I have no idea what you're talking about. — Michael
If someone wants to argue that my conclusion is false — Michael
His stipulation that the lamp is on (or off) at t1 is inconsistent with the premises of the problem. — Michael
Certainly, the lamp must be on or off at t1 (provided that it hasn't gone up in a metaphysical puff of smoke in the interval), but nothing we are told implies which it is to be. — Benecerraf
Not quite. The lamp is not defined as on or off. It's just that the rules don't apply at 12:00. But tertium non datur does apply. So it must be (either on or off).But, if the button is pushed at t1/2, t3/4, t7/8, and so on ad infinitum then the lamp is neither on nor off at t1. This is the contradiction. — Michael
It's just that the rules don't apply at 12:00. — Ludwig V
The fact that the conjunction of these premises with the performance of a supertask entails a contradiction is proof that the supertask is impossible, not proof that we can dispense with the premises at 12:00. — Michael
C3 says it's not. — Michael
Your arbitrary stipulation that the lamp is on or off at midnight is inconsistent with P1-P4. — Michael
The lamp can only ever be on iff the button is pushed when the lamp is off to turn it on. The lamp can only ever be off iff either it is never turned on or the button is pushed when the lamp is on to turn it off. Midnight is no exception. — Michael
That seems to be true, so Benacerraf is right. — Ludwig V
Doesn't it follow that both outcomes are consistent with the rules of the problem? — Ludwig V
If both outcomes are consistent with the rules of the problem, doesn't that imply that they are not self-consistent (contradict each other)? If so, Michael is right. — Ludwig V
But if they contradict each other, doesn't ex falso quodlibet applies (logical explosion)? — Ludwig V
The logical explosion implies your conclusion, that justifies your plate of spaghetti, doesn't it? So you are right.
End of discussion? Maybe. — Ludwig V
The rules must be consistent with each other where they apply. The problem is that the rules don't apply to the limit, because the limit is not generated by the function, that is, it is not defined by the function. — Ludwig V
The limit is defined, however, as part of the function, along with the starting-point and the divisor to be applied at each stage. In that sense, they are all arbitrary. But the idea that they could all be replaced by a plate of spaghetti is, I think, I mistake. — Ludwig V
Don't we need to say that these numbers are not defined by the function, but are assigned a role in the function when the function is defined, which is not quite the same as "arbitrary"? The range of arbitrary here, has to be limited to natural numbers; plates of spaghetti are neither numbers nor, from some points of view, natural. — Ludwig V
I'd find it helpful if you would write down a complete description of your version of the problem in one place, rather than pointing me to P1 here and C3 there. Just write down a complete description of the problem for my reference please. — fishfry
Why? I drive down the road and come to a fork. One day I turn left. Then next day I drive down the same road and turn right.? What logical inconsistency do you see to there being multiple possible outcomes to a process that are inconsistent with each other, but each consistent with the rules of the game? — fishfry
None. I'm afraid I'm indulging in double-think in this discussion. I can't make sense of the imaginary lamp. Either it is just a picturesque way of dressing up the abstract structure of the mathematics or it is a physical hypothesis. Some time ago I asked @michael why he didn't just run his computer program. He replied that a computer couldn't execute in the programme in less than some minute fraction of a second, so it wouldn't give an answer. Which was the answer I expected. The computer program was just another way of dressing up the mathematical structure. So I translate all talk of the lamp into abstract structure in which "0, 1, 0, 1, ..." is aligned with "1, 1/2, 1/4, ...".What rule of the problem constrains the terminal state of the lamp? — fishfry
I agree. But I have some other problems about this. I'll have to come back to this later. Sorry.In other words he (sc. Benacerraf) is making the the point that for all we know, the lamp is not even constrained to be either on or off at the terminal state. And why should it be so constrained? — fishfry
I'm afraid I couldn't follow your account of this. I'll have to take another look at it later on. But I'm not sure that the project of trying to articulate the Venn diagram is necessarily the best way to go. It may be constraining, rather than guiding, your thinking. — Ludwig V
Yes, but are the philosophers who want to make synthetic necessity among them? — Ludwig V
However, preserving those concepts doesn't seem to me particularly important. I would be quite happy to abandon all of them. — Ludwig V
Mackie has famously suggested that causes form a family of 'inus' conditions, where an inus condition is 'an insufficient but non-redundant part of an unnecessary but sufficient condition'.
The lamp problem is best modeled as a function defined on the ordinal w+1 — fishfry
But there is no immediate predecessor state to the state at 12:00, so I find it difficult to conceive also requiring that the state at 12:00 is determined by an immediate predecessor state that does not exist. — TonesInDeepFreeze
The sum is not the total addition of all the entries, but the limit of the total addition of all the entries. The total addition of all the entries up to a specific point will converge on/with the sum. — Ludwig V
You're putting the cart before the horse. — Michael
Before we even consider if and when we push the button it is established that the lamp can only ever be on if the button is pushed when the lamp is off to turn it on. The lamp cannot spontaneously and without cause be on. — Michael
Then, we may consider that the problem itself is impossible in the sense that it requires:
(1) a state requires an immediate predecessor state
(2) there is a state at 12:00
(3) there is no predecessor state to the state at 12:00 — TonesInDeepFreeze
If we agree that (1) (2) (3) are together impossible, then we can infer anything from the assumption that they are possible. — TonesInDeepFreeze
Thank you.We just need to say that the infinite sum is the limit of the sequence of finite sums. — TonesInDeepFreeze
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