So we can reject (1) and be left with a consistent set of two premises. — TonesInDeepFreeze

To reject (1) is to claim that the lamp can spontaneously and without cause be on at 12:00. This is impossible. The lamp can only be on at 12:00 if the button was pushed when the lamp was off to turn it on.

We must reject (3), and so reject the possibility of the supertask.

To reject (1) is to claim that the lamp can spontaneously and without cause be on at 12:00. — Michael

It rejects that having an on/off state is determined by an immediate predecessor state, so when we reject that premise, it is not ruled out that the state at 12:00 is determined in another way. (Yes, this is similar to Benacerraf.)

The lamp can only be on at 12:00 if the button was pushed when the lamp was off to turn it on. — Michael

You're reiterating a premise that we are free to reject since it is impossible given the other premises. Just reiterating it like that is begging the question in this context.

You're reiterating a premise that we are free to reject. Just reiterating it like that is begging the question in this context. — TonesInDeepFreeze

It's not a premise. It's a conclusion derived from our premises. See the argument above.

You are not including the premise "The lamp can only be on if immediately preceding it was off. And the lamp can be off only if immediately preceding it was on"?

You are not including the premise "The lamp can only be on if immediately preceding it was off. And the lamp an be off only if immediately preceding it was on"? — TonesInDeepFreeze

No, I've linked you to the argument. You can read the premises there.

P1. Nothing happens to the lamp except what is caused to happen to it by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at 10:00

From these we can then deduce:

C1. The lamp is either on or off at all tn >= 10:00
C2. The lamp is on at some tn > 10:00 iff the button was pushed at some ti > 10:00 and <= tn to turn it on and not then pushed at some tj > ti and <= tn to turn it off
C3. If the lamp is on at some tn > 10:00 then the lamp is off at some tm > tn iff the button was pushed at some ti > tn and <= tm to turn it off and not then pushed at some tj > ti and <= tm to turn it on

From these we can then deduce:

C4. If the button is only ever pushed at 11:00 then the lamp is on at 12:00
C5. If the button is only ever pushed at 11:00 and 11:30 then the lamp is off at 12:00
C6. If the button is only ever pushed at 11:00, 11:30, 11:45, and so on ad infinitum, then the lamp is neither on nor off at 12:00 [contradiction] — Michael

C1 is a premise. It is the premise that the lamp has only two states. But that's not a substantive problem; only that I'm mentioning that it is a premise.

C2 and C3 together seem to include "only by an immediate predecessor". But I don't see a valid inference for them from the premises. It seems to me that the premises don't preclude that the button can be pushed at 12:00 without there being an immediate predecessor state.

Thank you.

C1 is a premise. — TonesInDeepFreeze

It’s not, it’s a valid inference from the premises.

As per P4, the lamp starts off. As per P2, pushing the button will turn it on. As per P3, pushing the button again will then turn it off. As per P2, pushing the button again will then turn it on. And so on ad infinitum. And as per P1 there's no third party magic or God or gremlin that can cause the lamp to vanish in a puff of smoke or turn into a plate of spaghetti. So the lamp can only ever be either off or on. There's no mechanism by which the lamp can be anything else.

It seems to me that the premises don't preclude that the button can be pushed at 12:00 without there be an immediate predecessor state. — TonesInDeepFreeze

Firstly, as per the antecedent of C6, the button is not pushed at 12:00. Thomson is addressing what happens if the supertask is performed, not what happens if the supertask and some independent subsequent task is performed.

Secondly, pushing the button at 12:00 will only turn the lamp on if the lamp is off when the button is pushed. So to say that the button is pushed at 12:00 to turn the lamp on is to say that if the button is not pushed at 12:00 then the lamp would be off at 12:00. But that's not possible because of C3.

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

Possible outcomes can indeed be inconsistent with each other. But if they are inconsistent with each other, they can't both be actual at the same time. You can't drive down the road and turn left and right at the same time.

Benecerraf explicitly says: "... 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) ..."
In other words he 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

There is more to this than meets the eye, I think. Benecerraf's quotation is somewhat hedged. And "for all we know" hints at unexpressed complexities, I'm interested in all that. See below.

