I did assume you were endorsing the view you quoted. I should have been a bit more cautious. But I wasn't annoyed or anything. You don't seem particularly upset, so I'm not sure whether an apology is appropriate?

I don't think it is obvious what label we should have for this phenomenon. But I do think there are problems with the proposal. For the record:-

1 I don't see this use of "metaphysical" has any strong relationship to the traditional meaning of the term, so the choice could be regarded as misleading. "New metaphysics" might work.
2 If this is "synthetic necessity", I wonder how we might define "contingent" - the opposite of "necessary" in the traditional structure of these terms.
3 This is classified as "a posteriori" because it is contingent on the relevant rules existing and applying. But all analytic truths are contingent on the relevant definitions (rules of language) existing and applying, so does the term "a priori" have any use?

The following pseudocode provides a demonstration of C3 and is the correct way to interpret the logic of Thomson's lamp: — Michael

Why don't you just run the code and see?
You call it a pseudo-code, but one assumes that you could turn it into real code.

Why don't you just run the code and see? — Ludwig V

Because in reality a computer cannot perform two consecutive operations within 10^-44 seconds.

But we don't need to run the code. We can understand the logic of it and so know that there is no consistent answer. It cannot be true, cannot be false, and cannot be anything other than true or false.

But we don't need to run the code. We can understand the logic of it and so know that there is no consistent answer. It cannot be true, cannot be false, and cannot be anything other than true or false. — Michael

Fair enough. But that's a much better description, IMO. What more is there to say?

The crank needs to follow the conversation but first he needs to learn some basic mathematics.

The matter under consideration is whether time can be divisible ad infinitum.

Density is a property of orderings. An ordering is dense if and only if between any two points there is another point. If time is divisible ad infinitum, then the ordering of points of time is dense.

I have not taken a position on the matter of whether time can be divisible ad infinitum. I only pointed out that, pertaining to the lamp puzzle, Michael's argument is not to refute continuousness but rather to refute density (to be clear, Michael is not the crank).

But, of course, the crank in his unfocused and ignorant way just sees some words on a screen, doesn't know what they mean and then lashes out wildly.

The sum is not the total addition of all the entries, but the limit of the total addition of all the entries. — Ludwig V

No, the infinite sum is the limit of the sequence of the finite sums.

No, the infinite sum is the limit of the sequence of the finite sums. — TonesInDeepFreeze

Yes. Sorry. Careless.

1 I don't see this use of "metaphysical" has any strong relationship to the traditional meaning of the term, so the choice could be regarded as misleading. "New metaphysics" might work. — Ludwig V

Yes, the usage of the term falls out of the traditional. If we want to really go back, metaphysikē is whatever Aristotle wrote (catalogued as such) after his physics. The crux is that when we are talking about metaphysical possibility, it is not automatic that we are talking about possibility in metaphysics, but something related, though different altogether, that makes a noun by itself, metaphysical-possibility, or new(-)metaphysics as you say — likewise logical-possibility does not immediately overlap with possibility in logic, if such a phrase even makes sense immediately. Case in point, the arrangement of games, as described in Toulmin's article, seems to have nothing to do with 'metaphysics' as traditionally used.

2 If this is "synthetic necessity", I wonder how we might define "contingent" - the opposite of "necessary" in the traditional structure of these terms. — Ludwig V

"Metaphysical possibility" is sought as distinct from logical and physical possibility. Using the same Venn diagram I started the thread with, metaphysically contigent must also be logically contigent (Earth is the third planet from the Sun), but it doesn't inherently matter whether it is physically necessary (electric permeability ε) or contigent (kangaroos are digitigrades). Metaphysically contigent includes, at least, the set of all physical contigency. Strictly speaking, and using the definition I tried to work with, metaphysical contigency is everything where there is a possible world where X is not the case; those would also be synthetic contigencies — especially if we take analytic statements to be necessary, otherwise further definition is needed.

