Realism and anti-reductionism

The ordinary sentence "There are four chairs in my office" is true (in its ordinary context). Furthermore, its being true tells us very little about fundamental ontology. Fundamental physical reality could be made out of a single field, a handful of fields, particles in three-dimensional space, particles in ten-dimensional space, a single vector in a Hilbert space, etc., and yet the sentence could be true.

An interesting consequence: Even if in fact physical reality is made out of particles in three-dimensional space, we should not analyze the sentence to mean that there are four disjoint pluralities of particles each arranged chairwise in my office. For if that were what the sentence meant, it would tell us about which of the fundamental physical ontologies is correct. Rather, the sentence is true because of a certain arrangement of particles (or fields or whatever).

If there is such a broad range of fundamental ontologies that "There are four chairs in my office" is compatible with, it seems that the sentence should also be compatible with various sceptical scenarios, such as that I am a brain in a vat being fed data from a computer simulation. In that case, the chair sentence would be true due to facts about the computer simulation, in much the way that "There are four chairs in this Minecraft house" is true. It would be very difficult to be open to a wide variety of fundamental physics stories about the chair sentence without being open to the sentence being true in virtue of facts about a computer simulation.

But now suppose that the same kind of thing is true for other sentences about physical things like tables, dogs, trees, human bodies, etc.: each of these sentences can be made true by a wide array of physical ontologies. Then it seems that nothing we say about physical things rules out sceptical scenarios: yes, I know I have two hands, but my having two hands could be grounded by facts about a computer simulation. At this point the meaningfulness of the sceptical question whether I know I am not a brain in a vat is breaking down. And with it, realism is breaking down.

In order for the sceptical question to make sense, we need the possibility of saying things that cannot simply be made true by a very wide variety of physical theories, since such things will also be made true by computer simulations. This gives us an interesting anti-reductionist argument. If the statement "I have two hands" is to be understood reductively (and I include non-Aristotelian functionalist views as reductive), then it could still be literally true in the brain-in-a-vat scenario. But if anti-reductionism about hands is true, then the statement wouldn't be true in the brain-in-a-vat scenario. And so I can deny that I am in that scenario simply by saying "I have two hands."

But maybe I am moving too fast here. Maybe "I have two hands" could be literally true in a brain-in-a-vat scenario. Suppose that the anti-reductionism consists of there being Aristotelian forms of hands (presumably accidental forms). But if, for all we know, the form of a hand can inform a bunch of particles, a fact about a vector or the region of a field, then the form of a hand can also inform an aspect of a computer simulation. And so, for all we know, I can literally and non-reductively have hands even if I am a brain in a vat. I am not sure, however, that I need to worry about this. What is important is form, not the precise material substrate. If physical reality is the memory of a giant computer but it isn't a mere simulation but is in fact informed by a multiplicity of substantial and accidental forms corresponding to people, trees, hands, hearts, etc., and these forms are real entities, then the scenario does not seem to me to be a sceptical scenario.

Accretion, excretion and four-dimensionalism

Suppose we are four-dimensional. Parthood simpliciter then is an eternal relation between, typically, four-dimensional entities. My heart is a four-dimensional object that is eternally a part of me, who am another four-dimensional object.

But there is surely also such a thing as having a part at a time t. Thus, in utero my umbilical cord was a part of me, but it no longer is. What does it mean to have a part at a time? Here is the simplest thing to say:

1. x is a part of y at t if and only if x is a part of y and both x and y exist at t.

But (1) then has a very interesting metaphysical consequence that only a few Aristotelian philosophers endorse: parts cannot survive being accreted by or excreted from the whole. For if, say, my finger survived its removal from the whole (and not just because I became a scattered object), there would be a time at which my finger would exist but wouldn’t be a part of me. And that violates (1) together with the eternality of parthood simpliciter.

This may seem to be a reductio of (1). But if we reject (1), what do we put in its place, assuming four-dimensionalism? I suspect we will have to posit a second relation of parthood, parthood-at-a-time, which is not reducible to parthood simpliciter. And that seems to be unduly complex.

