John Lennon once explained that he had known he was a genius since about the age of twelve: “I thought, ‘I’m a genius or I’m mad. Which is it? I can’t be mad because nobody’s put me away. Therefore I must be a genius.’”

On the face of it, this is a simple example of the restricting the options (or “false dilemma”) fallacy. The starting point of the argument, “I’m a genius or I’m mad”, discounts for no good reason the possibility of sane normality; plenty of people are neither mad nor a genius, so why couldn’t John Lennon have been one of those?

However, the argument isn’t about people in general, but about John Lennon specifically. He didn’t reason “Everyone is either a genius or mad, and I’m not mad so I must be a genius”; he reasoned “I’m either a genius or I’m mad, and I’m not mad so I must be a genius”. If there were some feature of John Lennon that justified this narrowing of the options then the argument might just go through.

So why think that John Lennon must either have been mad or a genius?

Well, how about the fact that he was already convinced he was a genius? He described his sense that he was smarter than everyone else, including his teachers, and his constant mystification that no one else had noticed this. Given that he thought like that, either he was right (in which case he really was a genius) or he was wrong (in which case he was mad in at least some mild sense of suffering from persistent delusions of grandeur).

So perhaps, at a push, we can grant Lennon the first part of the argument; either he was a genius, or he was mad (in some sense).

However, the argument still presents us with a false dilemma, it still commits the restricting the options fallacy.

The problem is with the penultimate step in the argument: “I can’t be mad because nobody’s put me away”. This short sentence contains a whole sub-argument of its own, with the same structure as the main argument. Filling in the implicit reasoning in the sub-argument, we get this: “Either I’m not mad or they would have put me away, but they haven’t put me away so I must not be mad.”

Again, Lennon begins by narrowing the options to two: “Either I’m not mad or they would have put me away.” This time, however, there aren’t good grounds for doing so. Suffering from persistent delusions of grandeur need not be particularly dangerous, so it would be perfectly possible for Lennon to have been mad in that sense without getting locked up.

The argument fails, therefore, not because Lennon can’t prove that he isn’t sane but normal, but because he can’t prove that he isn’t mad.