Friday, September 06, 2013

A Path to the Solution of the Conjunction Paradox?

The "conjunction paradox" has bedeviled legal theorists for years. It still does. See, e.g., Kevin M. Clermont, Death of Paradox: The Killer Logic Beneath the Standards of Proof Cornell Law Faculty Publications, Scholarship@Cornell Law: A Digital Repository (Feb 1, 2013). Some legal theorists who doubt that the standard probability calculus is a satisfactory "model" of judicial proof are particularly prone to flog their opponents with this paradox.


But what is the conjunction paradox? According to Professor Clermont, id. at 1106, Professor Charles Nesson constructed "the best-known statement of the infamous conjunction paradox." Clermont quotes (id.) Nesson thus (original article: Charles Nesson, "The Evidence or the Event? On Judicial Proof and the Acceptability of Verdicts, 98 Harvard Law Review 1357, 1385-88 (1985) footnotes omitted)):


"We purport to decide civil cases according to a more-probable-than-not 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 more-probable-than-not 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."


The conjunction paradox persists even if we assume juries are told to assess the probability of a whole claim (or affirmative defense) as well as the probability of each element of the claim (or affirmative defense).


The conjunction paradox persists even if we assume that there are some partial dependencies among the elements of a claim (or affirmative defense).


The conjunction paradox takes an acute form in criminal cases - in which, let us assume, jurors are instructed they must find that proof establishes each element of the charge beyond a reasonable doubt and also the entire charge beyond a reasonable doubt. The problem here is that if the charge has two or more essential elements and if a numerical value such as a .95 probability is assigned to "beyond a reasonable doubt," it seems to follow, by probability logic - specifically the product rule, contrary to the instructions jurors are actually given,  that they must find that the probability of at least one of those elements must be very substantially above .95. If not, it seems to follow - by the probability logic - that jurors can return a guilty verdict even if they believe that the probability of the existence of all the essential elements taken together is less than .95.

I have made some entirely unsatisfactory attempts to develop a solution to the conjunction paradox. See, e.g.,

Sunday, April 18, 2010

Cf. my unsuccessful attempt to begin to find another way through the muddle:

I am now going to try to begin work toward very different kind of solution. Consider the following (fragmentary and tentative!) gambit:



Take the following possible situation (Situation X):

At about 3:00 p.m., on June 5, 2013, (a) James Jones becomes angry at Valiant Victim, (b) decides to hurt Valiant Victim, (c) picks up a knife, and (d) stabs Valiant Victim.

We can think of the Situation X as a set of distinct events a – d .

We can also think of the above situation Situation X as one event - as a single event - that has a number of parts or features (e.g., the events or features above, a – d ).

Probability theory alone does not tell us which of these two ways we should or must think of Situation X.


If we can or should think of Situation X with features a – d (only) as a single (possible) event, we are free to think of the question of the probability of Situation X without believing that we must or should think of the probabilities of each of the events or features a – d.


  • It is quite true that it is possible think of Situation X as a (possible) compound event that is nothing more than the collection of (possibly-connected) distinct events a – d over time.
  • If we think of Situation X in (only) this way, we can and surely must (at the very least) think of the separate probabilties of each of the events a – d if we wish to assess the probability ofSituation X.
  • Given the hypothesized way we are now thinking of Situation X, if we do not ponder these separate probabilities, we cannot possibly assess the probability of Situation X. This is because we think of Situation X as nothing more than the conjunction (over time) of events a – d.
  • But it is also true that it is possible to think of Situation X(only) as a single event with the features a – d over time. In that event, if we wish to assess the probability of Situation X, the structure of our thinking (or imagination) about Situation Xdoes not force or drive us to ponder the separate probability of each a – d. Indeed, if we think of Situation X as nothing more than a single event, we cannot readily imagine the possibilty of separate assessments of the probability of each a – d.


But a question: In a trial we take (and must take) evidence about matters such as events or elements a – d. Does it follow that we must think of (factual) hypotheses such as Situation X as being nothing more than a possible composite event consisting of events a – d?


Answer: I don't think so. The evidence in question may generate in our minds the hypothesis Situation X with the features a – d. Cf. P. Tillers & D. Schum, A Theory of Preliminary Fact Investigation, 24 University of California at Davis Law Review 931 (1991) (arguing in part that evidence serves to generate and refine hypotheses as well as to prove or disprove formulated hypotheses). But once a factual hypothesis is crystallized in our minds, we are free (as a logical matter) to think or imagine that the evidence about a – d is nothing more than evidence about the entire hypothesis Situation X.

But note (and this is an important note, a very important refinement): As I have hinted above - by using words such as “only” and “just” - I think it is both possible and likely that our minds shift between thinking of a factual scenario such as Situation X as being, one the one hand, a single event and, on the other hand, a composite event (i.e., an event consisting of distinct sub-events). If this is the case, it follows that when we shift from one way of thinking about a factual scenario to the other, the way that probability theory is applied to the situation also changes and must change. And that's perfectly fine and appropriate. Probability theory is a formal tool that does not, by itself, specify or even suggest how possible events in the world should or must be carved up.



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