Yes, but are the philosophers who want to make synthetic necessity among them?
— Ludwig V
I don't get it. There is something missing in this phrase. — Lionino

I was commenting on

Some philosophers make away with both the a posteriori / a priori and analytic/synthetic distinctions, — Lionino

I'll try again. "Is it the case that all the philosophers who want to make away with those distinctions the same as those who want to define synthetic necessary truths"



I looked again at Benecerraf's article and found what I was looking for. His position is much more nuanced than I thought. Selective quotation is not ideal, but my summary would likely be worse. So here goes:-

A "swindle" has taken place, and we have been the victims. Somehow, all was going along swimmingly, and suddenly we find ourselves drowning in contradiction with no idea of how we got there. We are told that the concept of a super-task is to blame, but we are not told what about it has such dire consequences. We are sufficiently sophisticated mathematically to know that the concept of infinity is not at fault (or if it is, a lot more than the future of super-tasks is at stake). — Benacerraf on Supertasks - The Journal of Philosophy, 1962, p. 781

I suspect that, by and large, it is principally compound expressions that suffer the fate I attribute to 'completed infinite sequence of tasks' ..... What seems most notable about such compounds is the fact that one component (e.g., 'infinite sequence') draws the conditions connected with its applicability from an area so disparate from that associated with the other components that the criteria normally employed fail to apply. We have what appears to be a conceptual mismatch. Sequences of tasks do not exhibit the characteristics of sequences that lend themselves to proofs of infinity. And since there seems to be an
upper bound on our ability to discriminate (intervals, say) and none on how finely we cut the task, it appears that we should never be in a position to claim that a super-task had been performed. But even if this is true, it only takes account of one kind of super-task, and, as I argue above, it hardly establishes that even this kind constitutes a logical impossibility. — Benacerraf on Supertasks - The Journal of Philosophy, 1962, p. 783/4

To look at the matter diachronically and therefore, I think, a little more soundly, we can see our present situation as akin to that of speakers of English long before electronic computers of the degree of complexity presently commonplace when confronted with the question of thinking robots (or, for that matter, just plain thoughtless robots, I suspect). They were as unthinkable as thinking stones. Now they are much less so. I am not sure that even then they constituted a logical contradiction. However, I would not resist as violently an account which implied that the expression 'thinking robot' had changed in meaning to some degree in the interim. Viewed as I suggest we view them, questions of meaning are very much questions of degree-in the sense that although relative to one statement of meaning there may be a more or less sharp boundary established, no statement of meaning (viewing things synchronically now) is uniquely correct. Other hypotheses, and therefore other lines may be just as reasonable in the light of the evidence. The statement of the meaning of a word is a hypothesis designed to explain a welter of linguistic facts-and it is a commonplace that where hypotheses are in question many are always possible. — Benacerraf on Supertasks - The Journal of Philosophy, 1962, p. 784

Therefore, I see two obstacles in the way of showing that supertasks are logically impossible. The first is that relevant conditions associated with the words and the syntactic structure involved must be found to have been deviated from; and it must be argued that these conditions are sufficiently central to be included in any reasonable account of the meaning of the expression. The second is simply my empirical conjecture that there are no such conditions: that in fact the concept of super-task is of the kind I have been describing above, one suffering from the infirmity of mismatched conditions. — Benacerraf on Supertasks - The Journal of Philosophy, 1962, p. 784

The bolded sentence expresses my preferred diagnosis. (Which, by the way, is channelling Ryle. I think Benecerraf must have know that - look at the date of the article.) In the light of the various further supertasks that have been developed, a conclusive refutation seems as unlikely for the supertask problems as it is for the Gettier problems. But this is a good candidate.

These are our premises before we even consider if and when we push the button:

P1. Nothing happens to the lamp except what is caused to happen to it by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at 10:00 — Michael

I repeat: Please post a complete description of the problem.

In P1 I have no idea what "the lamp" is. It's perfectly clear that you are adding hidden assumptions to make your view of the problem work out. I'm trying to get you to make these assumptions explicit.

So I ask you to please post a complete description of the problem, as if I'd never heard of it.

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. — Michael

That is far from "established." That's why I keep asking you to write out a complete description of the problem, as if I've never heard of it before. That's the only way to make plain the unspoken assumption's you're adding that are not in Thomson's original formulation.

Benacerraf makes this point. He specifically says that Thomson's argument is invalid, unless additional assumptions are made. You have clearly made additional assumptions, and I'm asking you to make them plain by writing out a complete description of this problem.

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, ...". — Ludwig V

Same way I see it. The sequence 0, 1, 0, 1, ... has no limit, so one terminal state is as good as any other.

