Conditional vs. means-limited intentions

This morning, I set out to walk to the Philosophy Department. If asked my intention, I might have said that it was to reach the Department. And in actual fact I did reach it. Suppose, however, that as I was walking, my wife phoned me to inform me of a serious family emergency that required me to turn back, and that I did in fact turn back.

Here’s a puzzle. The family emergency in this (fortunately) hypothetical scenario seems to have frustrated my intention to reach the Department. On the other hand, surely I did not intend to reach the Department no matter what. That would have been quite wicked (imagine that I could only reach the Department by murdering someone). If I did not intend to reach the Department no matter what, it seems that my intention was conditional, such as to reach the Department barring the unforeseen. But the unforeseen happened, so my conditional intention wasn’t frustrated—it was mooted. If I intend to fail a student if he doesn’t turn in his homework, and he turns in his homework, my intention is not frustrated. So my intention was frustrated and not frustrated, it seems.

Perhaps rather than my intention being frustrated, it was my desire to reach the Department that was frustrated. But that need not be the case. Suppose, contrary to fact, that I was dreading my logic class today and would have appreciated any good excuse to bail on it. Then either I had no desire to reach the Department or my desire was conditional again: to reach the Department unless I can get out of my logic class. In neither case was my desire frustrated.

Let me try a different solution. I intended to reach the Department by morally licit means. The phone call made it impossible for me to reach the Department by morally licit means—reaching the Department would have required me to neglect my family. My intention wasn’t relevantly conditional, but included a stipulation as to the means. Thus my intention was frustrated when it became impossible for me to reach the Department by morally licit means.

The above suggests that our intentions should generally be thus limited in respect of means, unless the means are explicitly specified all the way down (and they probably never are). Otherwise, our intention wickedly commits us to wicked courses of action in some possible circumstances. Of course, the limitation, just as the intention itself, will typically be implicit.

The many-worlds interpretation and probability

The probability of a proposition p equals the probability that p is true. I have argued that this principle refutes open future views. It’s interesting that it also refutes the many-worlds interpretation of quantum mechanics.

Suppose that I have prepared an electron mixed spin state 2^−1/2|↑⟩+2^−1/2|↓⟩ and we are about to measure whether the spin is up or down. The Born rule says that I should assign probability 1/2 to each of the two possible measurements. But by the many-worlds interpretation, the world splits into two (or more—but I will ignore that complication as nothing hangs on the number, or even the number being being well-defined) branches: in one an electron in a spin-up state is observed and in the other one in a spin down state is observed. Now consider these two propositions:

1. I will observe a spin-up state.

2. I will observe a spin-down state.

Given the many-worlds interpretation, metaphysically reality is symmetric with respect to these two propositions, as reality includes branches with both observes and with the observer standing in the same relationship to me. Hence, either both are true or neither is true on the correct reading of the many-worlds metaphysics: both are true if the observer in both branches counts as me, and otherwise both propositions are false. If the correct reading of the metaphysics is that both are true, then the probability of each being true is 1, and hence by the principle I started the post with, the probability that I will observe a spin-up state is 1 and so is the probability that I will observe a spin-down state. If the correct reading of the metaphysics is that neither is true, then the probability of truth for each will be 0, and hence the probability of my making either observation is 0.

So, the probabilities of (1) and (2) are 0 or 1. In neither case are they 1/2, which is what the Born rule stipulates.

This seems to me to be a stronger argument than the more common argument against the many-worlds interpretation that all branches should have equal probability, and hence would violate the Born rule in cases where the quantum state has unequal weights. For the usual argument depends on indifference, which is a dubious principle.

A hypothesis about authority and duties of care

Parents have the authority to command their children and parents have a special duty to care for children. Officers have special duties of care for those under their command. The state likewise has special duties of care for those under its jurisdiction. Special duties of care do not imply authority: adult siblings have special duties of care to one another but do not have jurisdiction over one another. But we can hypothesize that authority implies special duties of care.

