S5 has as an axiom that ◊□p ⊢ □p.

Therefore, under S5, these cannot both be true:

1. ◊□q
2. ◊□¬q

Therefore, under S5, this is not true:

3. ¬□¬p ⊢ ◊□p

This then relates to the post above.

Assuming that (a) means (b), (b) needs to be justified. Given that (3) is false, this is false:

4. ¬□¬∃xC(x) ⊢ ◊□∃xC(x)

So ◊□∃xC(x) must be justified some other way for a modal ontological argument to work.

So we both agree that modal ontological arguments like Gödel's fail to prove the existence of God.

See the opening post, where Gödel's argument is presented. See line C:



These are the kinds of modal ontological arguments that I am addressing.

The implications of the natural English propositions and the implications of the modal logic propositions diverge drastically, and it would be silly to prefer the modal logic to the natural English. That would be to let the tail wag the dog, as I argued — Leontiskos

I'm addressing modal ontological arguments. These arguments try to use modal logic to prove the existence of God.

No one thinks creation was necessary. It seems that you have gotten your theology from Richard Dawkins. — Leontiskos

It was just an example. Replace with "omnipotence", "omniscience", or whatever you want.

Then the modal logic fails to translate, because <it is possible that there exists a necessary being> does not mean <it is possibly necessary that there is a being>. — Leontiskos

Modal ontological arguments try to use modal logic to prove the existence of God. In particular, they use S5's axiom that ◊□p ⊢ □p.

At their most fundamental, their premises take the following form:

1. X is God if and only if X necessarily exists and has properties A, B, and C¹.
2. It is possible that God exists.

To prevent equivocation, we must use (1) to unpack (2), reformulating the argument as such:

1. X is God if and only if X necessarily exists and has properties A, B, and C.
3. It is possible that there exists some X such that X necessarily exists and has properties A, B, and C.

The phrase "it is possible that there exists some X such that X necessarily exists" is somewhat ambiguous. To address this ambiguity, we should perhaps reformulate the argument as such:

1. X is God if and only if X necessarily exists and has properties A, B, and C.
4. It is possibly necessary that there exists some X such that X has properties A, B, and C.

We can then use S5's axiom that ◊□p ⊢ □p to present the following modal ontological argument:

1. X is God if and only if X necessarily exists and has properties A, B, and C.
4. It is possibly necessary that there exists some X such that X has properties A, B, and C.
5. Therefore, there necessarily exists some X such that X has properties A, B, and C.

This argument is valid under S5. However, (4) needs to be justified; it is not true a priori.

If, as you claim, (3) and (4) are not equivalent, then prima facie one cannot derive (5) from (3), and so something other than S5 is required.

_{¹ The particular properties differ across arguments; we need not make them explicit here.}

Hence, if it is not necessary that there is a god, then there is no god. — Banno

Both this claim and the claim that God is necessary amuse/confuse me.

Imagine that some intelligent, all powerful, all knowing, creator of the universe actually does exist, but that because it doesn't necessarily exist then we refuse to call it God, as if the name we give it is what matters.

So, X is God if and only if X has all positive properties.
Necessary existence is a positive property.
Being all powerful is a positive property.
Being all knowing is a positive property.
Therefore, X is God if and only if X necessarily exists, is all powerful, is all knowing, etc.

Now, what does "God possibly exists" mean? In modal logic we would say ◊∃xG(x) which translates to "it is possible that there exists an X such that X is God."

Using the definition above, this means:

It is possible that there exists an X such that X necessarily exists, is all powerful, is all knowing, etc.

But what does this mean? In modal logic we would say ◊□∃x(P(x) ∧ K(x) ∧ ...) which translates to "it is possibly necessary that there exists an X such that X is all powerful, is all knowing, etc."

Notice how "it is possible that there exists an X such that X necessarily exists ..." becomes "it is possibly necessary that there exists an X such that X ...". This step is required to make use of S5's axiom that ◊□p ⊢ p. But it also removes necessary existence as a predicate.

All we are left with is the claim that it is possibly necessary that there exists an X such that X is all powerful, is all knowing, etc. This is a claim that needs to be justified; it isn't true by definition.