I agree. But I have some other problems about this. I'll have to come back to this later. Sorry. — Ludwig V

This is regarding the puff of smoke or the plate of spaghetti. And that's why I mention Cinderella's coach. Nobody ever complains about that. Why is the lamp constrained to be off or on, when it's a fictitious lamp in the first place?

Possible outcomes can indeed be inconsistent with each other. But if they are inconsistent with each other, they can't both be actual at the same time. You can't drive down the road and turn left and right at the same time. — Ludwig V

Different copies of the lamp, or same experiment run at different times. Why is this unclear? Benacerraf didn't say Aladdin and Bernard each see different things at the same time with the same lamp. Why are you objecting to an argument nobody made?

You have a hidden element here, known as freedom of choice. The "multiple possible outcomes" are only the result of this hidden premise, you have freedom to choose. That premise overrules "the rules of the game", such that the two are inconsistent. In other words, by allowing freedom of choice, you allow for something which is not "consistent with the rules of the game", this is something outside the rules, the capacity to choose without rules. — Metaphysician Undercover

Didn't follow that.

The claim was that multiple possible outcomes of a process is inconsistent. Not so. Each outcome is consistent with the rules of the problem. There's nothing inconsistent about a lamp being on sometimes and off other times.

The claim was that multiple possible outcomes of a process is inconsistent. Not so. Each outcome is consistent with the rules of the problem. There's nothing inconsistent about a lamp being on sometimes and off other times. — fishfry

The claim was directed at your example of choosing a direction at a fork in the road. The only way that you could have multiple possible outcomes is by assuming a principle that overrules the rules, i.e. transcends the rules. Freedom of choice, allows you to choose rather than follow a rule. If your example is analogous, then multiple possible outcomes being consistent with the rules, implies that choice is allowed, i.e. the rules allow one to transcend the rules. Strictly speaking the actions taken when the rules are transcended are not consistent with the rules, because these actions transcend the rules. The rules may allow for such acts, acts outside the system of rules, but the particular acts taken cannot be said to be consistent with the rules because they are outside the system.

The claim was directed at your example of choosing a direction at a fork in the road. The only way that you could have multiple possible outcomes is by assuming a principle that overrules the rules, i.e. transcends the rules. Freedom of choice, allows you to choose rather than follow a rule. If your example is analogous, then multiple possible outcomes being consistent with the rules, implies that choice is allowed, i.e. the rules allow one to transcend the rules. Strictly speaking the actions taken when the rules are transcended are not consistent with the rules, because these actions transcend the rules. The rules may allow for such acts, acts outside the system of rules, but the particular acts taken cannot be said to be consistent with the rules because they are outside the system. — Metaphysician Undercover

The terminal state of the lamp is not defined, so it may be on or off. What on earth is wrong about that? If you flip a coin it might be heads or tails. That doesn't mean it can't be both at different times. Why is Benacerraf's point confusing to people?

P1. Nothing happens to the lamp except what is caused to happen to it by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at 10:00 — Michael

(1) Why not use 11:00 rather than 10:00? Usually the problem concerns 11:00 to 12:00, which is tidy for the halvings of the durations. (I'll use 11:00.)

(2) We could do without pushing a button and even the lamp. We could couch it in more abstract terms.

(3) Your premises have only one necessary condition for the lamp changing state, which is P1.

This is consistent with your premises:

The lamp is off at 11:00. The button is pushed at 12:00 and the lamp goes on.

So I think you've left out a lot of what you need in your premises.

I have no idea what "the lamp" is. — fishfry

It doesn't matter to me what the lamp is.

I can regard the problem abstractly, in terms just of:

time
two states
sufficient and necessary conditions for changing states

Mentioning a lamp and it being on or off and a button that is pushed are, for me, all just visualization aids that are dispensable. Moreover, they seem to interfere sometimes when people get hung up on how to relate such a hypothetical lamp and button with actual lamps and buttons.

Moreover, they seem to interfere sometimes when people get hung up on how to relate such a hypothetical lamp and button with actual lamps and buttons. — TonesInDeepFreeze

Yes, another one of @Micheal's conceptual confusions. That's why I mention Cinderella's coach. The lamp is a fairy tale, and it's a fallacy to try to reason about how it works.

EDIT LATER: Disregard this post. I hope to post a revision.

I'll try this:

Suppose:

There are two states F and N.

At any moment either F is active or N is active and not both.