Why would that be so? One possibility is that authority always arises out of special duties of care: in some cases, in order to properly care for y one must have authority over y. That fits neatly with the parent-child case, but doesn't fit with the military case, where the authority seems explanatory of the duties of care, or at least not posterior to it. But in the military case we might say this: in paradigmatic cases (putting to one side the case of mercenaries), the officer's authority derives from the state's authority. And the state's authority may well arise out of special duties of care for its citizens, whom the state can thus induct, voluntarily or not, into the military.

This more general pattern can fit cases which don't fit the simple version of the authority-care hypothesis. For instance, perhaps, a judge has commanding authority over a convicted prisoner but does not have special duties of care for the prisoner. But the judge's authority derives from the state's authority, which is explained by the state's special duties towards its citizens. So the more refined hypothesis is something like this: The authority to command is connected with special duties of care, but the special duties of care need not be had by the one who has the authority to command--the authority to command may have been deputized from another who had both the authority and the special duties.

But what about this case: Sometimes a state will imprison those who are not under its care but who have harmed its citizens. One example is prisoners of war. Another is the case of seizing a criminal from another country, as in the case of Manuel Noriega. I could wimpily say that the hypothesis is just a general rule with exceptions. But perhaps what I should instead say is that the case of prisoners of war and criminals seized from abroad is not a case of authority to command and hence no exception to the hypothesis. While an imprisoned citizen does violate duties of obedience to the state in escaping, the prisoner of war or criminal seized from abroad do not violate any such duties of obedience in escaping. There may be a limited commanding authority, however, derived from duties of care. Thus, an officer in charge of a prisoner of war camp might have commanding authority in respect of keeping order at food lines. And in even other cases there may be moral reasons to obey not because of authority but in order to maintain order, which is good in itself.

So let's suppose the hypothesis is correct. We now come to two of the most interesting cases: God and self. If the hypothesis is true, then God's absolute commanding authority over us derives from God's duty to love us. That's surprising, but may be right. The case of self is even more interesting. While we may not, strictly speaking, have commanding authority over ourselves (though "promises to self" might be an example), the authority we have over ourselves goes beyond most cases of commanding authority. Does that authority, too, derive from duties to care for ourselves? I like that idea, but many will not like the idea of duties to care for ourselves.

From suicide to slavery

I've been thinking about an argument with this logical form:

1. If suicide is permissible, then slavery is permissible.
2. Slavery is not permissible.
3. So, suicide is not permissible.

Of course, the most controversial premise is (1), though I could also imagine a defender of suicide denying (2) in the case of voluntary enslavement. One reason to accept (1) is something like this:

4. If suicide is permissible, then we have ultimate authority over our own lives.
5. If we have ultimate authority over our own lives, then it is permissible and valid for us to sell ourselves into slavery.
6. If it is permissible and valid for us to sell ourselves into slavery, then slavery is permissible.
7. So, if suicide is permissible, then slavery is permissible.

By "valid", I mean that the sale would actually work: that authority over our lives would be transferred to another. The notion of "ultimate authority" is rather foggy and I think (4) and (5) can be questioned. But I still think it's an argument worth developing, as all three premises (4)-(6) have some plausibility.

Another line of thought in favor of (1) is:

8. If suicide is permissible, it is permissible and valid to deputize another to unconditionally kill one.
9. If it is permissible and valid to deputize another to unconditionally kill one, it is permissible and valid to deputize another to kill one at will.
10. If it is permissible and valid to deputize another to kill one at will, then it is permissible and valid to sell oneself into slavery.
11. If it is permissible and valid to sell oneself into slavery, then slavery is permissible.

Here, valid deputization is a deputization that actually succeeds in giving the other the requisite authority. The thought behind (10) is that if one give life-and-death authority over oneself to another, one can a fortiori give the other kinds of authority that define the master-slave relationship.

