One possible example suggested by Parfit is "FutureTuesday Indifferance". Suppose Bob cares about his future self and in particular prefers not to have pain in the future, as normal people do--except for any future Tuesday. He does not care at if he is going to suffer pain in a future Tuesday. He is happy to choose on Sunday to have an extremely painful operation scheduled for Tuesday instead of a mild one for either Monday or Wednesday. It is not that he does not suffer the pain when Tuesday comes: he is only (irreducibly, arbitrarily) indifferent to pain on Tuesdays that are in the future. Parfit argues that Bob has irrational preferences.

Of course this is a very contrived and psychologically unrealistic example. But Parfit argues that, once it is granted that brute preferences can be irrational, then the question is open whether many of our actual ones are (maybe because they are subtly grounded on distinctions that when examined are as arbitrary as that between future Tuesday and Monday).