3 This is classified as "a posteriori" because it is contingent on the relevant rules existing and applying. But all analytic truths are contingent on the relevant definitions (rules of language) existing and applying, so does the term "a priori" have any use? — Ludwig V

Some philosophers make away with both the a posteriori / a priori and analytic/synthetic distinctions, Kripke gave a strong blow against the idea that a posteriori propositions are always contigent. I don't feel informed enough yet, if ever, to make a statement about it. I take the traditional view and accept all those distinctions.

Case in point, the arrangement of games, as described in Toulmin's article, seems to have nothing to do with 'metaphysics' as traditionally used. — Lionino

Yes. Exactly.

"Metaphysical possibility" is sought as distinct from logical and physical possibility. — Lionino

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.

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

Yes, but are the philosophers who want to make synthetic necessity among them?

I take the traditional view and accept all those distinctions. — Lionino

I'm afraid that I was brought up in the tradition that says that the analytic/synthetic distinction is the only one that means anything. My dilemma is that can see some sense in the point that that one only applies to language, where a priori applies to knowledge and contingent applies to "the world".

Density is a property of orderings. An ordering is dense if and only if between any two points there is another point. If time is divisible ad infinitum, then the ordering of points of time is dense. — TonesInDeepFreeze

As usual, you're making things up to suit your purpose. "Dense order" is a property of the elements of sets, commonly numbers, and never "points". Points do not make up the elements of a set, neither does time make up the elements of a set. You continue in your sophistry, taking a definition from set theory which is specifically applicable only to the elements of a set, and applying this definition to points and time. TIDF, the epitome of a sophist, taking a term with a specific articulated definition, designed for a very specific application, and applying it somewhere else, where it is not suited. How do you propose that a multitude of "points" could be the distinct elements of a set?

But it is still the case that the lamp cannot arbitrarily be on (whether at midnight or any other time). It can only be turned on by pushing a button when it is off. You continually ignore this fact when you talk about the mathematical value ω. — Michael

I've been thinking about your button. I was going to try to address your argument on your own terms, but first I thought I'd go back to Benecerraf's paper to review his argument, which is the one that makes sense to me. He directly addresses some of your concerns.

A note on notation. You like 12:30-1:00 and I prefer 1/2, 3/4, 7/8, ... We can convert between my notation and yours as 12:30 + that fraction of 30 minutes. So my t = 1/2 is your 12:45. Hope that's acceptable. Benecerraf uses my notation, or rather he says that $t_0$ and $t_1$ are the initial and terminal times of the lamp. He doesn't go into any more detail, but this corresponds to my idea of 1/2, 3/4, 7/8, ...; 1 as the times at which we observe the lamp.

I find it helpful to analogize with the mathematical sequence 1/2, 3/4, 7/8, ..., which converges to 1.

Now to Benecerraf's argument.

He posits that two individuals, Aladdin and Bernard, each "perform the supertask." At time $t_1$, the terminal state, Aladdin finds the lamp on, and Bernard finds it off.

Benacerraf argues that neither outcome is inconsistent with the rules of the problem, for the reason that the rules are defined at each of the times t = 1/2, 3/4, 7/8, etc., but NOT at t = 1.

He writes: [Benecerraf's italics as indicated, but I didn't mark up all of them]

I submit that neither description [Aladdin or Bernard's - ed] is self-contradictory, or, more cautiously, that Thomson's argument shows neither description to be self-contradictory (although possibly some other argument might).