Suicide can be murder

Joe thinks chess is an evil game and creates a killer robot tasked with killing the greatest chess player on earth, whoever it might be. The robot succeeds with the task. Joe is clearly a murderer, not merely an attempted murderer. But suppose Joe is the greatest chess player on earth, though he had no suspicion of this fact. Then Joe has committed suicide. And is a murderer. Hence a suicide can be a kind of murder.

(I actually think suicide is generally murder. But that's a stronger claim.)

Uncountably many coin tosses and a technical problem for the many-minds interpretation

Suppose I toss infinitely many fair coins. By the Law of Large Numbers (and assuming an appropriate version of the Axiom of Choice), with probability one I will have infinitely many heads and infinitely many tails. But what if I toss an uncountable infinity of fair coins, and I want to know whether I will get uncountably many heads and uncountably many tails? Intuitively, surely I will. It would seem really unfair if I only got countably many heads and all but uncountably many were tails (or the other way around)!

But the usual mathematical model for this situation offers no such guarantee. Let I be an uncountable index set, let Ω = {H, T}^I be the space of coin toss outcomes indexed by I, and let P be the completion of the product P₀ of I-many fair coin flip measures on {H, T}. Let U_X be the subset of Ω where there are uncountably many Xs (where X = H or X = T). It can be proved (see Appendix) that U_X is saturated nonmeasurable, i.e., any measurable subset of it has zero probability and any measurable superset of it has probability one.

The same is true (and with the same proof) if we ask what the probability is that there is some specific cardinality κ of heads (or of tails), where ℵ₀ < κ ≤ ∥I∥. Again, we come up against a saturated nonmeasurable set of outcomes.

So what? Well, this leads to a mildly interesting technical problem for the Albert-Loewer many-minds interpretation of quantum mechanics. Albert and Loewer want their many-minds interpretation to allow for the supervenience of minds on the wavefunction. This requires that every branch of the multiverse always be populated by the same cardinality of minds. To ensure this, they prepopulate every branch with continuum-many minds, in the hope that every branching will keep continuum-many minds in every branch. But the above result shows that there is no guarantee of this. If continuum-many minds each, as it were, flip a coin whether to take branch H or branch T, we cannot even say that it’s likely that continuum-many will take each branch.

This might seem very interesting: it might seem to entirely undercut the physicalist supervenience behind their story. But that’s going too far. For it may be that the completion of the product measure on the coin-flip space {H, T}^I isn’t the right model. It may be that we should extend the measure to assign probability 1 to U_H and to U_T. This can, indeed, be mathematically done. (I haven’t checked to make sure that one can keep all the intuitive symmetries in the probability space, though.) Though in fact we can extend to assign other probabilities, too (though it then may be impossible to keep the symmetries).

Appendix: Sketch of proof that U_X is saturated measurable: Suppose that A is a non-empty P₀-measurable set. Then A is defined by a constraint on countably many of the factors in Ω. But no such constraint suffices can ensure that every member of A has uncountably many Xs (unless it ensures that A is empty), and hence it is not the case that A ⊆ U_X. It follows that the only subsets of U_X that are P-measurable have null measure. But for exactly the same kind of reason, a non-empty P₀-measurable set A cannot be a subset of the complement Ω − U_X either. For then the constraint on countably many factors would have to guarantee the lack of uncountably many Xs, while allowing A to be non-empty, and that’s impossible. It follows that the only supersets of U_X that are P-measurable have full measure.

From modal to many-minds interpretations of quantum mechanics

I should warn readers that all my posts on quantum stuff are very, very sketchy. I'm very much learning the material.

In my exploration of many-minds interpretations of quantum mechanics, I’ve been trying to figure out how the many-minds dynamics could work without using problematic notion of a “local” or “effective” wavefunction. Here’s a way I like.

1. Start with a privileged set O of commuting observables whose values would be sufficient to ground the phenomenal states of all minded beings. Perhaps particle positions will do.

