Thursday, September 3, 2015

Thomson's lamp and two counterfactauls

Thomson's lamp toggles each time you press the button and nothing else affects its state. The lamp is on at noon, and then a supertask consisting of infinitely many button presses that completes by 1 pm, and the question is whether the light is on or off at 1 pm. There is no contradiction yet. But now add these two claims:

  1. The state of the lamp at 1 pm would not be affected by shifting the times at which the button presses happen, if (a) all the button presses happen between noon and 1 pm, and (b) we ensure that no two button presses happen simultaneously.
  2. If we removed one button press from the sequence of button presses between noon and 1 pm, the state of the lamp at 1 pm would not change.
Given this intuition, we do have a problem. Suppose that our sequence of supertask button presses occurs at 12:30, 12:45, 12:52.5, and so on. Then shift this sequence of button presses forward in time, so that now the sequence is at 12:45, 12:52.5, 12:45.25,and so on. By (1) this wouldn't affect the outcome, but by (2) it would as we will have gotten rid of the first button press. That's a contradiction.

So if we think Thomson's lamp is possible--which I do not--we need to deny at least one of the two counterfactuals. I think the best move would be simply to deny both (1) and (2), on the grounds that the connection between the state of the lamp at 1 pm and the button presses must be indeterministic.

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