Complete conditional probabilities, infinitesimal probabilities and two easy Frankenstein facts

Suppose we have a complete finitely-additive conditional probability P(⋅|⋅) on some algebra F of events on a space Ω (i.e., P is a Popper function with all non-empty sets regular), so that P(A|B) is defined for every non-empty B.

Here’s a curious thing: there is a very large disconnect between how P behaves when conditioning on sets of zero measure and when conditioning on sets of non-zero measure.

Here’s one way to see the disconnect. Consider any other complete finitely additive conditional probability Q(⋅|⋅) on the same algebra F, and suppose that Q assigns unconditional measure zero to everything that P does: i.e., if P(A|Ω)=0, then Q(A|Ω)=0.

Then, Frankenstein-fashion, we can sew P and Q into a new conditional probability R where R(A|B)=P(A|B) if P(A|Ω)>0 and R(A|B)=Q(A|B) if P(A|Ω)=0. In other words, R behaves exactly like P when conditioning on non-zero measure sets and exactly like Q when conditioning on zero measure sets.

[To check that R is a conditional probability, the one non-trivial condition to check here is that R(A ∩ B|C)=R(A|C)R(B|A ∩ C). Suppose P(C|Ω)=0. Then the equality follows from the corresponding equality for Q. Suppose P(C|Ω)>0. If P(A ∩ C|Ω)>0 as well, then our equality follows from the corresponding equality for P. Suppose now that P(A ∩ C|Ω)=0. Then the equality to be demonstrated is equivalent to P(A ∩ B|C)=P(A|C)Q(B|A ∩ C). But if P(A ∩ C|Ω)=0, then P(A ∩ B|C)=0 and P(A|C)=0.]

There is an analogous Frankenstein fact for infinitesimal probabilities. Let P and Q be any two finitely-additive probabilities with values in some hyperreal field, and suppose that Q is tiny whenever P is tiny, where a hyperreal is “tiny” provided that it is zero or infinitesimal. Then there is a Frankenstein probability R = Std P + Inf Q, where Std x and Inf x are the standard and infinitesimal parts of a finite hyperreal. (The fact that Q is tiny when P is tiny is used to show that R is non-negative.) This R then has the same large-scale (i.e., standard scale) behavior as P and small-scale behavior as Q.

In other words, as we depart from classical probability, we get a nearly complete disconnect between small-scale and large-scale behavior.

Arbitrariness, regularity and comparative probabilities

In “Underdetermination of infinitesimal probabilities”, following a referee’s suggestion, I allow that qualitative (i.e., comparative) probabilities may escape arbitrariness problems for infinitesimal probabilities. But I now think this may be wrong.

Consider an infinite line of independent fair coins, numbered with the integers ..., −2, −1, 0, 1, 2, .... Let R_n be the hypothesis that coins n, n + 1, ... are all heads. Let L_n be the hypothesis that coins ..., n − 2, n − 1, n are all heads.

Suppose “being less likely or equally likely” is transitive, reflexive and total. Write A < B for A being less likely than B, A ≈ B for A and B being equally likely, and A ⪅ B for A being less likely or equally like than B.

We have strong regularity provided that if event A is a proper subset of event B, then A is strictly less likely than B.

Given strong regularity, the events L_n are strictly decreasing in probability: ... > L₋₂ > L₋₁ > L₀ > L₁ > ... and the events R_n are strictly increasing in probability: ... < R₋₂ < R₁ < R₀ < R₁ < ....

Theorem: Given strong regularity, exactly one of the following options is true:

1. For all n and m, L_n < R_m (heads-runs are right-biased)

2. For all n and m, R_m < L_n (heads-runs are left-biased)

3. There is a unique n such that for all m ≤ n we have R_m ⪅ L_m and for all m > n we have L_m < R_m (there is a switch-over point at n).

But if (1) or (2) is true, it is difficult to see what objective reality could possibly ground whether heads-runs are right-biased or left-biased in our infinite sequence of coin tosses. And if (3) is true, it is difficult to see what objective reality could possibly make n be a switch-over point. The choice between left- and right-bias seems completely arbitrary, and the choice of a switch-over point is also arbitrary.

