I have no idea what you're talking about.

I have no idea what you're talking about. — Michael

And that is understood. Well and good. This is, after all, the only rational non-ill-intended platform of debate or intellectual opposition. Reasonable inquiry itself, even.

I suppose, to simplify, how would you best explain the reasoning behind your recent argument/reply/rebuttal or opposing position to someone who is either: unaware of the argument, that is to say, does not have the fortune to be in the posses ion of the knowledge you hold (yet is capable of so). Surely your questioner is of sound and rational mind and morality. So, if he is incorrect, and you assert the position you remain the opposite, how does one become equals on the same page of truth and rationale you see?

I can't preempt someone's disagreement. If someone wants to argue that my conclusion is false then they need to tell me which step in the argument they disagree with, and why. Only then can I respond.

If someone wants to argue that my conclusion is false — Michael

Surely there are many ways to assist a friend that their current action or belief that will assuredly guide future actions are "lacking", that is to say, have reasonable methods of improvement not currently taken.

Things are not alwaysfalse or valid. Binary enslavement. It is not always "my way or the highway", that is to say, perhaps one there is more that one can contribute to a certain goal or ideal than can be immediately ascertained. Is this false or true?

I'm really sorry, but my fat thumb syndrome struck and my last message got posted before I had finished with it. This version is finished.

His stipulation that the lamp is on (or off) at t1 is inconsistent with the premises of the problem. — Michael

Benecerraf's sentence is not exactly that:-

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

Aren't you forgetting tertium non datur?

The rules of the problem stipulate whether the lamp is on or off at 11:00, 11:30, 11:45, and so on ad infinitum, but not whether it is on or off at 12:00.

I grant you that they will tell you whether it is on or off at any specific time before 12:00, but they does not tell you whether it is on or off at 12:00.

The lamp is either on or off at t₁.

But if the button is pushed at t_1/2, t_3/4, t_7/8, and so on ad infinitum, then the lamp is neither on nor off at t₁. This is the contradiction.

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

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).

It's just that the rules don't apply at 12:00. — Ludwig V

Yes they do. P1-P3 are always true. C1-C3 follow from P1-P4 and explicitly apply at all times >= 10:00.

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.

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

Sorry. I have something else to do. I didn't expect to convince you, but our discussion has helped to confirm my opinion.

C3 says it's not. — Michael

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.

Also, can you please note any divergences between Michael's lamp and Thomson's.

Your arbitrary stipulation that the lamp is on or off at midnight is inconsistent with P1-P4. — Michael

Mine, yes. But Benacerraf's also. I wish you'd carefully study his argument and respond to it. I mention this "appeal to authority" because Benacerraf is a big time mathematical philosopher, not just some anon rando such as my humble self.

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

Benacerraf addressed this point. Your assumption about what happens at the limit point is either (a) not part of Thomson's original formulation; or (b) logically inconsistent.

That seems to be true, so Benacerraf is right. — Ludwig V

Seems that way to me as well. @Michael is wishing that a limit point had an immediate predecessor, but it doesn't, and he's stuck there.

Doesn't it follow that both outcomes are consistent with the rules of the problem? — Ludwig V

Yes, Benacerraf and I conclude that.

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

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?

But if they contradict each other, doesn't ex falso quodlibet applies (logical explosion)? — Ludwig V

Not at all. What logical inconsistency is there if I turn left today and right at the same junction tomorrow?

Is geometry inconsistent because it allows both Euclidean and non-Euclidean geometry? They are inconsistent with each other; but they are each consistent in themselves.

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

As far as I'm concerned this is a solved problem. And I do believe that if @Micheal would stop falsely imagining that there's an immediate predecessor to a limit point, he'd agree.

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

Yes. Benacerraf explicitly says that. He says it's a mistake to assume that the terminal state bears any "logical relation," his words, to the preceding sequence of states.

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

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?

I am indeed gratified to see Benacerraf making this point.

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

Why? It's an imaginary lamp to start with. I have previously analogized it to Cinderella's coach, which turns into a pumpkin at exactly midnight, but is a beautiful jewel-encrusted coach at and time prior to midnight.

