> It's different because by adopting C, Bentham prevents the mugger from mugging, which is a better world than one where the mugger goes on mugging.
This assumption is wrong. You are assuming that the mugger is also a utilitiarian, so will do cost-benefit analysis, and thus decide not to mug. But that is not necessarily true.
If the mugger mugs anyway, despite mugging being "suboptimal," Bentham ends up in a situation where he has exactly the same choice: either lose $10, or have the mugger cut off their own finger. If Bentham is to follow (act-)utilitarianism precisely, he must pay the mugger $10. (Act-)utilitarianism says that the only thing that matters is the utility of the outcome of your action. It does not matter that Bentham previously committed to not paying the mugger; the fact is, after the mugger "threatens" Bentham, if Bentham does not pay the mugger, total utility is less than if he does pay. So Bentham must break his promise, despite "committing" not to. (Assuming this is some one-off instance and not some kind of iterated game; iteration makes things more complicated.)
If everyone were a utilitarian, then there would be far fewer objections to utilitarianism. (E.g. instead asking people in wealthy countries to donate 90% of their income to charity, we could probably get away with ~5-10%.) Bentham's mugging is a specific objection to utilitarianism that shows how utilitarians are vulnerable to manipulation by people who do not subscribe to utilitarianism.
Also, to be precise, Bentham's mugging does not show a contradiction. It's showing an unintuitive consequence of utilitarianism. That's not the same thing as a contradiction. (If you want to see a contradiction, Stocker has a different critique: https://www.jstor.org/stable/2025782.)
Except that eventually the mugger will run out of fingers (and/or reattaching them will eventually start not working out), so the mugger will be forced to stop mugging. Well, ok, they could start cutting off toes, or threaten some other form of bodily mutilation.
But regardless, giving in to the mugger enables the mugger to continue mugging indefinitely. Not giving in -- assuming the mugger goes through with whatever self-mutilation they've threatened -- will eventually cause the mugger to stop mugging. This would be a net positive, better than allowing the mugger to mug indefinitely.
On top of that, I was disappointed that, in the story, Bentham does actually bring up the idea that capitulating could encourage copycats to run similar schemes, but this rationale for not cooperating is hand-waved away. This is pretty standard "don't negotiate with terrorists" stuff. Giving in just tells the mugger -- and other potential muggers -- that this strategy works. Surely it's more utilitarian to stamp this out at the source, even if it costs the original mugger some fingers.
(But I guess this is in part the point of Bentham being an act utilitarian in the first encounter, as he wouldn't consider the larger implications of his actions, just on the specific, immediate result of the action in front of him.)
This assumption is wrong. You are assuming that the mugger is also a utilitiarian, so will do cost-benefit analysis, and thus decide not to mug. But that is not necessarily true.
If the mugger mugs anyway, despite mugging being "suboptimal," Bentham ends up in a situation where he has exactly the same choice: either lose $10, or have the mugger cut off their own finger. If Bentham is to follow (act-)utilitarianism precisely, he must pay the mugger $10. (Act-)utilitarianism says that the only thing that matters is the utility of the outcome of your action. It does not matter that Bentham previously committed to not paying the mugger; the fact is, after the mugger "threatens" Bentham, if Bentham does not pay the mugger, total utility is less than if he does pay. So Bentham must break his promise, despite "committing" not to. (Assuming this is some one-off instance and not some kind of iterated game; iteration makes things more complicated.)
(In fact, this specific objection -- that utilitarianism requires people to "give up" their commitments -- is at the foundation of another critique of utilitarianism by Williams: https://123philosophy.files.wordpress.com/2018/12/bernard-wi...)
If everyone were a utilitarian, then there would be far fewer objections to utilitarianism. (E.g. instead asking people in wealthy countries to donate 90% of their income to charity, we could probably get away with ~5-10%.) Bentham's mugging is a specific objection to utilitarianism that shows how utilitarians are vulnerable to manipulation by people who do not subscribe to utilitarianism.
Also, to be precise, Bentham's mugging does not show a contradiction. It's showing an unintuitive consequence of utilitarianism. That's not the same thing as a contradiction. (If you want to see a contradiction, Stocker has a different critique: https://www.jstor.org/stable/2025782.)