The Theorem follows from the following lemma:

Lemma: Let (S, ⪅) be a totally preordered set, and let L_n and R_n be sequences of members of S as n ranges over the integers. Suppose L_n is strictly decreasing and R_n is strictly increasing. Then exactly one of the following is true:

1. For all n and m, L_n < R_m

2. For all n and m, R_m < L_n

3. There is a unique n such that for all m ≤ n we have R_m ≤ L_m and for all m > n we have L_m < R_m.

Proof of Lemma: Suppose that for all n we have L_n < R_n. I claim that (1) is true. For suppose that (1) is false and hence (by totality) we have R_m ⪅ L_n for some m and n. Then m ≠ n, since L_n < R_n. Either m < n or n < m. If m < n, then R_m ⪅ L_n < L_m, and if m > n, then R_n < R_m ⪅ L_n. In either case we have a violation of the fact that L_n′ < R_n′ for all n′.

Now suppose that for all n we have R_n < L_n. I now claim that (2) is true. For if (2) is false, we have L_n ⪅ R_m for some m ≠ n. Suppose m < n. Then L_n ⪅ R_m < R_n, a contradiction. And if n < m, then L_m < L_n ⪅ R_m, also a contradiction.

Let A(n) be the statement that for all m ≤ n we have R_m ⪅ L_m and for all m > n we have L_m < R_m. There is at most one n such that A(n). For suppose that we had A(n) and A(n′) and that n < n′. Then by A(n) we have L_n′ < R_n′ (since n′>n), and by A(n′) we have R_n′ ⪅ L_n′, resulting in a contradiction.

Assume (1) and (2) are false. By what we have shown earlier and totality, if (2) is not true, there is an n such that L_n ⪅ R_n. Note that if L_n ⪅ R_n, then L_m < L_n ⪅ R_n < R_m for all m > n. Hence, either L_n ⪅ R_n is true for all n or else there is a smallest n for which it’s true. If it’s true for all n, then for all n we have L_n + 1 < L_n ⪅ R_n < R_n + 1, and hence for all n we have L_n < R_n, which we saw would imply (1), which we assumed to be false.

So suppose that n is the smallest integer for which L_n ⪅ R_n. Thus, R_m < L_m whenever m < n. Moreover, since L_i is strictly decreasing and R_i is strictly increasing, we have L_m < R_n whenever m > n. There are now two possibilities. Either L_n < R_n or L_n ≈ R_n. If L_n ≈ R_n, then we have A(n) and the proof is complete. Suppose now that L_n < R_n. Then we have A(n − 1) and the proof is also complete.

Symmetry, regularity and qualitative probability

Let ⪅ be a qualitative probability comparison for some collection F of subsets of a space Ω. Say that A ≈ B iff A ⪅ B and B ⪅ A, and that A < B provided that A ⪅ B but not B ⪅ A. Minimally suppose that ⪅ is a partial preorder (i.e., transitive and reflexive). Say it’s total provided that for all A and B either A ⪅ B or B ⪅ A. Suppose that G is a group of symmetries acting on Ω, and that F is G-invariant in the sense that gA ∈ F for all g ∈ G. Then we can define:

1. ⪅ is strongly G-invariant provided that for all A in G and all g in G we have A ≈ gA, and

2. ⪅ is weakly G-invariant provided that for all A and B in G and all g in G we have A ⪅ B iff gA ⪅ gB.

There is some reason to be suspicious of strong G-invariance. For in some interesting cases, say where Ω is a circle that G is the set of all rotations, there will be cases where gA is a proper subset of A, and by regularity we would expect to have gA < A rather than gA ≈ A. But weak G-invariance seems harder to question.

Say that g ∈ G is of order n provided that gⁿ = e, where e is identity. However, we also have:

Lemma 1. If ⪅ is total, g is of order 2, and A ⪅ B implies gA ⪅ gB for all A and B, then A ≈ gA for all A.

Proof: Since ⪅ is a total order, either A ⪅ gA or gA ⪅ A. Suppose A ⪅ gA. Then gA ⪅ g²A. But g²A = A. Hence gA ≈ A. Similarly, if gA ⪅ A, then g²A ≈ gA. But g²A = A, so gA ≈ A.

So, we have:

Proposition 1. If ⪅ is total and G is a group generated by elements of order 2, then weak G-invariance entails strong G-invariance.

Say that ⪅ is strongly regular provided that if A is a proper subset of B, then A < B. Weak regularity would say that if B is non-empty then ∅ < B. Weak regularity together with an appropriate additivity condition will imply strong regularity (details left to the reader).