What rule of the problem constrains the terminal state of the lamp? Thomson gives no such constraint. On the other hand @Michael may indeed have added his own such constraint, making the problem impossible. So Michael has not solved Thomson's lamp, he's solved Michael's lamp, by adding a condition that isn't present in the original problem.

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

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

From these we can then deduce:

C1. The lamp is either on or off at all t_n >= 10:00
C2. The lamp is on at some t_n > 10:00 iff the button was pushed at some t_i > 10:00 and <= t_n to turn it on and not then pushed at some t_j > t_i and <= t_n to turn it off
C3. If the lamp is on at some t_n > 10:00 then the lamp is off at some t_m > t_n iff the button was pushed at some t_i > t_n and <= t_m to turn it off and not then pushed at some t_j > t_i and <= t_m 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]

Benacerraf's "solution" – to arbitrarily stipulate that the lamp is on/off at 12:00 – is inconsistent with C2 and C3. Even our hypothetical lamp cannot spontaneously and without cause be on.

His "solution" doesn't even answer the question, which asks what happens to the lamp if we push its button an infinite number of times. We want to understand the causal consequence of having performed the supertask. Having the lamp spontaneously and without cause be on after having performed the supertask does not tell us what having performed the supertask entails.

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

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.

What rule of the problem constrains the terminal state of the lamp? — 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, ...".

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

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

It is the point of this thread as briefly stated in the OP itself https://thephilosophyforum.com/discussion/14855/metaphysically-impossible-but-logically-possible/p1

As to the issue with Venn diagrams, that is something that other users articulated too. Nevertheless, I think this stems from the definition of metaphysics we are going with. In some thread me and Josh had a short debate about it. If we see metaphysics as broadly synonymous with ontology, the Venn diagram linked there seems to be unproblematic: surely we agree that every physical possibility is also a logical possibility, so one encompasses the other; I think we can also agree that if something is ontologically possible, it is also logically possible, so one encompasses the other; using an etymological fallacy, metaphysics (ontology) is everything beyond physics, so metaphysical possibility encompasses physical possibility.

But it is mostly a musing of mine. There is a lot of talk about the spheres of possibility. I don't expect others to read it fully, but all I can do is recommend the following article: https://plato.stanford.edu/entries/modality-epistemology/ The beginning is the most important part of the article anyway.

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.

However, preserving those concepts doesn't seem to me particularly important. I would be quite happy to abandon all of them. — Ludwig V

I don't disagree, especially when names such as "logically-possible" seem to actually be conceptually/analytically-possible instead of anything to do with logic. The SEP article claims that the epistemology of possibility is the primordial issue at stake in issues such as mind-body dualism and the Berkeley's argument for Idealism. Though this could be true in a sense, I myself think that the issue is a semantic one and, if it is a metaphysical issue in any capacity, it is a derivative issue, not a primordial one — just like p-zombies is completely derivative from dualism/physicalism.

On the talk of necessity and contigency, I recently came across something interesting, "inus" conditions, where necessity and sufficiency blend:

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 butting of heads over Benacerraf can be reduced at least somewhat if we look closely at the premises. Two options:

(1) We do not make explicit the premise that the state at 12:00 is determined by an immediate predecessor state.

That tends to favor Benacerraf.

(2) We do make explicit the premise that the state at 12:00 is determined by an immediate predecessor state.

That tends to disfavor Benacerraf.

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.

The lamp problem is best modeled as a function defined on the ordinal w+1 — fishfry

I understand the idea that the domain is w+1 and I too mentioned it a while back.

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

You're putting the cart before the horse.

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.

Our problem is that if (a) a supertask is performed and if (b) the lamp is on after we stop pushing the button then (c) the lamp is spontaneously and without cause on.

@fishfry and Benacerraf fail to acknowledge that (c) is impossible, proving that (a) and (b) are incompatible.

If (a) is true then (b) is false. If (b) is true then (a) is false. The same reasoning holds when we claim that the lamp is off after we stop pushing the button.