According to Thomson, Aladdin 's lamp cannot be on at $t_1$, because Aladdin turned it off after each time he turned it on. But this is true only of instants before $t_1$! From this it follows only that there is no time between $t_0$ and $t_1$ at which the lamp was on and which was not followed by a time also before $t_1$ at which it was off. Nothing whatever has been said about the lamp at $t_1$ or later. And similarly with Bernard's lamp. The only reasons Thomson gives for supposing that his lamp will not be off at $t_1$ are ones which hold only for times before $t_1$. The explanation is quite simply that Thomson's instructions do not cover the state of the lamp at $t_1$, although they do tell us what will be its state at every instant between $t_0$ and $t_1$ (including $t_0$). Certainly, the lamp must be on or off at $t_1$ (provided that it hasn't gone up in a metaphysical puff of smoke in the interval), but nothingf we are told implies which it is to be. The arguments to the effect that it can't be either just have no bearing on the case. To suppose that they do is to suppose that a description of the physical state of the lamp at $t_1$ (with respect to the property of being on or off) is a logical consequence of a description of its state (with respect to the samne property) at times prior to $t_1$. I don't know whether this is true or not, and in section II I shall briefly investigate some matters that bear on this issue. But, true or not, the argument is invalid without the addition of a premise to that effect.
— Benecerraf

I could not have said that any better. Though I'd have inserted some paragraphs. I left it as is since that's what he wrote.

I note in passing that he's anticipating my plate of spaghetti. When he says, "... the lamp must be on or off at $t_1$ (provided that it hasn't gone up in a metaphysical puff of smoke in the interval) ..." he is making the point that the terminal state of the lamp is entirely arbitrary. There are no constraints on the terminal state in Thomson's description of the problem, not even the necessity of being either off or on.

He also denies that the terminal state of the lamp is a logical function of the prior states. He is directly addressing your belief that the terminal state must somehow arise from what's gone before. He says that without additional assumptions, the argument is invalid.

Can you respond to Benecerraf's argument? Clearly he is responding to Thomson's version, and perhaps you have additional assumptions.

My sense of your view is that you intuitively wish that the limit 1 had an immediate predecessor among 1/2, 3/4, 7/8, ... But of course it doesn't. Limits are like that. They don't have immediate predecessors. In fact if you plotted the points 1/2, 3/4, 7/8, ..., and 1 on the number line, you can see that any step left from 1, no matter how tiny, necessarily jumps over all but finitely many of the terms of the sequence.

It's true that if the terminal state is on, then it was off at some time in the past. And if it's off, then it was on at some time in the past. Benecerraf points that out as if anticipating your thought process.

I address it all here. P1 is an implicit premise in Thomson's argument. He is asking "what happens to a lamp if we push its button an infinite number of times?", not "what happens to a lamp if we push its button an infinite number of times and then some arbitrary thing happens to it?".

If your only way to make sense of a supertask is by introducing God or magic to fix the problem at the end then you haven't made sense of a supertask at all. You might as well try to resolve something like the grandfather paradox by doing the same. Time travel into the past isn't physically possible but granting its possibility for the sake of argument doesn't then entail that anything goes, but that seems to be your approach to this issue.

I address it all here. P1 is an implicit premise in Thomson's argument. He is asking "what happens to a lamp if we push its button an infinite number of times?", not "what happens to a lamp if we push its button an infinite number of times and then some arbitrary thing happens to it?". — Michael

P1 says, "Nothing happens to the lamp except what is caused to happen to it by pushing the button"

I do not know if that is something Thomson said or if you added that.

But in that case, then you are subject to Benacerraf's point. You haven't said ANYTHING about the terminal state. I don't know if a button is pushed or not at the terminal time. Who says it's not? You've just got your imagination running away with you about things that don't make sense.

If nothing happens to the lamp unless the button was pushed; then at the terminal time, if anything at all has happened to it -- turned on, turned off, turned into a plate of spaghetti, or as Benecerraf himself allows, just disappeared entirely -- then a button was pushed. If you say so.

There's no contradiction. You said if something happened, a button was pushed. Ok. Something happened and a button was pushed. What of it?

If your only way to make sense of a supertask is by introducing God or magic to fix the problem at the end then you haven't made sense of a supertask at all. — Michael

This is something you're adding, I don't know where you're getting it.

P1 is your own rule. If something happened to the lamp, a button was pushed. So at the terminal state if ANYTHING happened, a button was pushed. You're right. P1 is satisfied. And the lamp is on. Or the lamp is off. Or the lamp has turned into a plate of spaghetti.

