Cora Diamond has an interesting paper giving a supposed (emphasis should be placed on this word) counterexample to utilitarianism. She writes
We start with a philosophical story of the usual sort. The ocean liner has sunk, and you can save only one of two people struggling in the water nearby. One has one leg; you have gathered from what you heard during the voyage that the loss of the leg occurred some time ago. The other person has two legs. You know virtually nothing else about them. They appear to be about the same age. They are both men or both women; it does not matter which. Many contemporary versions of utilitarianism imply that it would normally be wrong for you to choose to save the one-legged person." They have an even stronger consequence: that to choose to save the one-legged person is not just wrong, but is wrong for almost the same reasons for which it is wrong to cause a person to lose his leg in the first place. (1 am excluding cases in which there was some good reason for causing someone to lose his leg, e. g., in order to save his life. I mean to include cases in which the loss of a leg is caused by negligence, or by more spectacular wrongdoing, like terrorist acts or deliberate attempts to maim
I don’t think our intuitions about the strength of reasons is very reliable. As I’ve argued previously — those intuitions are just wildly off (also see Singer’s drowning child thought experiment for more on this).
But an argument can be given for the same strength of reasons.
The reasons to save a leg and to save a life with an extra leg over one without one are equally strong if saving the life that’s missing a leg plus sacrificing another random extra leg of a person would be just as good as saving the life that is missing a leg.
saving the life that’s missing a leg plus sacrificing another random extra leg of a person would be just as good as saving the life that is missing a leg.
Therefore, the reasons to save a leg and to save a life with an extra leg over one without one are equally strong.
1 is obvious — the reasons are equally strong if sacrificing a leg makes the scenarios precisely equal. This can be easily seen with an example — if we think that you have a 25 unit of reason force reason to save a leg and 5000 to save a life with two legs and 4975 to save a life with one leg, then the difference between your reasons to save a two life leg and a one life leg would be 25 — precisely equal to your reasons to save a leg from destruction.
2 is also plausible — after you take the action in both cases there will be a person still alive but missing a leg. One would rationally be indifferent from behind the veil of ignorance.
I don’t think our intuitions about the strength of our reasons are very reliable — especially in cases where we have many reasons pointing in different directions. Your reason to save 100,000 rather than 99,999 is just as strong as your reason to save one — yet that doesn’t seem intuitive. When we consider how strong a reason is, our assessment of its strength is often affected by the other reasons in the scenario — hence this weird leg-based intuition.
One practical issue worth flagging is adjustment costs. Cost (losing a leg) = cost (adjusting to one leg) + cost (having only one leg). Since we're told that "the loss of the leg occurred some time ago", the adjustment cost has already been paid. So that's a reason why your premise (2) isn't quite right: sacrificing "another random extra leg" would lead to additional adjustment costs that aren't present in saving the person who is already used to living with one leg. (It's equally a reason why Diamond may be mistaken to attribute to utilitarians the view that "to save the one-legged person... is wrong for almost the same reasons for which it is wrong to cause a person to lose his leg in the first place", depending on how much weight is put on "almost" here, and the empirical question of how the two aforementioned costs compare.)
But it's easy enough to tweak the case. Just make the "random extra leg sacrifice" involve *moving forwards in time* a loss of leg that would eventually occur anyway (perhaps very near their end of life, otherwise), so that a corresponding number of two-legged life-years get converted into one-legged life-years, without changing the number of people who have to go through the adjustment at some point in their life.