This post continues the line of investigation from an earlier post. There I supposed that you're a police officer and you saw that Glossop was very likely to be about to murder Fink-Nottle, and the only way to stop him was to shoot Glossop in the head. Glossop was pointing a shotgun at Fink-Nottle, and your credence that he was about to shoot was 0.9999. I took it for granted that under circumstances like that it was permissible to kill Glossop. But on deontological grounds, I also assumed it was wrong to kill one person to save 9999 innocents. This did not seem to fit with standard decision theory, but I managed to make it fit.
I now consider a variant. In Gotham there are right now 10000 setups like the Glossop and Fink-Nottle story, each observed by a different police officer. You are the police chief. You know that in 9999 of the cases, the person in the Glossop role is about the murder the person in the Fink-Nottle role. But in the remaining one case, they are just amateur actors rehearsing a scene.
Should you order by radio all of your officers to shoot the Glossops in the head? There are arguments on both sides.
Pro: Each of the 10000 cases is just like the original case. So if in the original case, the officer should shoot Glossop, each of your 10000 officers should shoot. But if that's what each should do, that's what you should tell each to do.
Con: You know that by ordering all 10000 Glossops to be shot, you will be killing one innocent man (and 9999 guilty ones, but surely that doesn't make it better) in order to save 9999 Fink-Nottles. By deontology, this is wrong.
I think the Con argument is flawed. The death of the innocent Glossop differs from the deaths of the guilty Glossops. The deaths of the guilty Glossops are a means to saving the Fink-Nottles. The death of the one innocent Glossop isn't a means to saving any of the Fink-Nottles. We should take your issuing of the order "Shoot" to be intended to kill the 9999 guilty Glossops, with the death of the innocent Glossop a side-effect. (Admittedly, this is a bit awkward because you are ordering the innocent Glossop to be shot.) The side-effect is not intended, proportionality is met if the ratio 9999:1 is high enough (if not, change the numbers), and it seems likely that the Principle of Double Effect will justify the action.
I think the Pro argument, perhaps with some refinement, is convincing. This means that deontologists must find a way to block the Con argument. But it seems that the only relevant difference between this case and cases where ordering shooting is wrong on deontological grounds is the structure of intentions. So the above case provides support for the kind of focus on intentions that is essential to the Principle of Double Effect.