You might as well try to resolve something like the grandfather paradox by using the same. Time travel into the past isn't physically possible but granting its possibility for the sake of argument doesn't then entail that anything goes, but that seems to be your approach to this issue. — Michael

I'd invite you to engage more deeply with Benacerraf's argument. I was indeed gratified that he even made my point about the spaghetti. "... provided that it hasn't gone up in a metaphysical puff of smoke in the interval ..." The terminal state is arbitrary.

I don't know if a button is pushed or not at the terminal time. Who says it's not? — fishfry

C3 says it's not. If the button is only ever pushed at 23:00, 23:30, 23:45, and so on ad infinitum, then ipso facto the button is not pushed at midnight.

The terminal state is arbitrary. — fishfry

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

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.

Benacerraf argues that neither outcome is inconsistent with the rules of the problem, — fishfry

That seems to be true, so Benacerraf is right.
Doesn't it follow that both outcomes are consistent with the rules of the problem?
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.
But if they contradict each other, doesn't ex falso quodlibet applies (logical explosion)?
The logical explosion implies your conclusion, that justifies your plate of spaghetti, doesn't it? So you are right.
End of discussion? Maybe.

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.

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. 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.

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

It's not true, and so he's not right.

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" is inconsistent with C2 and C3.

Even some subsequent midnight button push is of no help because of C2 and C3. — Michael

You missed out "The lamp is either on or off at all times."

You missed out "The lamp is either on or off at all times." — Ludwig V

The laws of noncontradiction and excluded middle are implied.

The laws of noncontradiction and excluded middle are implied. — Michael

Clearly not by you. Could've easily included it. So why didn't you?

Surely you must believe your interlocutor has a point in his statement other than that of a any random dreg of society: "you're wrong".

He has a point, a truth and testimony, an entire world he wishes you to experience

— Michael

If the button is only ever pushed at 23:00, 23:30, 23:45, and so on ad infinitum, then the lamp is neither on nor off at midnight — Michael

So we can agree that the consequent is false. Ex falso quodlibet, so a plate of spaghetti.

Clearly not by you. Could've easily included it. So why didn't you? — Outlander

Yes by me. I didn't include them because they're implied. But if you insist on making it explicit then I will.

So we can agree that the consequent is false. — Ludwig V

Yes. And therefore the antecedent is necessarily false. The supertask is metaphysically impossible.

Yes. And therefore the antecedent is necessarily false. Supertasks are metaphysically impossible. — Michael

I don't know about metaphysically possible or impossible. Logically impossible, certainly. So what are you arguing about?

So what are you arguing about? — Ludwig V

That the supertask is metaphysically impossible.

That supertasks are metaphysically impossible. — Michael

Is is not the case that "logically impossible" implies "metaphysically impossible"?

Is is not the case that "logically impossible" implies "metaphysically impossible"? — Ludwig V

Yes.

So what are you arguing about?

So what are you arguing about? — Ludwig V

I have already said. I am arguing that the supertask is metaphysically impossible.

Am I contradicting you?

Am I contradicting you? — Ludwig V

You were when you said this:

Benacerraf argues that neither outcome is inconsistent with the rules of the problem,
— fishfry

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

Benacerraf is not right. His stipulation that the lamp is on (or off) at t₁ is inconsistent with the premises of the problem.

I am arguing that the supertask is metaphysically impossible. — Michael

As you understand it, of course. And, make no mistake, I would be among the first to swear your understanding and take of things is valid, but also the first to defend the idea that things can change.

Surely some things must remain consistent, for is this not the basis of sanity, after all?

But there still remains some outlier arguments you believe to be irrelevant, or in your words "implied". This is all the opposing party wishes to acknowledge, I do believe.

In simple terms, perhaps, a remote possibility exists, the individual who declared such a statement, has different views and relevant perceptions toward said statement than you hold. Nothing more. Simply the remote possibility. Not that one is "wrong" but simply there is more to the intended truth than one currently perceives. Surely such a thing is possible, no?