So I propose that the four-dimensionalist embrace (1) and conclude to the thesis that parts cannot survive their accretion or excretion.

Dualist survivalism

According to dualist survivalism, at death our bodies perish but we continue to exist with nothing but a soul (until, Christians believe, the resurrection of the dead, when we regain our bodies).

A lot of the arguments against dualist survivalism focus on how we could exist as mere souls. First, such existence seems to violate weak supplementation: my souls is proper part of me, but if the body perished, my soul would be my only part—and yet it would still be a proper part (since identity is necessary). Second, it seems to be an essential property of animals that they are embodied, an essential property of humans that they are animals, and an essential property of us that we are humans.

There are answers to these kinds of worries in the literature, but I want to note that things become much simpler for the dualist survivalist if she accepts a four-dimensionalism that says that we are four-dimensional beings (this won't be endurantist, but it might not be perdurantist either).

First, there will be a time t after my death (and before the resurrection) such that the only proper part of mine that is located at t is my soul. However, the soul won’t be my only part. My arms, legs and brain are eternally my parts. It’s just that they aren’t located at t, as the only proper part of me that is located at t is my soul. There is no violation of weak supplementation. (We still get a violation of weak supplementation for the derived relation of parthood-at-t, where x is a part-at-t of y provided that x is a part of y and both x and y exist at t. But why think there is weak supplementation for parthood-at-t? We certainly wouldn’t expect weak supplementation to hold for parthood-at-z, where z is a spatial location and x is a part-at-z of y provided that x is a part of y and both x and y are located at z.)

Second, it need not follow from its being an essential property of animals that they are embodied that they have bodies at every time at which they exist. Compare: It may be an essential property of a cell that it is nucleated. But the cell is bigger spatially than the nucleus, so it had better not follow that the nucleus exists at every spatial location at which the cell does. So why think that the body needs to exist at every temporal location at which the animal does? Why can’t the animal be bigger temporally than its body?

Of course, those given to three-dimensional thinking will say that I am missing crucial differences between space and time.

Three strengths of desire

Plausibly, having satisfied desires contributes to my well-being and having unsatisfied desires contributes to my ill-being, at least in the case of rational desires. But there are infinitely many things that I’d like to know and only finitely many that I do know, and my desire here is rational. So my desire and knowledge state contributes infinite misery to me. But it does not. So something’s gone wrong.

That’s too quick. Maybe the things that I know are things that I more strongly desire to know than the things that I don’t know, to such a degree that the contribution to my well-being from the finite number of things I know outweighs the contribution to my ill-being from the infinite number of things I don’t know. In my case, I think this objection holds, since I take myself to know the central truths of the Christian faith, and I take that to make me know things that I most want to know: who I am, what I should do, what the point of my life is, etc. And this may well outweigh the infinitely many things that I don’t know.

Yes, but I can tweak the argument. Consider some area of my knowledge. Perhaps my knowledge of noncommutative geometry. There is way more that I don’t know than that I know, and I can’t say that the things that I do know are ones that I desire so much more strongly to know than the ones I don’t know so as to balance them out. But I don’t think I am made more miserable by my desire and knowledge state with respect to noncommutative geometry. If I neither knew anything nor cared to know anything about noncommutative geometry, I wouldn’t be any better off.

Thinking about this suggests there are three different strengths in a desire:

1. S_p: preferential strength, determined by which things one is inclined to choose over which.

2. S_h: happiness strength, determined by how happy having the desire fulfilled makes one.

3. S_m: misery strength, determined by how miserable having the desire unfulfilled makes one.

It is natural to hypothesize that (a) the contribution to well-being is S_h when the desire is fulfilled and −S_m when it is unfulfilled, and (b) in a rational agent, S_p = S_h + S_m. As a result of (b), one can have the same preferential strength, but differently divided between the happiness and misery strengths. For instance, there may be a degree of pain such that the preferential strength of my desire not to have that pain equals the preferential strength of my desire to know whether the Goldbach Conjecture is true. I would be indifferent whether to avoid the pain or learn whether the Goldbach Conjecture is true. But they are differently divided: in the pain case S_m >> S_h and in the Goldbach case S_m << S_h.

