Q&A

# What was Justice Scalia's mathematical mistake in Penry v. Lynaugh (1989)?

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Please see the bolden phrase below. Please don't hesitate to reduce the amount of quotation, which I know is lengthy, whilst preserving enough context.

But even granting this point, Scalia writes, state legislatures have not demonstrated a national consensus against execution of the mentally retarded, as the precedent of Penry requires:

The Court pays lip service to these precedents as it miraculously extracts a “national consensus” forbidding execution of the mentally retarded . . . from the fact that 18 States—less than half (47%) of the 38 States that permit capital punishment (for whom the issue exists)—have very recently enacted legislation barring execution of the mentally retarded. . . . That bare number of States alone—18—should be enough to convince any reasonable person that no “national consensus” exists. How is it possible that agreement among 47% of the death penalty jurisdictions amounts to “consensus”?

The majority’s ruling does the math differently. By their reckoning, there are thirty states that prohibit execution of the mentally retarded: the eighteen mentioned by Scalia and the twelve that prohibit capital punishment entirely. That makes thirty out of fifty, a substantial majority.
Which fraction is correct? Akhil and Vikram Amar, brothers and constitutional law professors, explain why the majority has the better of it on mathematical grounds. Imagine, they ask, a scenario in which forty-seven state legislatures have outlawed capital punishment, but two of the three nonconforming states allow execution of mentally retarded convicts. In this case, it’s hard to deny that the national standard of decency excludes the death penalty in general, and the death penalty for the mentally retarded even more so. To conclude otherwise concedes an awful lot of moral authority to the three states out of step with the national mood. The right fraction to consider here is 48 out of 50, not 1 out of 3.
In real life, though, there is plainly no national consensus against the death penalty itself. This confers a certain appeal to Scalia’s argument. It’s the twelve states that forbid the death penalty* that are out of step with general national opinion in favor of capital punishment; if they don’t think executions should be allowed at all, how can they be said to have an opinion about which executions are permissible?
Scalia’s mistake is the same one that constantly trips up attempts to make sense of public opinion; the inconsistency of aggregate judgments. [Emphasis mine] Break it down like this. How many states believed in 2002 that capital punishment was morally unacceptable? On the evidence of legislation, only twelve. In other words, the majority of states, thirty-eight out of fifty, hold capital punishment to be morally acceptable.
Now, how many states think that executing a mentally retarded criminal is worse, legally speaking, than executing anyone else? Certainly the twenty states that are okay with both practices can’t be counted among this number. Neither can the twelve states where capital punishment is categorically forbidden. There are only eighteen states that draw the relevant legal distinction; more than when Penry was decided, but still a small minority.
The majority of states, thirty-two out of fifty, hold capital punishment for mentally retarded criminals in the same legal standing as capital punishment generally.†
Putting those statements together seems like a matter of simple logic: if the majority thinks capital punishment in general is fine, and if the majority thinks capital punishment for mentally retarded criminals is no worse than capital punishment in general, then the majority must approve of capital punishment for mentally retarded criminals.
But this is wrong. As we’ve seen, “the majority” isn’t a unified entity that follows logical rules. Remember, the majority of voters didn’t want George H. W. Bush to be re-elected in 1992, and the majority of voters didn’t want Bill Clinton to take over Bush’s job; but, much as H. Ross Perot might have wished it, it doesn’t follow that the majority wanted neither Bush nor Clinton in the Oval Office.

• Since 2002, the number has risen to seventeen.
† This is not precisely Scalia's computation; Scalia didn't go so far as to assert that the no-death-penalty states thought execution of mentally retarded criminals no worse than execution in general. Rather, his argument amounts to the claim that we have no information about their opinions in this matter, so we shouldn't count these states in our tally.

Ellenberg, How Not to Be Wrong (2014), pages 373-374.

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Consider the universal quantifier. I'll write it as: $\mathsf{forall}\ x.P(x)$ to be read as "for all $x$ in the domain $P(x)$ holds". We then have the logical identity $$(\mathsf{forall}\ x.P(x)) \land (\mathsf{forall}\ x.Q(x)) \to (\mathsf{forall}\ x. P(x) \land Q(x))$$ where $\land$ is logical conjunction and $\to$ is logical implication. Let's have the domain be people and $P(x)$ mean "$x$ want $A$" and $Q(x)$ mean "$x$ wants $B$". We can then read the above condition as: "if everybody wants $A$ and everybody wants $B$, then everybody wants $A$ and $B$".

If, however, we consider a majority quantifier, write it as $\mathsf{most}\ x. P(x)$, meaning "$P(x)$ holds for 'most' $x$", we do not have an identity similar to the above. Using the same intepretation as before, we'd get a statement like "if most people want $A$ and most people want $B$, then most people want $A$ and $B$" which does NOT hold in general. The people who want both is the intersection between the people who want each individually, and while this intersection is guaranteed to be non-empty it's certainly not guaranteed to be a majority. It's easy to imagine a situation where three out of five of your friends want to eat pizza for lunch, and three out of five want to go to the park in the evening, but only one wants to do both.

Scalia's argument could then potentially be interpreted as: "Most states think the death penalty should be allowed and most states treat these groups the same, therefore most states think the death penalty should be allowed with no distinction between these groups of people." This statement doesn't follow.

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