2. Now let’s suppose we have a modal interpretation with O as the privileged set of observables and with an appropriate dynamics (like the one here).

3. Attach immaterial minds to the systems described by O, and have them travel along with the systems.

4. But now do a switcheroo: instead of supposing the observables in O to describe physical reality, ground their values in properties of the minds. If O is phenomenally distinguishable, i.e., if any two distinct assignments of values to the observables in O will result in different ensembles of phenomenal states, then we don’t need to posit properties of minds over and beyond the phenomenal ones here. But if O is too rich to be phenomenally distinguishable, we will need to suppose unconscious properties of the minds to ground the values of the observables in O.

This yields something closer to the Squires-Barrett Traveling Minds variant of Albert and Loewer’s Single Mind View, rather than the many-minds view. (In particular, there is no problem of meeting “mindless hulks”.)

If O determines all particle positions, then the result is a Leibnizified Bohm-like theory where particle positions are grounded in the properties of monads (minds).

If we want many-minds, then we just do the above uncountably infinitely often for each different assignment of values to members of O.

History matters for probabilities

Suppose at t₁ there are countably infinitely many people with red hats and countably infinitely many people with black hats. You’re one of them and you can’t see anybody hat (including your own). What probability you should attach to the proposition that your hat is red depends on the causal history rather than on what the world is like at t₁.

For consider two causal histories, each of which results in the same time slice at t₁. In the first history, we start with infinitely many hatless people at t₀, and for each one we flip a fair coin to see if they get a red or a black hat. Then we arrange the people in a (bidirectionally infinite) line of alternating hat colors. In the second history, we start the same way, but now our coin is unfair, so any given person has only a 1/4 chance of getting a red hat and a 3/4 chance of getting a black hat. But again after the fact the people are arranged in an infinite line of alternating hat colors.

In the two scenarios, the outputs at t₁ are relevantly alike—an infinite line of people of alternating hat colors—but what probability you should assign to the proposition that your hat is red depends on which causal history actually took place. So probabilities don’t just depend on how things are now, but also on how things were. At least when we’re dealing with infinities.

Multi-Bohm

I am exploring what seem to me to be under-explored parts of the logical space of interpretations of Quantum Mechanics. I may be wasting my time: there may be good reasons why those parts of logical space are not explored much. But I am also hoping that such exploration will broaden my mind.

So, here’s a curious interpretation: multi-Bohm. Assume no collapse as in Everett. At any given time t, there is the set S_t of all particle position assignments compatible with the value of the wavefunction ψ(t) (we can extend to spin and other things in the same way that Bohm gets extended to spin and other things). Typically, this set will include every possible position assignment, and will have continuum cardinality.

Now on Bohm’s interpretation, one member of S_t is privileged: it is the actual positions of the particles. But drop that privileging. Suppose instead that all the assignments of S_t are on par. Then S_t gives us a synchronic decomposition of the Everettian multiverse into "branches". Now stitch the synchronic decomposition into trajectories using the guiding equation: a position assignment s_t ∈ S_t is part of the same trajectory as a position assignment s_t′ ∈ S_t′ if and only if the guiding equation evolves s_t into s_t′ over the time span from t to t′ given the actual wavefunction ψ.

We can think of the above as a story with infinitely many (continuum many) parallel Bohmian universes. But that bloats the ontology by including infinitely many ensembles of particles. Since the wavefunction fully determines the sets S_t of position assignments (or so I assume—there are some worries about null-measure stuff that I am not perfectly sure of), we can stop thinking about real particles and just as a way of speaking superimposed on top of the many-worlds interpretation.

This means that we can interpret the many-worlds interpretation not as a branching-worlds story, but as a deterministic parallel worlds reading. For given the two-way (I assume) determinism in the guiding equation, the trajectories never meet: the branches always stay separate and parallel. Moreover, the probability problem of the many-worlds interpretation is unsolved, and so we cannot say that the story fits better with one set of experimental results rather than another.