There might be some desires where S_m = 0. In those cases we think “It would be nice…” For instance, I might have a desire that some celebrity be my friend. Here, S_m = 0: I am in no way made miserable by having that desire be unfulfilled, although the desire might have significant preferential strength—there might be significant goods I would be willing trade for that friendship. On the other hand, when I desire that a colleague be my friend, quite likely S_m >> 0: I would pine if the friendship weren’t there.

(We might think a hedonist has a story about all this: S_h measures how pleasant it is to have the desire fulfilled and S_m measures how painful the unfulfilled desire is. But that story is mistaken. For instance, consider my desire that people not say bad things behind my back in such a way that I never find out. Here, S_m >> 0, but there is no pain in having the desire unfulfilled, since when it’s unfulfilled I don’t know about it.)

"Should know"

I’ve been thinking about the phrase “x should know that s”. (There is probably a literature on this, but blogging just wouldn’t be as much fun if one had to look up the literature!) We use this phrase—or its disjunctive variant “x knows or should know that s”—very readily, without its calling for much evidence about x.

• “As an engineer Alice should know that more redundancy was needed in this design.”

• “Bob knows or should know that his behavior is unprofessional for a librarian.”

• “Carl should have known that genocide is wrong.”

Here’s a sense of “x should know that s”: x has some relevant role R and it is normal for those in R to know that s under the relevant circumstances. In that sense, to say that x should know that s we don’t need to know anything specific about x’s history or mental state, other than that x has role R. Rather, we need to know about R: it is normal engineering practice to build in sufficient redundancy; librarians have an unwritten code of professional behavior; human beings normally have a moral law written in their hearts.

This role-based sense of “should know” is enough to justify treating x as a poor exemplar of the role R when x does not in fact know that s. When R is a contingent role, like engineer or librarian, it could be a sufficient for drumming x out of R.

But we sometimes seem use a “should know” claim to underwrite moral blame. And the normative story I just gave about “should know” isn’t strong enough for that. Alice might have had a really poor education as an engineer, and couldn’t have known better. If the education was sufficiently poor, we might kick her out of the profession, but we shouldn’t blame her morally.

Carl, of course, is a case apart. Carl’s ignorance makes him a defective human being, not just a defective engineer or librarian. Still a defective human being is not the same as a morally blameworthy human being. And in Carl’s case we can’t drum him out of the relevant role without being able to levy moral blame on him, as drumming him out of humanity is, presumably, capital punishment. However, we can lock him up for the protection of society.

On the other hand, we could take “x should know that s” as saying something about x’s state, like that it is x’s own fault if x doesn’t know. But in that case, I think people often use the phrase without sufficient justification. Yes, it’s normal to know that genocide is wrong. But we live in a fallen world where people can fall very far short of what is normal through no fault of their own, by virtue of physical and mental disease, the intellectual influence of others, and so on.

I worry that in common use the phrase “x should know that s” has two rationally incompatible features:

• Our evidence only fits with the role-based normative reading.

• The conclusions only fit with the personal fault reading.

Two senses of "decide"

Suppose:

1. Alice sacrifices her life to protect her innocent comrades.

2. Bob decides that if he ever has the opportunity to sacrifice his life to protect his innocent comrades, he’ll do it.

We praise Alice. But as for Bob, while we commend his moral judgment, we think that he is not yet in the crucible of character. Bob’s resolve has not yet been tested. And it’s not just that it hasn’t been tested. Alice’s decision not only reveals but also constitutes her as a courageous individual. Bob’s decision falls short both in the revealing but also in the constituting department (it’s not his fault, of course, that the opportunity hasn’t come up).