Proposition 2. If G is generated by elements of order 2, and ⪅ is total and weakly G-invariant, then if there exists a g ∈ G and A ∈ F such that gA is a proper subset of A, then G is not strongly regular.

Proof: Strong regularity would require that gA < A, but that would contradict strong G-invariance which we have by Proposition 1.

Corollary 1. If F is a collection of subsets of the unit circle containing all countable sets and invariant under all reflections, and ⪅ is a total qualitative probability comparison weakly invariant under all reflections, then ⪅ is not strongly regular.

Proof: The group generated by all reflections includes all rotations. But there is a subset A of the circle and a rotation g such that gA is a proper subset of A. For instance, let A be the set of points at angles r, 2r, 3r, ... in degrees, where r is irrational, and let g be rotation by r. Then rA is the set of points at angles 2r, 3r, 4r, ... in degrees.

Now, imagine an infinite line on which there are infinitely many evenly spaced people, stretching out in both directions, each of whom flips a fair coin. Let Ω be the probability space describing these flips. Let H_n be the event that all the flips starting with person number n (i.e., n, n + 1, n + 2, ...) land heads. Suppose that F contains all the H_n and is invariant under all reflections of the situation (where we reflect the setup either about the point at which some person stands or at a point half-way between two neighboring people).

Corollary 2. If ⪅ is a total qualitative probability comparison weakly invariant under all reflections, then ⪅ is not strongly regular.

Proof: Let G be the group generated by the reflections. This group contains all translations. A non-trivial translation of H_n will either be a proper subset or a proper superset of H_n, depending on the direction of H_n. So by Proposition 2 we cannot have regularity.

Bibliographic note: Lemma 1 and Corollary 2 are analogous to Lemma 2 and Theorem 4 of this paper.

The choice between qualitative probabilities and generalized quantitative probabilities is illusory

There are two approaches to generalizing probabilities beyond the classical real-valued numerical approach.

1. Switch the values of the probability function from real numbers to values in some other ordered algebraic entity (e.g., the hyperreals, the surreals, an arbitrary totally ordered monoid).

2. Switch to qualitative probabilities, where instead of assigning values to events, one just compares the events probabilistically (this one is at least as likely than that one).

In the literature (including stuff I wrote myself!), the qualitative probability approach is treated as more general. But in fact, it’s not, at least not if we assume that the “at least as likely as” relation is transitive and reflexive, i.e., is a partial preorder. For suppose that we have a collection F of events and a partial preorder ⪅ on them. Say that A ≈ B iff A ⪅ B and B ⪅ A. Let V be the set of equivalence classes of events under the relation ≈, and define the partial order ≤ on V by [A]≤[B] iff A ⪅ B, where [A] is the equivalence class of A. (All this makes sense if ⪅ is a partial preorder.) Define P(A)=[A].

And that’s it! You now have a probability function P on F whose values are a partially ordered set V. So, what you do with qualitative comparisons you can do with values. Now that I’ve said it, it’s obvious to me and I’m kicking myself for not noticing it earlier. Perhaps some people in the field have noticed it and found it so obvious that it’s not worth saying.

And rather than thinking of there being a debate between probability functions with values and qualitative probabilities, we can now just stick to the probability function approach, and say that there is a serious debate as to what kind of a structure the values have: are they a real closed field, a totally ordered field, a totally ordered monoid (and if so, does its operation correspond to addition or to multiplication in the classical case), a mere partially ordered set, etc.?

The point here applies in other contexts, such as qualitative utilities, moral value comparisons, etc. Using the apparatus of set theory, we can replace a comparison relation by a value-assignment, which makes it more convenient to apply all the apparatus of ordered algebraic entities of different sorts.

As an application, in a recent post I said that given two very plausible axioms on probability functions, one cannot have a regular rotationally-invariant probability function on the measurable subsets of the circle. Using the above remarks, that point immediately extends to qualitative probabilities that satisfy the following two axioms:

1. If A and B are disjoint, A and C are disjoint, and B and C are equally likely, then A ∪ B and A ∪ C are equally likely, and

2. Ω − A and Ω − B are equally likely iff A and B are equally likely,

with rotational invariance being understood as saying that A is at least as likely as B if and only if ρA is at least as likely as B if and only if A is at least as likely as ρB for every rotation ρ, and with regularity understood as saying that every non-empty event is more likely than the empty event.

