Thomson's Lamp and change

Start with two Thomson's Lamps. They each have toggle switches and are on at 10 a.m. The switches are toggled at 10:30, 10:45, 10:52.5 and so on. Now suppose, as is surely possible, that regardless of what, if any, state the lamps would have had at 11 a.m., aliens come and instantaneously force the first lamp to be on at 11, and force the second to be off (say, by breaking the bulb!).

Now some but not all causal interactions are changes. Which lamp's on/off state did the aliens change? There are four possible answers:

1. The first but not the second
2. The second but not the first
3. Both
4. Neither.

Symmetry considerations rule (1) and (2) out of court. Can we say that in both cases, the aliens changed the on/off state of the lamp? Surely not. For if something can have only two states, it can't be that each of the two possible state inductions counts as a change. Moreover, if the lamp were to have changed state, what state did it change from? For inducing an on state only counts as a change if the induction starts with the lamp in an off state, and vice versa. But the induction didn't start with the lamp on, nor did it start with the lamp off. That leaves only last option: Neither.

But it seems that if an object has been persisting, and a causal interaction induced a state in that objection, that causal interaction either was a state-change or a state-maintenance. So if in neither case did the aliens change the state of the lamp, then it seems that in both cases they maintained the state. But we get analogues of (1)-(4), and analogues of the above arguments also lead to the conclusion that in neither case did the aliens maintain the state.

So the Thomson's Lamp story forces us to reject the dichotomy between state-change and state-maintenance.

Here's another curious thing. It seems that the following is true:

5. If an object has state A at t₁ and non-A at t₂, then the object's having state non-A at t₂ is the result of a change.

But applying (5) shows that both lamps' final states are the results of a change. But that change must have thus been from the opposite state. And yet the final state doesn't follow right after the opposite state.

One could use this as an argument against the possibility of infinitely subdivided time. Alternately, one could use this as an argument against principles like (5) and the idea that the concepts of change and maintenance are as widely applicable as we thought them to be.