This isn’t very attractive…

A probability problem for the many minds interpretation

The branching many worlds interpretation of quantum mechanics famously faces the problem of why it is appropriate to interpret the weights in the wavefunction as probabilities. The many minds interpretation is designed to solve this problem. Infinitely many minds traverse the branching multiverse, and move in accordance with objective chances defined by the wavefunction. It sounds like everything is just fine probabilistically: each mind's movements are nicely stochastic.

Yes, the movements are, but what about the initial setup of the minds? Presumably minds come on the scene when in some branch of the multiverse matter is so arranged as to form something like a brain. And infinitely minds come into existence then. But much earlier than evolution has managed to produce brains on earth, the branching multiverse will have branches with Boltzmann brains: brains that come into existence randomly out of quantum fluctuations in little bubbles of order. Some of these Boltzmann brains will have brain states exactly like ours. And each Boltzmann brain, on the many minds interpretation, will get infinitely many minds. So am I associated with a real brain or a Boltzmann brain? There are infinitely many minds with states like mind associated with a real brain and infinitely many minds with states like mine associated with a Boltzmann brain. The infinities are, presumably, the same. So how is it that I know that I have two hands?

Here's one move that a defender of many minds could make. Assume a pre-existing infinite bucket of minds, existing unconsciously apart from the physical universe. Consider all the brain-coming-into-existence events throughout the multiverse and across time. Assign probabilities to these brain-coming-into-existence events in proportion to the weight assigned the event by the wavefunction. Now add this dynamics: Whenever a brain-coming-into-existence event happens, each mind in the infinite bucket has the indicated probability of getting pulled out of the bucket and connected with this brain. Then as long as we're confident that, roughly, the number of Boltzmann brains weighted by the probabilities is less than the number of normal brains weighted by the probabilities, it seems all should be well.

There are potential technical problems with the normalization of the probabilities. Also, there is the issue that the metaphysics now seems excessively dualistic, in that we are supposing that bucket of minds independent of the physics from which the minds are pulled. Maybe one could just suppose a bucket of mind-haecceities? I am not sure.

This post is inspired by remarks Rob Koons made about retrospective probabilities in the many worlds interpretation. Essentially the point of this post is that that problem isn't solved by the many minds interpretation.

Sceptical scenarios and Quantum Mechanics

It's occurred to me that on most non-Bohmian interpretations of quantum mechanics, we end up being in a superposition of the kind of universe we think we're in and a sceptical scenario, e.g., some brain in a vat scenario, though details of the scenario may differ depending on an interpretation. All that's needed for that to happen is for there to have been a non-zero chance of a sceptical scenario with the same phenomenal states that we actually have. I don't know if this is a problem if the sceptical scenario is assigned low weight by the wavefunction.

Phenomenally accessible hidden variables

Consider Jeffrey Barrett's traveling minds interpretation of Quantum Mechanics (see also here). On this interpretation, minds traverse a branching Everett-style multiverse in accordance with the probabilities given by the Born rule. But unlike on the Albert-Loewer many-minds interpretation, the minds are constrained to travel together: they are always found in the same branch of the universe.

Here is something interesting about the position in logical space of this interpretation. It is a hidden-variables interpretation in the sense that it supposes that there are realities that cannot be reduced to the wavefunction. The hidden variables on this story correspond to facts about brain states. For instance, the wavefunction may place my brain in a superposition of a brain state in which I feel I am sitting with a state in which I feel I am standing, but the minds (jointly) pick out a branch of the wavefunction--the one in which I feel I am sitting (and writing a post on quantum mechanics). Notice, however, that the hidden variables in this story are hidden from the wavefunction but not hidden from us: they are phenomenally accessible to us.

Interestingly, the Bohm interpretation can be seen also to be a hidden-variables interpretation where the variables are not entirely hidden from us. For presumably it is the "hidden" positions of the particles that determine the brain state that gives rise to my phenomenal state. So from my phenomenal state, I can tell something about the hidden variables--for instance, that they comprise a brain. Bohm is a paradigmatic hidden variable interpretation, and yet it does not actually hide the variables from us. So we need to be cautious about the phrase "hidden variables".