Gamifying planking

There is research that some video games can relieve pain. And boredom saps endurance. So the Stealth board product which lets you play games on your smartphone while doing core exercise on a specially shaped balance board seems to be a rather fine idea. But it's too expensive for me. So instead I cut a foam overlay with a phone slot and put it on my home-made balance board (with grippy skateboard tape). And the two free games for the Stealth board work great with this (I haven't bought their premium games).

Trivial Instructable here.

Result: I can plank more time on this than I can endure ordinary planking. Admittedly, the fact that the games involve some motion make the exercise less static. But it's still probably good core exercise.

Things God can't cause

It is widely held by Thomists that anything that a creature can cause can be caused directly by God without any creaturely involvement. Now suppose Sam thinks the following thought:

1. All my thoughts are caused in part by me.

But if God directly causes thought (1) in Sam, then God will directly be causing Sam to think a falsehood (since it will be false that all of Sam’s thoughts are caused in part by Sam). This seems contrary to divine truthfulness.

In an earlier post, I suggest that the Thomist retreat to the principle that for anything a creature can cause, God can directly cause something qualitatively identical to it. But that won't help with this problem, since anything qualitatively identical to (1) but caused directly by God will be false.

I suppose the Thomist can make a similar move here to the one Thomas makes when he says that God can do evil in the sense that he has the power to do it, but not the will. So we would deny that it is possible for God to directly cause anything that a creature can (since God’s goodness rules out God’s causation of some such things), but affirm that he has the power to do so.

Catastrophic decisions

Kirk has come to a planet with two intelligent species in the universe, the Oligons and the Pollakons. There are a million Oligons and a trillion (i.e., million million) Pollakons. They are technologically unsophisticated, live equally happy lives on the same planet, but have no interaction with each other, and the universal translator is currently broken so Kirk can’t communicate with them either. A giant planetoid is about to graze the planet in a way that is certain to wipe out the Pollakons but leave the Oligons, given their different ecological niche, largely unaffected. Kirk can try to redirect the planetoid with his tractor beam. Spock’s accurate calculations give the following probabilities:

• 1 in 1000 chance that the planetoid will now miss the planet and the Oligons and Pollakons will continue to live their happy lives;

• 999 in 1000 chance that the planetoid will wipe out both the Oligons and the Pollakons.

If Kirk doesn’t redirect, expected utility is 10⁶ happy lives (the Oligons). If Kirk does redirect, expected utility is (1/1000)(10¹² + 10⁶)=10⁹ + 10³ happy lives. So, expected utility clearly favors redirecting.

But redirecting just seems wrong. Kirk is nearly certain—99.9%—that redirecting will not help the Pollakons but will wipe out the Oligons.

Perhaps the reason intuition seems to favor not redirecting is that we have a moral bias in favor of non-interference. So let’s turn the story around. Kirk sees the planetoid coming towards the planet. Spock tells him that it has a 1/1000 chance that nothing bad will happen, and a 999/1000 chance that it will wipe out all life on the planet. But Spock also tells him that he can beam the Oligons—but not the Pollakons, who are made of a type of matter incapable of beaming—to the Enterprise. Spock, however, also tells Kirk that beaming the Oligons on board will require the Enterprise to come closer to the planet, which will gravitationally affect the planetoid’s path in such a way that the 1/1000 chance of nothing bad happening will disappear, and the Pollakons will now be certain, and not merely 999/1000 likely, to die.

Things are indeed a bit less clear to me now. I am inclined to think Kirk should rescue the Oligons (this may require Double Effect), but I am worried that I am irrationally neglecting small probabilities. Still, I am inclined to think Kirk should rescue. If that intuition is correct, then even in other-concerning decisions, and even when we have no relevant deontological worries, we should not go with expected utilities.

But now suppose that Kirk over his career will visit a million such planets. Then a policy of non-redirection in the original scenario or of rescue in the modified scenario would be disastrous by the Law of Large Numbers: those 1/1000 events would happen a number of times, and many, many lives will be lost. If we’re talking about long-term policies, then, it seems that Kirk should have a policy of going with expected utilities (barring deontological concerns). But for single-shot decisions, I think it’s different.