For every natural number n>0, there is a time T(1/(2^n)).

At time T(0), F is the active state.

For every natural number n>0, the active state changes at time T(1/(2^n)).

The active state changes only at time (T(1/(2^n)) for some natural number n.

Question: What is the active state at time T(1)?

I don't see it as a confusion of Michael. He is only rendering Thomson's setup. And I don't see Michael getting tripped up by the metaphorical use of a lamp and button. And I don't see Thomson as getting tripped up either.

That was a complete description. There are no hidden assumptions.

P1-P4 are our premises. C1-C3 follow. And then C4-C6 follow.

P1 is implicit in Thomson's argument. Using the principle of charity you should infer it. As neither you nor Benacerraf have done so I have had to make it explicit.

As a comparison, consider the following:

The lamp is off at 10:00. The button is pushed 10¹⁰⁰¹⁰⁰ times between 10:00 and 10:01. Is the lamp on or off at 10:02?

Any reasonable person should infer that nothing else happens between 10:01 and 10:02. Even though this is a physically impossible imaginary lamp, and even though I haven't told you what happens at 10:02, it is poor reasoning to respond to the question by claiming that the lamp can turn into a plate of spaghetti. The correct answer is that because 10¹⁰⁰¹⁰⁰ is an even number, the lamp will be off at 10:02.

There is no Supreme Button Pusher arbitrarily willing the lamp to be on or turning it into a pumpkin. There is only us pushing the button once, twice, or an infinite number of times, where pushing it when the lamp is off turns the lamp on and pushing it when the lamp is on turns the lamp off.

This is consistent with your premises:

The lamp is off at 11:00. The button is pushed at 12:00 and the lamp goes on. — TonesInDeepFreeze

Yes, this is where we have C4 and C5:

C4. If the button is only ever pushed at 11:00 then the lamp is on at 12:00
C5. If the button is only ever pushed at 11:00 and 11:30 then the lamp is off at 12:00

It is only when we change from having pushed the button a finite number of times to having pushed the button an infinite number of times that we are met with a contradiction:

C6. If the button is only ever pushed at 11:00, 11:30, 11:45, and so on ad infinitum, then the lamp is neither on nor off at 12:00

For every $n \in \mathbb{N_0}$ it is possible to have pushed the button $n$ times (and every $n \in \mathbb{N_0}$ is finite), but it is not possible to have pushed the button $\aleph_0$ times.

If you flip a coin it might be heads or tails. That doesn't mean it can't be both at different times. — fishfry

This doesn't make sense. Each flip of the coin is an individual act, and it has a single outcome. Once the outcome is achieved, that outcome stands until there is another flip. The outcome "can't be both at different times", because a different outcome requires a different flip. However, there can be different outcomes from different flips.

This is regarding the puff of smoke or the plate of spaghetti. And that's why I mention Cinderella's coach. Nobody ever complains about that. Why is the lamp constrained to be off or on, when it's a fictitious lamp in the first place? — fishfry

I think the problem is precisely that there is nothing to constrain the lamp and we want to find something. In theory, we could stipulate either - or Cinderella's coach. But we mostly think in the context of "If it were real, then..." Fiction doesn't work unless you are willing to do that. It's about whether you choose to play the game and how to apply the rules of the game.

The terminal state of the lamp is not defined, so it may be on or off. What on earth is wrong about that? — fishfry

This seems to be more in tune with common sense, for what it's worth. The question is, why? I think it is because of the dressing up of the abstract structure. We assume the lamp has existed before the sequence and will continue to exist after it. So the fact that the sequence does not define it does not close the question and we want to move from the possible to the actual. But it is not clear how to do that - and we don't want to simply stipulate it. Perhaps that's because defining the limit of the convergent sequence as 1 - or 0, which have a role in defining the sequence in the first place, invites us to think in the context of the natural numbers (or actual lamps), whereas defining ω as the limit of the natural numbers does not.

Strictly speaking the actions taken when the rules are transcended are not consistent with the rules, because these actions transcend the rules. The rules may allow for such acts, acts outside the system of rules, but the particular acts taken cannot be said to be consistent with the rules because they are outside the system. — Metaphysician Undercover

I hope you meant that actions taken outside the system are neither consistent nor inconsistent with the rules. Could we not express this by saying that the rules don't apply, or that it is not clear how to apply the rules, in the new context?