To me, this is circumvented by D1, defining God as having all positive properties. — Lionino

Here are three different claims:

1. If X is God then X has all positive properties
2. If X has all positive properties then X is God
3. X is God if and only if X has all positive properties

Which of these is meant by "God is defined as having all positive properties"?

The true value of Gödel's work is that it manages to prove that atheists will reject a mathematically unobjectionable proof if it proves something that they disagree with. — Tarskian

It's not a mathematically unobjectionable proof.

In its simplest form it is:

◊p
p ≔ □q
∴ ◊□q
∴ q

But given the second line, this is equivalent to:

◊□q
∴ q

Which begs the question.

I'll translate it into English for ease.

Neither of these are contradictions:

1. There exists a unique creator god who performs miracles
2. There exists a unique creator god who does not perform miracles

But they cannot both be true. Therefore, under S5, at least one of these is false:

3. It is possibly necessary that there exists a unique creator god who performs miracles
4. It is possibly necessary that there exists a unique creator god who does not perform miracles

Therefore, we cannot just assume that because some X is not a contradiction that it is possibly necessary. We need actual evidence or reasoning to support such a claim.

◊~p → □◊~p (5 axiom)
◊~p → ~◊~◊~p (Definition of □)
~~◊~◊~p → ~◊~p (Contraposition)
◊~◊~p → ~◊~p (Double negation)
◊□p → □p (Definition of □)

Question: Do you put the same constraint on Cinderella's coach? Why or why not? Want to understand your answer. — fishfry

I don't understand your question.

Asking me why I'm using P1 as a premise is as nonsensical as asking me why I'm using P2 as a premise. They are just the premises of the thought experiment. The intention is to not allow for the lamp to be off, for the button to be pushed just once, turning the lamp on – and then for the lamp to be off.

We are trying to understand what it means to perform a supertask, and so we must assert that nothing other than the supertask occurs. There are no spontaneous, uncaused events. If we cannot make sense of what the performance of the supertask (and only the supertask) causes to happen to the lamp then we must accept that the supertask is metaphysically impossible.

So I don't see the point of your argument here — Ludwig V

Benacerref claimed that the supertask being performed and then the lamp being on is not a contradiction. I am trying to prove that it is (or rather that Thomson already proved this).

The lamp cannot be on after the performance of the supertask and cannot be off after the performance of the supertask – precisely because there is no final button push and because the lamp cannot spontaneously and without cause be either on or off.

The pseudocode I provided before shows this. Its logic does not allow for echo isLampOn to either be determined to output true or false or to arbitrarily output true or false. Therefore, we must accept that it is impossible in principle for while (true) { ... } to ever complete.

And so we must accept that it is impossible in principle for a supertask to be performed.

If the last stage of the supertask was on... — Ludwig V

A supertask has no last stage. Again to quote Thomson, "I did not ever turn it on without at once turning it off [and] I did in the first place turn it on, and thereafter I never turned it off without at once turning it on."

Therefore, if the lamp is on after having performed the supertask then the lamp being on has nothing to do with me having pushed the button to turn it on. The lamp being on would be spontaneous and without cause, which just isn't possible given our premises.

The problem is that whether the supertasks can be performed is not really the issue. — Ludwig V

Yes, it is. Thomson's argument attempts to prove that supertasks cannot be performed.

The premises don't not specify that the button is ever pushed. — TonesInDeepFreeze

If the button is never pushed then as per P1 and P4 the lamp will forever be off, consistent with C1.

The premises do not specify that there are only two states, unless, in this very hypothetical context we are clear that 'Off' is defined as 'not On', though it does seem reasonable that that is implicit. — TonesInDeepFreeze

Yes, "off" means "not on". The lamp's bulb is either emitting light or not emitting light.

See also the first part of my response to fishfry above.

So, the first question to consider is:

1. If something is possibly necessary, is it necessary?

Under S5 (one type of modal logic), the answer is "yes". Ontological arguments depend on this. They all reduce to the claim that because God is possibly necessary, God is necessary.

If we reject S5 then the answer is "no" and all ontological arguments fail.

But let's assume S5 and that the answer is "yes".

