Legal Logic

In my last year of college, I briefly flirted with the idea of going to law school. The reasoning seemed kind of fun. Lots of people (some might say too many) were doing it. But I chose technology, deciding that for the most part lawyers divide an existing wealth pie and take a slice for themselves: a zero sum game for society. A career in technology promised the possibility of increasing the size of the pie. Of course, my analysis was simplistic, but I still think fundamentally correct. Technology opens new worlds. Legal reasoning retreads paths that the Mesopotamians blazed more than 4,000 years ago. Still, my career has since brought me in close contact with very smart lawyers. I have seen how they argue and how their arguments affect society. I offer you a small axiomatization of what I think is going on and then propose a modest reform.

Suppose there is a new device (or drug) D aimed at sickness S. Legal logic plays with the following predicates:

• mayuse(x,D) - Person x may use device D (i.e., D is available).

• hurtby(x, D) - Person x has been hurt by device D.

• newdevice(D) - Device D is new.

• sick(x,S) - Person x is sick with sickness S.

• sueandwin(x) - Person x sues someone with a good probability of winning.

This is first order predicate logic so:

• “ThereExists x such that …” means at least one person x has the property represented by the ellipses.

• “ForAll x …” means that all people have the property represented by the ellipses.

• A→B means that if A holds, then B must hold. If A doesn’t hold, then B need not either.

• Rule 1) ForAll x hurtby(x,D)→sueandwin(x)

In words: If any person x is hurt by D, then x sues and will probably win.

• Rule 2) ThereExists x such that hurtby(x,D) & sueandwin(x)→ForAll y not mayuse(y,D)

In words: If at least one person x is hurt by D and sues successfully, then nobody is able to use device D. Admittedly, this is extreme, but in the United States, at least, we have almost arrived at that point.

• Rule 3) ForAll x sick(x,S) & not mayuse(x,D)→die(x)

In words: If x has sickness S and x doesn’t have device D, then x will die. So, we’re assuming the device D is important.

Warm-Up

Let’s see the consequences of some starting assumptions. Suppose we assume that D is new and there is an axiom:

 ForAll Y newdevice(Y)→ThereExists x hurtby(x, Y)

What are the consequences?

Solution to Warm-Up

Given the axiom, the invention of D will inevitably lead to lawsuits and no savings in lives based on the following rules.

 ThereExists x hurtby(x,D)→sueandwin(x) ThereExists x such that hurtby(x,D) & sueandwin(x)→ ForAll y not mayuse(y,D) ForAll x sick(x,S) & not mayuse(x,D)→die(x)

But this state of affairs is clearly undesirable. If D hurts very few people and saves many more, then D is overall good for society, at least in some utilitarian sense. Is there any way to take this into account in the lawsuits?

That is, suppose the judge at a trial could inform juries of the global cost-benefit history of the device. If the device has helped many more people than have been hurt, then the jury might take this into account. This could reduce the probability of success in lawsuits.

So, we enhance our logic with new predicates:

• “MoreThanFraction(f) x …” means that more than a fraction f, including x, of people have the property represented by the ellipses.

• This then allows us to define a new rule 1 based on a fraction of, say, .02 (2.0%) for some device that is risky but overall very helpful:

• Rule 1) MoreThanFraction(0.02) x hurtby(x,D)→sueandwin(x)

In words: If more than 2% of the people treated by device D are hurt, then those people can sue and win. For someone to sue and win, then that person must have been hurt and more than 2% must have been hurt.

Using this rule and assuming only one person in a thousand is hurt, you can see that there are no lawsuits.

This may be the right idea, but a few problems remain. First, even if the effective probability of being hurt is in fact one quarter the threshold (say 0.005 in this case), the early history of the device could exceed the threshold.

1. If each patient has a 0.005 probability of being hurt, how likely is it to get one failure within the first 50 patients?

Hint: 

You can solve this using probability theory or by writing a small program that simulates this probability.

But overall success might cause us to miss a real lemon. After all, failures aren’t necessarily independent. In the hands of some hospitals, for example, device D might be far more dangerous than in others. Suppose only 10 people out of 2,000 have been hurt overall, but all 10 were hurt in the same hospital, which had treated only 100 patients. What should be done? This is where your advice as a mathematician comes in. But first, suppose that the threshold for successful lawsuits is reduced to 1% per hospital.

2. Suppose that 18 people out of 4,000 have been hurt by the device. There are eight hospitals, each of which has treated 500 patients. What is the likelihood that at least one hospital exceeds a 1% test, i.e., has 6 patients who were hurt, even if the underlying chance of failure were in fact independent of hospital and is 0.005 per patient?

3. Suppose the distribution of patients were much more skewed, with seven hospitals having treated only 200 patients each and the remaining one having treated all the rest. How likely is it that at least one of those smaller hospitals had hurt three people or more under the same conditions (i.e., 18 people who were hurt overall and the underlying chance of failure is 0.005 per patient)?

So far, these cases have made the lawsuits look frivolous. After all, even if the ground probability of being hurt is 0.005 and the hospitals are all equally careful, hospitals are quite likely to be vulnerable to lawsuits based on the threshold test. But suppose we fix this problem by increasing the threshold back to 2%. If the higher threshold test is met or exceeded in some hospital, then the likelihood that the 0.005 model is correct becomes so small that a statistically informed jury would infer that the hospital had not done its job.

4. Suppose we say that if a single hospital has more than 2% of bad outcomes, then that is a bad hospital. Assume that each hospital treats 500 patients. What is the likelihood for at least one hospital to hurt 10 or more people if the 0.005 hurt probability held and there were no bias in any hospital?

5. How would you answer the last question in the case that seven hospitals each treat only 200 patients and one treats 3,600?

The bottom line: If a hospital wants to offer the use of a new device, it should use it a lot.