Now compare Alice and Bob to Carl:

3. Carl knows that tomorrow he’ll have the opportunity to sacrifice his life to protect his innocent comrades, and he decides he will make the sacrifice.

Carl is more like Bob than like Alice. It’s true that Carl’s decision is unconditional while Bob’s is conditional. But even though Carl’s decision is unconditional, it’s not final. Carl knows (at least on the most obvious way of spelling out the story) that he will have another opportunity to decide come tomorrow, just as Bob will still have to make a final decision once the opportunity comes up.

I am not sure how much Bob and Carl actually count as deciding. They are figuring out what would or will (respectively) be the thing to do. They are making a prediction (hypothetical or future-oriented) about their action. They may even be trying by an act of will to form their character so as to determine that they would or will make the sacrifice. But if they know how human beings function, they know that their attempt is very unlikely to be successful: they would or will still have a real choice to make. And in the end it probably wouldn’t surprise us too much if, put to the test, Bob and Carl failed to make the sacrifice.

Alice did something decisive. Bob and Carl have yet to do so. There is an important sense in which only Alice decided to sacrifice her life.

The above were cases of laudable action. But what about the negative side? We could suppose that David steals from his employer; Erin decides that she will steal if she has the opportunity; and Frank knows he’ll have the opportunity to steal and decides he’ll take it.

I think we’ll blame Erin and Frank much more than we’ll praise Bob and Carl (this is an empirical prediction—feel free to test it). But I think that’s wrong. Erin and Frank haven’t yet gone into the relevant crucible of character, just as Bob and Carl haven’t. Bob and Carl may be praiseworthy for their present state; Erin and Frank may be blameworthy for theirs. But the praise and the blame shouldn’t go quite as far as in the case of Alice and David, respectively. (Of course, any one of the six people might for some other reason, say ignorance, fail to be blameworthy or praiseworthy.)

This is closely to connected to my previous post.

Two senses of "intend"?

Consider these sentences:

1. Intending to kill the wolverine, Alice pulled the trigger

and:

2. Intending to get to the mall, Bob started his car.

If Alice pulls the trigger intending to kill the wolverine and the wolverine survives, then necessarily Alice’s action is a failure.

But suppose that Bob intends to get to the mall, starts his car, changes his mind, and drives off for a hike in the woods. None of the actions described is a failure. He just changed his mind.

If nanoseconds after the bullet leaving the muzzle Alice changed her mind, and it so happens the wolverine survived, it is still true that Alice’s action failed. Given her intention, she tried to kill the wolverine, and failed.

In the change of mind case, Bob, however, didn’t try to get to the mall. Rather, he tried to start to get to the mall, and he also started trying to get to the mall. His trying to start was successful—he did start to get to the mall. But it makes no sense to attribute either success or failure to a mere start of trying.

There seems to be a moral difference, too. Suppose that killing the wolverine and getting to the mall are both wrong (maybe the wolverine is no danger to Alice, and Bob has promised his girlfriend not to go to this mall). Then Alice gets the opprobrium of being an attempted wolverine killer by virtue of (1), while Bob isn’t yet an attempted mall visitor by virtue of (2)—only when he strives to propel his body through the door does he become an attempted mall visitor. Even if killing the wolverine and getting to the mall are equally wrong, Bob has done something less bad—for the action he took in virtue of (2) was open to the possibility of changing his mind, as bringing it to completion would require further voluntary decisions. What Bob did was still wicked, but less so than what Alice did.

Action (1) commits Alice to killing the wolverine: if the wolverine fails to die, Alice is still an attempted wolverine killer. But Bob has undertaken no commitment to visiting the mall by starting the car.

This suggests to me that perhaps “intends” may be used in different senses in (1) and (2). In (1), it may be an “intends” that commits Alice to wolverine killing; in (2), it may be an “intends” that only commits Bob to starting trying to visit the mall. In (1), we have an intending that p that constitutes an action as a trying to bring it about that p.

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.