Thomson's lamp

Thomson's lamp has an on-off switch. It begins in the "off" position. At noon the switch is toggled, and the lamp comes on. Half a minute later, the switch is toggled, and the light goes off. A quarter of a minute later, the switch is toggled again, and the light comes on. And so on. There are no other switch flippings than these, and the switch survives at least until 12:01 pm. At 12:01 pm, is the switch on or off?

As paradoxes go, this one seems really flimsy. As best I can see, the argument to a paradox is something like this:

1. Time is actually infinitely subdivided.
2. If time is actually infinitely subdivided, the story of Thomson's lamp is possible.
3. Necessarily, if the story is true, then the switch is either on or off at 12:01.
4. Necessarily, if the story is true, then the switch is not on at 12:01.
5. Necessarily, if the story is true, then the switch is not off at 12:01.

The argument for (4) is, presumably, that after every time the switch is on, there is a next time when it is off, and the argument for (5) is similar.

There are a couple of ways of showing what's wrong with the argument. Here is one. In order to argue for (4) and (5), it needs to be a part of the story that

6. At each time t after noon, the position of the switch is the result of the last switch-flipping event prior to t.

For suppose that we deny this. Then we can allow that the switch is on at 12:01, but not due to any switch-flipping event. Or we can allow that the switch is off at 12:01, but not due to any switch-flipping event. After all, perhaps, the switch, instead of being flipped, just undergoes a quantum leap from one position to another.

Fine, then, says the paradoxer: Add (6) to the story.

However, now (2) becomes false. The defender of actually infinitely subdivided time can simply deny (2), since the story is plainly inconsistent: the position of the switch at 12:01 is determined by the last flip before 12:01, but there is no last flip before 12:01. It is a story as plainly inconsistent as this one: "Whether there is an obligatory side of the street to drive on is determined by the content of the will of the king of France. And France is a monarchy." The question of the position of the switch is rather like asking: "If atheism were true, would God want us to be atheists?"

Perhaps the paradoxer will say that that was her whole point, but nonetheless the defender of actually infinitely subdivided time has to affirm that this inconsistent story is possible. But why? It is an easy game to construct inconsistent stories by including a stipulation that something is sufficient to determine something, and then adding to the story something that denies the existence of the determiner. In addition to my present king of France story, consider this one:"A lamp with an on-off switch that can only have two positions, on and off, is produced ex nihilo by God at t₀. The position of the switch at any time is fully determined by how it has last been flipped." Then ask: What is the position of the switch at t₀? Obviously, we have an inconsistency in the story—if the lamp came into existence ex nihilo at t₀ it came into existence with the switch in a particular position, but that position was not determined by a flipping.

But does not the defender of actually infinitely subdivided time think that a lamp switch's being flipped in the supertask way is possible? Certainly. But she has to hold that this is only possible in those worlds in which either something other than the last flip determines the position of the switch at 12:01 or the Principle of Sufficient Reason is violated (I don't think there are any such) or both.