Tuesday, October 10, 2023

A variant of Thomson's Lamp

In the classic Thomson’s Lamp paradox, the lamp has a switch such that each time you press it, it toggles between on and off. The lamp starts turned off, say, before 10:00, and then the switch is pressed at 10:00, 10:30, 10:45, 10:52.5, 10:56.25, and so on ad infinitum. And the puzzle is: Is it on or off at 11? It’s a puzzle, but not obviously a paradox.

But here’s an interesting variant. Instead of a switch that toggles on or off each time you press, you have a standard slider switch, with an off position and an on position. Before 10:00, the lamp is off. At 10:00, 10:45, 10:56.25, and so on, the switch is pushed forcefully all the way to the on side. At 10:30, 10:52.5, and so on, the switch is pushed forcefully all the way to the off side.

The difference between the slider and toggle versions is this. Intuitively, in the toggle version, each switch press is relevant to the outcome—intuitively, it reverses what the outcome would be. In the slider variant, however, each slider movement becomes irrelevant as soon as the next time happens. At 10:45, the switch is pushed to the on side, and at 10:52.5, it is pushed to the off side. But if you skipped the 10:45 push, it doesn’t matter—the 10:52.5 push ensures that the switch is off, regardless of what happened at 10:45 or earlier.

Thus, on the slider version, each of the switch slides is causally irrelevant to the outcome at 11. But now we have a plausible principle:

  1. If between t0 and t1 a sequence of actions each of which is causally irrelevant to the state at t1 takes place, and nothing else relevant to the state takes place, the state does not change between t0 and t1.

Letting t0 be 9:59 and t1 be 11:00, it follows from (1) that the lamp is off at 11:00 since it’s off at 10:00, since in between the lamp is subjected to a sequence of caually irrelevant actions.

Letting t0 be 10:01 and t1 still be 11:00, it follows from (1) that the lamp is on at 11:00, since it’s on at 10:01 and is subjected to a sequence of causally irrelevant actions.

So it’s on and off at 11:00. Now that’s a paradox!


SMatthewStolte said...

Limit the scope of your principle to cases where the state at t_0 is causally relevant to the state at t_1. The paradox is that you are treating the initial state as both relevant and irrelevant to the end state. I don’t think the principle sounds as plausible when you remove this scope limitation. It sounds awfully weird to say, “Even if t_0 is causally irrelevant to the state at t_1, it can determine the state at t_1 if nothing else is causally relevant to the state at t_1.”

IanS said...

Note that in this version the lamp adds nothing. Only the position of the slider matters. This is different from the standard version.

I take the paradox as a reason to reject (1), which seems contrived in any case. You could use a different phrasing: given the setup, the action at 10:00 ‘does not determine’ the state at 11:00. Similarly for the action at 10:15 etc. Then a principle similar to (1) would say that the whole setup ‘does not determine’ the state at 11:00. Which is true and not paradoxical.

Here is a variation. Suppose that the action a 10:30 is to cancel whatever was done (or not) at 10:00. (If the slider was moved at 10:00, it will be changed back to its pre-10:00 position at 10:30. If it was left unchanged at 10:00, it will again be left unchanged at 10:30.) Similarly, the action at 10:52.5 cancels the action at 10:45, etc. Then the pair of actions at 10:00 and 10:30 is ‘causally inert’, in a stronger sense that in the post, on the state after 10:30. Similarly for the other pairs. It seems that, with this stronger sense of ‘causally inert’, a principle like (1) could apply without paradox. Or could it?

Heavenly Philosophy said...


You seem to echo this here.

IanS said...

Here’s a more direct response. Why is each the actions said to be ‘causally irrelevant’ to the state at 11:00? Because for each action, there is a later action that makes it irrelevant. But for the whole sequence, there is no such action. So (1), at least with ‘causally irrelevant’ interpreted as in the post, is false.