it? Or am I being unfair on Texans?
Stipulation: Dan = Don
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