The next questions are:

2. Is it possible that there necessarily exists a God who is unique and performs miracles?
3. Is it possible that there necessarily exists a God who is unique and does not perform miracles?

If we accept S5 and if (2) and (3) are both true then it is both the case that there necessarily exists a God who is unique and performs miracles and that there necessarily exists a God who is unique and does not perform miracles.

This is a contradiction. Therefore, (2) and (3) cannot both be true.

Therefore, either:

4. It is not possible that there necessarily exists a God who is unique and performs miracles, or
5. It is not possible that there necessarily exists a God who is unique and does not perform miracles

Even though "God is unique and performs miracles" is not a contradiction, it might not be possibly necessary, and even though "God is unique and does not perform miracles" is not a contradiction, it might not be possibly necessary.

Therefore, one cannot claim that because some definition of God is consistent then it is possibly necessary.

Therefore, the claim that God is possibly necessary begs the question, and as such all ontological arguments fail.

Perfectly clear that you have stated nothing about 10:02. For all we know it turns into a pumpkin. — fishfry

As per P1, the lamp cannot spontaneously and without cause turn into a pumpkin, and there cannot be a god or wizard or gremlin magically turning the lamp into a plate of spaghetti.

And then as per P2, P3, and P4, pushing the button can never cause the lamp to vanish in a puff of smoke.

So the lamp can never turn into a pumpkin. It can never turn into a plate of spaghetti. It can never vanish in a puff of smoke. It can only ever be either off or on.

Before we even consider a supertask, do you at least understand that if the button is pushed to turn the lamp on (and then not pushed again) then the lamp stays on?

Do you at least accept these?

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

Can't you see why I'm demanding that you write out, in one place, your entire description of the problem. That way you would be able to catch yourself making stuff up as you go. — fishfry

I did so. It's here.

Quite so. But how does it help when we are thinking about an infinite sequence? As I understand it, the point is that the sequence cannot define it's own limit. — Ludwig V

That's precisely the problem. Both of these things are true:

1. The lamp can never spontaneously and without cause be on
2. If the supertask is performed, and if the lamp is on after the performance of the supertask, then the lamp being on after the performance of the supertask is spontaneous and without cause.

Therefore we must accept that the supertask cannot be performed.

And even if we were to grant an alternate account that allows for the lamp to spontaneously and without cause be on, doing so does not answer Thomson's question. He wants to know what the performance of the supertask causes to happen to the lamp. Having some subsequent, independent, spontaneous, acausal event after the performance of the supertask does not tell us what the performance of the supertask causes to happen to the lamp. It's a red herring.

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.

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.

Consider this:

1. ∃xF(x) → ∃x∀y(F(y) ↔ (x = y))

If we take F(x) to mean something like "x is the only unicorn" then (1) is true.

Now consider these:

2. ◇∃x(F(x) ∧ A(x))
3. ◇∃x(F(x) ∧ ¬A(x))

If take A(x) to mean something like "x is male" then both (2) and (3) are true.

Now consider these:

4. ◇□∃x(F(x) ∧ A(x))
5. ◇□∃x(F(x) ∧ ¬A(x))

Under S5, ◇□p ⊢ □p, and so these entail:

6. □∃x(F(x) ∧ A(x))
7. □∃x(F(x) ∧ ¬A(x))

(6) and (7) cannot both be true, and so therefore (2) does not entail (4) and (3) does not entail (5):

8. ◇∃xP(x) ⊬ ◇□∃xP(x).

This is where modal ontological arguments commit a sleight of hand. To claim that it is possible that God₁ exists, where necessary existence is one of God₁'s properties, is to claim that it is possibly necessary that God₂ exists, where necessary existence is not one of God₂'s properties.

The claim that it is possibly necessary that God₂ exists isn't true a priori, and so the claim that it is possible that God₁ exists isn't true a priori. As it stands it begs the question.

Or we have to reject S5, but if we reject S5 then modal ontological arguments are invalid because “possibly necessary” wouldn’t entail “necessary”.

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.

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.

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.

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.

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.

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.

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.

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?

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.

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.

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.

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.

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.

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 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.

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.

I have no idea what you're talking about.

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.

So what are you arguing about? — Ludwig V

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