This line of thought suggests two things to me:

• maximization of expected utilities in ordinary circumstances has something to do with limit laws like the Law of Large Numbers, and

• we need a moral theory on which we can morally bind ourselves to a policy, in a way that lets the policy override genuine moral concerns that would be decisive absent the policy (cf. this post on promises).

Regularity and rotational invariance

Suppose that we have some sort of (not merely real-valued) probability assignment P to the Lebesgue measurable subsets of the unit circle Ω.

Theorem: Suppose that the probability values are rotationally invariant (P(A)=P(ρA) for any rotation ρ) and satisfy the two axioms:

1. If A and B are disjoint, A and C are disjoint, and P(B)=P(C), then P(A ∪ B)=P(A ∪ C)

2. P(Ω − A)=P(Ω − B) if and only if P(A)=P(B).

Then P(A)=P(∅) for every singleton A.

In other words, we cannot have regularity (non-empty sets having different probability from empty sets) if we have the additivity-type condition (1), the complement condition (2) and rotational invariance.

Proof: Fix an irrational number r and let B be the set of points at angles in degrees r, 2r, 3r, .... Let x₀ be the point at angle 0. Then B and C = B ∪ {x₀} are rotationally equivalent (you get the former from the latter by rotating by r degrees). So, P(B)=P(C). Let A = Ω − C. Then A and B are disjoint as are A and C. Hence, P(A ∪ B)=P(A ∪ C). But A ∪ C = Ω. So, P(A ∪ B)=P(Ω) by axiom 1. But A ∪ B = Ω − {x₀}. So, P({x₀}) = P(∅) by axiom 2. But all singletons are rotationally equivalent so they all have the same measure.

This result is a variant of the results here.

Causation and Thomism

Assume a Thomistic metaphysics, including the primary/secondary causation model from Aquinas. Thus, whenever a created cause has an effect, it has the effect it does only because God, through primary causation, cooperates with the created cause. If God did not cooperate with the created cause, the creature’s secondary causal power would be impotent to produce the effect. On the other hand, God can directly cause something by primary causation without the secondary causes doing anything.

Suppose that I strike a match, but God doesn’t cooperate in the frictional causing of fire. Then the match is struck, and does everything a struck match does, except that no fire results. Now imagine these two scenarios:

1. I strike a match which, in the ordinary way and with God’s cooperation, causes fire.

2. I strike a match, but God does not cooperate with me; however, God miraculously causes a fire just like the one in (1).

These two scenarios are different. So they must differ in something. They cannot differ in God, since that would violate divine simplicity, a core commitment of Thomistic metaphysics. So they must differ on the side of creatures. If so, they differ in the match striking or in the fire (or, more precisely, in the match and in that which is on fire, given a substance-accident ontology), God’s cooperation or lack thereof making the match-striking or the fire different.

Suppose God’s cooperation makes the match-striking different. Then in scenario (1), created reality includes the event of divinely-cooperated-match-striking. This event surely doesn’t need any further divine cooperation, or else we’d have a regress. But no item in created reality is sufficient on its own to produce an effect witout God’s cooperation being added to it on the primary/secondary causation model.

So, it is the fire that must be made different by God’s cooperation. In scenario (1), the fire is caused by the match and God while in (2), it is caused by God alone, and that makes the fire different between the scenarios. I would like to say that the esse of the fire is different in the two cases: in the ordinary case its esse is at least in part being caused by the match with God’s cooperation while in the other case the match doesn’t enter into its esse. But the details don’t matter for this post: what matters is that the fire is different between scenarios (1) and (2).

But this has an unfortunate consequence: If the fire must be different in some metaphysical way in cases (1) and (2), it follows that God cannot directly and independently of creation cause the same effect as the match caused. And this violates the Thomistic principle that whatever a finite cause suffices for can be produced by God directly. God cannot directly produce a fire-caused-by-the-match; that would be a contradiction.

So we have a conflict between a number of Thomistic principles:

3. Divine simplicity

4. Divine omnipotence

5. The primary/secondary causation model

6. God can directly cause anything he can cause in cooperation with a creature.

It seems to me that 3-5 are more central to Thomism than 6. So, I am inclined to reject 6, perhaps replacing it with the weaker principle that for any item x that God can cause in cooperation with a creature, God can cause an item x^* which is qualitatively just like x.

