# Alito on the death penalty, and deontic logic.

In a 2015 death penalty case Alito wrote: “Because it is settled that capital punishment is constitutional, [i]t necessarily follows that there must be a [constitutional] means of carrying it out.” I think Alito’s reasoning is formally incorrect.

Let: $e$ = We execute Jones. $p$ = We cause Jones excruciating pain.

Suppose it is a brute fact that if we execute Jones then we cause Jones excruciating pain: ie, $e \rightarrow p$.

Introduce $PE$ as a deontic operator for “it is constitutionally permissible that” and $OB$ for “it is constitutionally obligatory that.” These should function as a deontic operators in standard deontic logic (SDL). Alito’s claim is that the following is an inconsistent triad: $\{ PE(e), \neg PE(p), e \rightarrow p \}$

These would be inconsistent if $\neg PE(p), e \rightarrow p \vdash \neg PE(e)$.

What would an argument for this look like. Maybe the following:

der(A)
1. $\neg PE(p)$ assumption
2. $e \rightarrow p$ assumption
3. $PE(e \rightarrow p)$ from 2, deontic logic
4. $PE(e) \rightarrow PE(p)$ from 3, deontic logic
5. $\neg PE(e)$ from 1 and 4, modus tollens

I suppose we might have to grant that brute facts must be permissible: ie that step 3 is valid. $PE(e \rightarrow p)$ is true because $p \rightarrow e$ is true in the actual world and hence must be accepted as permissible.

But I don’t think that step 4 is valid. Supposing we grant that $P(e)$ is true, this just means there’s some normatively admissible world, maybe some world where all feasible means of execution does not cause excruciating pain, maybe not the actual world, where $e$ is made true by our action. This is consistent with there being no normatively admissible world where $p$ is made true by our action, including the actual world.

Maybe the thought is that the derivation should go like this:

der(B)
1. $\neg PE(p)$ assumption
2. $e \rightarrow p$ assumption
3. $OB(e \rightarrow p)$ from 2, deontic logic
4. $OB(e) \rightarrow OB(p)$ from 3, deontic logic
5. $PE(e) \rightarrow PE(p)$ from 4, deontic logic
6. $\neg PE(e)$ from 1 and 5, modus tollens

The reason for doing this is to try to use the distributivity of \rightarrow OB$(analogue of modal S5) to get get around the problem of the non-distributivity of \rightarrow PE$ in the failed der(a), and into a position to do the modus tollens Alito desires. There is a rule of SDL that says that if a proposition is a theorem then it is obligatory. Maybe this could be used to justify step 3. This would be a mistake; $e \rightarrow p$ is not a theorem. Also, I think that the step from 4 to 5 would be invalid. It would work if $\vdash PE(e) \rightarrow OB(e)$ and $\vdash OB(p) \rightarrow PE(p)$, but the former is clearly not a theorem.

If I’ve got this right, I’ve really only shown that two attempts to reconstruct Alito’s thinking are dead ends. This doesn’t show that there is no valid reconstruction, but to me it doesn’t look promising. Alito seems to have confused the condition of there being some normatively permissible world in which $e$ is made true by our action with the condition that the actual world must be a normatively permissible world in which e is made true by our action, but the latter condition is precisely what is made false if the only actual means to $e$ is $p$ and $\neg PE(p)$.