I think the Albert-Loewer many-minds interpretation is also a hidden variable theory. The variables are the states of the many minds. But there is a difference between the Barrett and Albert-Lower interpretations, on the one hand, and the Bohm interpretation, on the other. In the Bohm interpretation, the hidden variables are a part of physical reality. On the mind-based interpretations, the hidden variables are a part of mental reality. In all cases, we have at least partial access to the hidden variables.

Talk video: Marriage is a natural kind

I just noticed that my recent SCP talk arguing that marriage is a natural kind has been posted. Slides (which I think don't show up on the video) are here.

The DIY urge, Satan's sin and Pelagianism

I've got a big DIY urge. My motivations usually include being too cheap to buy something (typically because I'm saving up for something else--right now, a 3D printer). A fair amount of the time there is vanity--wanting to brag online, say. Sometimes perhaps there is a minor motivation (which really should be much stronger) to repair things rather than wastefully throwing them out. And sometimes the activity itself is very pleasant (I really enjoy using power tools like a sewing machine, a drill press or a stand mixer; I like the smell of solder rosin or freshly cut softwood wafting in the air). But I think often the strongest motivation is the intrinsic pull of doing things myself.

According to Aquinas, that motivation is why Satan sinned. He wanted the good things that God was going to give to him, but he didn't want them from God--he wanted getting them himself. In other words, the first sin is Pelagianism.

This makes me a bit worried about my DIY urge. Is it an echo of the Satanic pride that led to the downfall of the universe?

Not necessarily. Aquinas' discussion of the first sin is driven by two theses: (a) Satan was very smart and (b) Satan's motivations were good. So Aquinas needs needs to identify a good motivation that led him to sin, not simply by a stupid mistake. It is thus central to Aquinas' story that the DIY urge that Satan had was a good motivation: there is a genuine good in achieving good things by oneself. But in order to achieve that good, Satan refused God's gift of grace, settling for (lesser, presumably) goods that he could get by himself.

The fundamental motivation behind the DIY urge is good, thus. But there is a serious danger that it misses what St. John Paul II called our "nuptial nature": that it is our nature to give ourselves to others and to receive others' gift of themselves. Satan refused God's gift. The parallel danger in the DIY case is that it not turn into a refusal of the gift of others' creativity and labor, a refusal to acknowledge that (to use older language) we are social animals.

Of course, the products of commerce are not gifts personally directed to us. (After all, we have to pay for them!) But there is a sense in which they still have some gift-like nature. People have chosen not to be subsistence farmers, but to make stuff for others. There is an imperfect duty somewhere around here to participate in the back-and-forth of commerce, which bears some relevant resemblance to the back-and-forth of gift giving and reciprocation. And so, like all things, the DIY urge needs moderation, not just for reasons like not wasting time or avoiding vanity, but lest it become a denial of our social nature.

The camera as a temporal microscope

For a long time I wanted to take one of those high speed water splash pictures that I've seen other people take. I played around with controlling my Lumix GH3 camera (thanks, Dad!) via WiFi, and considered setting up a microcontroller and a light sensor to time a photo for when an object hits water, like I've seen in some instructions online.

But then I had a much, much simpler idea. Just take pictures in continuous mode. Then if I just drop an object in the water enough times, one of these times I'll capture a nice splash.

I set up a bowl in sunlight, put the camera on a tripod, set exposure time to 1/1300 s, focused manually, and started taking pictures while dropping a rubber ball (to avoid contortions, I used a wired remote). To my surprise, almost every run captured something nice-looking, even though I was only using 6 picture per second mode. And I didn't expect 1/1300 s to be good enough, but it was.

Eternalists like me can think of a still camera as a temporal microscope, stretching an image temporally.