It's noon. You and two other innocents, A and B, are imprisoned by a dictator in separate blast-proof cells. All the innocents are strangers, and you know of no morally relevant differences between them (whether absolutely or relative to you). A's and B's cells both contain bomb and timer apparatuses that A and B cannot do anything about. B's bomb timer is turned off. A's timer is set to blow her up at 1:00 pm. In your cell, there is a yummy mint on a weight-sensitive switch connected to the apparatus in B's cell. If the mint is removed, B's timer will be set to go off at 1:00 pm. The dictator will check up on the situation shortly before 1:00 pm, and will turn off A's timer if you've done something that caused B's timer to turn on. Anybody who survives past 1:00 pm will then be released.[note 1]
So you reason to yourself. "I like mints. If I eat the mint, I will cause B's death, but A will be saved. My causing of B's death will be non-intentional, and on balance the consequences to human life are neutral. But I get a mint out of it. So the Principle of Double Effect should permit me to eat the mint."
If this reasoning is good, the Principle of Double Effect is close to useless. Strict deontologists think it's wrong to kill one innocent to save millions. Most think it's wrong to kill one innocent to save two. But just about every deontologist will say that it's wrong to kill one innocent to save one innocent and one cat. Now, consider this case. The dictator hands you a gun, and tells you that if you don't kill innocent B, she'll kill innocent A and a cat. You clearly shouldn't. But if you thought it was acceptable to take the mint, then you could reason thus: "It would be interesting to see what a bullet hole in a shirt pocket looks like (and the shirt doesn't belong to B—it is prison attire, belonging to the dictator). If I aim the gun at B's shirt pocket, and press the trigger, the bullet will make a hole in the shirt pocket. And as an non-intended side-effect, it will subsequently cause B's death. But that's fine, because on balance the consequences to human life are neutral, as then B will be saved—plus a cat!" And since you can always think up some minor good that is served by pulling a trigger (finger exercise, practice aiming, etc.), you will get results any deontologist should reject.
So something is wrong with the reasoning—or Double Effect is wrong. I do not think, however, that Double Effect is wrong—I think it's indispensible. So what I will say is this. Double Effect requires that the evil effect not be intended and that there be a proportionality between the side-effect and the intended effect. What the above cases show is that, as a number of authors have noted, proportionality is not a matter of utilitarian calculation. Not only should we have on-balance positive consequences, but the intended effect should be a good proportionate to the foreseen evil. And the foreseen evil is not "that one person fewer will be alive than otherwise", but the foreseen evil is that a particular person should die. The deaths of different people are incommensurable evils even when we know no morally significant differences between the people.
In some cases the virtuous agent may count the numbers of people. But not in these cases. It is callous and unloving to get a mint or produce a bullet hole at the cost of B's death. It trivializes the value of B's life. There is a dilemma here. Either one is acting in the way that causes B's death for the sake of saving A, or not. If one is not, then B literally died so that one might have a mint or be intellectually gratified by the sight of a bullet hole. And so one trivializes B's life. If one is acting to save A, then one is not trivializing B's life. But in that case one is intending B's death, and deontology forbids that.
Here is a variant analysis that comes to the same thing, perhaps. There are cases where one can only do something in one of two ways: by intending a basic evil or by having a morally vicious set of intentions. The cases I gave are like that: one can only take the mint or produce the bullet hole by intending B's death or by having a set of intentions that trivialize B's life. In either case, one is unloving to B. It's hard to say which is the worse.
(This is related to the looping trolley case. There, I think one is either intending the absorption of kinetic energy by the one person, which is problematic, or one is intending a slight increase in length of life or slightly increase in probability of survival on the part of the five, which trivializes the death of the one.)