Changing from a finite number of button pushes to an infinite number of button pushes doesn't let you avoid this common sense reasoning. — Michael

No, but it might be the case that common sense reasoning doesn't apply or is misleading in the context of infinity.

There is only us pushing the button an infinite number of times, where pushing it when the lamp is off turns it on and pushing it when the lamp is on turns it off. — Michael

You can think about us doing that, but you can't limit our thinking to that context. That's where the problems start.

What is the causal consequence of us having done this (and only this)? — Michael

Us doing this is not an empirical possibility, so there can't be any causal consequences. But I think you mean to ask what outcome there can be if we think only in that context. Sadly, that context doesn't give us an answer - except possibly that the state of the lamp is both on and off or neither on nor off.

You can think about us doing that, but you can't limit our thinking to that context. That's where the problems start. — Ludwig V

The lamp is off at 10:00. I push the button at 10:01, turning the lamp on. Is the lamp on or off at 10:02?

The correct answer is "on".

You don't get to invent your own premises and stipulate that some magical gremlin turns the lamp into a plate of spaghetti at 10:02. In doing so you are no longer addressing the thought experiment that I have presented.

Us doing this is not an empirical possibility — Ludwig V

Neither is pushing the button 10¹⁰⁰¹⁰⁰ times within one minute, but we are still able to reason as if it were possible and deduce that the lamp would be off when we finish. That's just how thought experiments work.

You don't get to invent your own premises and stipulate that some magical gremlin turns the lamp into a plate of spaghetti at 10:02. In doing so you are no longer addressing the thought experiment that I have presented. — Michael

Your thought experiment, your rules. But whose thought experiment is Achilles' race and Thompson's lamp? I had the impression that they are Zeno's or Thompson's. What if there's something wrong with them, such as they contradict each other or lead to a self-contradictory conclusion?

Neither is pushing the button 10100100 times within one minute, but we are still able to reason as if it were possible and deduce that the lamp would be off when we finish. That's just how thought experiments work. — Michael

True. I wrote carelessly. What deduction do you make when you think about pushing the button after an infinite sequence, which is defined without completion, of button pushes within one minute. Oh, wait, I know.

If the button is only ever pushed at 11:00, 11:30, 11:45, and so on ad infinitum, then the lamp is neither on nor off at 12:00 — Michael

You know perfectly well that's self-contradictory, so necessarily false. Ex falso quodlibet otherwise known as logical explosion. Or your deduction is wrong. (But I don't think it is wrong - or at least, not any more wrong than the spaghetti).
Not my rules. Yours.

You know perfectly well that's self-contradictory, so necessarily false. — Ludwig V

Yes, this is how refutations by contadiction work.

I'll quote Thomson for you:

After I have completed the whole infinite sequence of jabs, i.e. at the end of the two minutes, is the lamp on or off? It seems impossible to answer this question. It cannot be on, because I did not ever turn it on without at once turning it off. It cannot be off, because I did in the first place turn it on, and thereafter I never turned it off without at once turning it on. But the lamp must be either on or off. This is a contradiction.

This is what proves that "super-tasks are not possible of performance". Any completed sequence of tasks is necessarily finite.

Any completed sequence of tasks is necessarily finite. — Michael

Quite so. And the phrase "completed sequence of tasks" is self-contradictory. So what do we need your argument for?

And the phrase "completed sequence of tasks" is self-contradictory. — Ludwig V

Did you mean that the phrase "completed infinite sequence of tasks" is self-contradictory? If so then yes.

So what do we need your argument for? — Ludwig V

To prove it. Those like Benacerraf and fishfry either claim that it isn't self-contradictory or that it hasn't been proven to be self-contradictory.

Did you mean that the phrase "completed infinite sequence of tasks" is self-contradictory? If so then yes. — Michael

You are right, of course. I'm glad you could decipher what I meant to say.

Those like Benacerraf and fishfry either claim that it isn't self-contradictory or that it hasn't been proven to be self-contradictory. — Michael

Benacerraf's position is a bit more complicated than that.

Thomson is ... successful in showing that arguments for the performability of super-tasks are invalid and ... nevertheless his own arguments against their possibility suffer the same fate. — Benacerraf on Supertasks

Those like Benacerraf and fishfry either claim that it isn't self-contradictory or that it hasn't been proven to be self-contradictory. — Michael

Thanks for clarifying that you meant self-contradictory. I've been wondering what your conclusion contradicted.