Perhaps I was too quick when I said that (1) and (2) must differ in the match or the fire. Perhaps they differ in something “in between” the match and the fire, a token causal relation. I think this is a problematic solution for two reasons. First, it is central to Aristotelianism that all that exists are substances and their accidents. The token causal relation, if it’s not “in” any creature, would violate this. Second, it seems that the match strike plus this “in between” thingy are now sufficient to produce the fire, or else we can run the above argument with the match strike and the “in between” thingy in place of the match strike. But no mere creature is sufficient to produce an effect without God’s cooperation.

Ataraxia

The stoics, the academic sceptics and the epicureans all to various degrees basically agreed—or at least largely lived as if they agreed—that happiness was ataraxia, imperturbable calm and tranquility. This is a useful and important corrective to our busy work and busy “leisure”. But at the sme time, it’s really a quite empty and negative picture of life’s fulfillment. It’s more like a picture of how to get done with life without too much misery.

Perhaps they had a part of the truth: perhaps what is truly worth having is imperturbably, calmly and tranquilly doing certain things, such as enjoying the companionship of those we love—God above all. But the ataraxia is just a mode of the worthwhile activity rather than the center of it.

Furthermore, perhaps these ancients were extensionally right: for perhaps the only way to have ataraxia is by being with God, since our hearts are restless apart from him. In that case, ataraxia isn’t happiness, or worth pursuing for its own sake, but is a sign of happiness.

Hangboard build

With our climbing gym still closed, I made a hangboard for myself and my son, a quick and inferior clone of the Lyons Edge Hideout. My build instructions are here (with kind permission from Lyons Edge).

Avoiding Dutch Books Despite Inconsistent Credences

Preprint is available here. Just accepted by Synthese.

Does supererogation always deserve praise?

Suppose that Bob spent a month making a birthday cake for Alice that was only slightly better than what was available in the store, and Bob did not enjoy the process at all. One can fill out the case in such a way that what Bob did was permissible. Moreover, it is was more burdensome to him than buying the slightly less good cake would have been, and it was better for Alice, so it looks like the action was supererogatory. Nonetheless, we wouldn’t praise this action: We would say that the action was insufficiently prudent. So, it seems that not every supererogatory action is praiseworthy.

Perhaps the problem is with my understanding of supererogation. If we add the necessary condition for supererogation that the action is on balance better than the relevant alternative, then we can avoid saying that Bob’s action is supererogatory, because it is not better on balance than the alternative. But I would rather avoid adding that a supererogatory action is on balance better than the alternative. For then it becomes mysterious how it can be permissible to do the alternative.

I am inclined to just bite the bullet and deny the supererogation always deserves praise.

Generalizing supererogation

My preferred way of understanding supererogation is that an action is supererogatory provided that it is permissible and more burdensome than some permissible alternative (see here for a defense). This suggests an interesting generalization. Let J denote an individual or a group (perhaps described relative to the agent). Then an action is J-supererogatory provided that it is is permissible and more burdensome for J than some permissible alternative.

Then supererogatory actions are, in the new terminology, agent-supererogatory. On the other hand, we have a new category of actions, others-supererogatory. These actions are permissible but more burdensome to others than some permissible alternative. An action can be both agent-supererogatory and others-supererogatory. For instance, suppose that by sacrificing two arms I can save two people from losing two arms each, but by sacrificing one arm I can save one person from losing one arm. And suppose I have no special duties here, so it is permissible for me to make no sacrifice at all. Then, the action of sacrificing one arm is agent-supererogatory (it is more burdensome than the permissible alternative of no sacrifice) and others-supererogatory (it is less burdensome than sacrificing both arms).

Supererogation and determinism

1. If at most one action is possible for one, that action is not supererogatory.

2. If determinism is true, then there is never more than one action possible for one.

3. So, if any action is supererogatory, determinism is false.

There is controversy over (2), but I don’t want to get into that in this post. What about (1)? Well, the standard story about supererogation is something like this: A supererogatory action is one that is better than, or perhaps more burdensome, that some permissible alternative. In any case, supererogatory actions are defined in contrast to a permissible alternative. But that permissible alternative has got to be possible for one to count as a genuine alternative. For instance, suppose I stay up all night with a sick friend. That’s better than going to sleep. But if there is loud music playing which would make it impossible for me to go to sleep and I am tied up so I can’t go elsewhere, then my staying up all night with the friend is not supererogatory.