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

We just need to say that the infinite sum is the limit of the sequence of finite sums.

Whether one agrees with Michael or not, at least he has been making a good faith argument and refining it along the way. And fishfry pertinently presents a dissent that deserves consideration.

But meanwhile there is the crank who posts pure garbage as he shows that he is not even following along in the discussion. In the course of exercising his personal umbrage, he terribly fouls up the subject matter and the path of the conversation here.

The comments below should not have to be belabored, but the crank's trash talk should not be left standing.

The problem under discussion itself is couched in terms of moments* in time and durations of time that are indexed numerically. And the problem supposes that the durations may be divided and that there are infinitely many divisions. *And it is common for people to refer to 'points in time' pretty much in the same sense of 'moments in time'. Again, l have not taken any position about the nature of time or the common understanding of time represented as points on a line. I am only taking the problem as it is presented. The premises and rubrics in the problem are not ones that I have endorsed nor disputed.

Michael claimed to prove that time is not continuous. I pointed out that, for his own purposes, he needs to prove that time is not dense. Again, I haven't taken a position on that question.

The problem presupposes that points in time are ordered, such that 11:00 precedes 11:30 precedes 11:45, etc. A dense order is one in which between any two different points there is yet another point. And the ordering of time in which the thought experiment supposes that time can be divided ad infinitum is a dense order. So, my remark about density is that for Michael's own purpose, he should be claiming to refute that time is dense in that sense and not the weaker claim that time is not continuous. Again, I haven't taken any position of my own about time; I have only taken the problem and poster's arguments on their own terms.

But the crank lashes out with stupid strawmen, getting the path of conversation abysmally mixed up. He irrationally snarls with nonsense projected onto me such as "time consists of two distinct substances", "infinitely dense particles", "the irrational space between the rational numbers".

The crank is a bane.

You're putting the cart before the horse. — Michael

A horse can push a cart, not only pull 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. The lamp cannot spontaneously and without cause be on. — Michael

I haven't refused that. But I suggest that 'immediate predecessor' is a good way of couching the matter. 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

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

Yes, that's the basic argument I've been making. The lamp being off must always precede it being on. Therefore, the lamp cannot be modelled over time by the infinite sequence off, on, off, ...; on.

That seems to drawing an inference from an impossibility.

If we agree that (1) (2) (3) are together impossible, then we can infer anything from the assumption that they are possible.

If we agree that (1) (2) (3) are together impossible, then we can infer anything from the assumption that they are possible. — TonesInDeepFreeze

If you're referring to the principle of explosion, then sure.

The point though is that Thomson shows that the lamp can neither be on nor off after having performed the supertask, which is a contradiction, and so that the supertask is impossible in principle to perform.

But, if I am not mistaken, your argument comes down to: From the assumption that (1) (2) (3) are together possible, we infer that time is not infinitely divisible, not merely that a certain supertask is impossible.

More like:

P1. The lamp being off must always precede it being on.
C1. Therefore, the lamp cannot be modelled over time by the infinite sequence off, on, off, ...; on.

P2. If time is infinitely divisible then the lamp can be modelled over time by the infinite sequence off, on, off, ...; on.
C2. Therefore, time is not infinitely divisible.

Although on this latter point perhaps this argument is more compelling.

We reject that it is possible for (1) (2) (3) to hold together. So we can reject (1) and be left with a consistent set of two premises. So it is not ruled out that it is possible that there are denumerably many alternating states and that time is infinitely divisible.

I don't understand your argument, or at least I don't think you understand my argument. Before we even address the infinite divisibility of time, the full argument related to Thomson's lamp is presented here.

Do you disagree with anything said there?

You can couch the hypothetical situation with whatever premises you like. In that sense it's not a matter of me agreeing or disagreeing. And the part about Benacerraf doesn't seem relevant to my previous post, since I am not using referencing Benacerraf's argument in my previous post.

Meanwhile, I don't know what is not understandable about my previous post.

We just need to say that the infinite sum is the limit of the sequence of finite sums. — TonesInDeepFreeze

Thank you.