Reasons of marriage

Suppose we take seriously the idea that when a couple marries, an entity with moral standing—the married couple—comes into existence. Then there might be cases where an action is good for the spouses but bad for the married couple, and the fact that it is bad for the married couple could provide a strong moral reason to refrain from an action even if the action is good for the spouses.

That said, I don't accept an ontology on which a new entity comes into existence when a couple marries. But something similar to the above could still be the case. There are two kinds of wellbeing: one may call them narrow and extended wellbeing. Extended wellbeing is flourishing you have in virtue things outside of you going right for you. For instance, when someone you love has a success, your extended wellbeing increases even before you find out about it. Likewise, our reputation is a matter of our extended wellbeing, though it also tends to instrumentally affect our narrow wellbeing.

It can be quite rational to engage in some actions that sacrifice narrow wellbeing for extended wellbeing (just as sometimes the opposite makes sense). Now, even if a new entity doesn't come into existence when a couple marries, the members of the couple acquire a new mode of extended wellbeing, a mode where they are well insofar as the marriage goes well and poorly insofar as the marriage goes poorly.

But this means that it could happen that it would be rational for the spouses to sacrifice the narrow wellbeing of both persons for the sake of the external wellbeing they have in virtue of their marriage. It could well be that destroying the marriage is on balance a harm to the spouses even if they no longer care about the marriage and its destruction makes them feel better, just as an action that destroys one's reputation may be a harm to one even if one no longer cares about one's reputation and enjoys ruining it.

Mistaken gratitude and an argument for Christianity

Suppose I thank you with sincerity and expansiveness for saving my life at the risk of your own, and continually praise you to others, trying to get the President to give you a medal. But you didn't actually do anything like saving my life. I am just quite mistaken. Surely you, like any other virtuous person, would be dying of embarrassment and would be doing your best to convince me that you had not done this and hence do not deserve the thanks and praise.

Of course, it is crucial that the praise and thanks be sincere. A virtuous person need not allow himself to be manipulated by insincere fulsomeness. And there will be exceptions. If you thought that my own mental state was too fragile to hear the truth, or if I was too irrational in my belief, you might leave me to my mistake. If you hadn't saved my life but unbeknownst to me had done something else for me that was of the same sort, then you might leave me mistaken as to the exact nature of what you did for me. And, finally, if you haven't yet saved my life, but have an opportunity to do so, you might save my life now or soon instead of correcting me. This would be especially true if you wanted a loving relationship with me, for a love based on such a mistake is little better than a forced love.

The fact that a virtuous person does not contradict great thanks and praise by people who are sincerely convinced that he has made a great sacrifice for them is strong evidence that he has made, or is going to make, either that sacrifice or one of at least the same order of magnitude. And if the praise and thanks comes from people who are rational and psychologically healthy, the evidence is even stronger. And, finally, the simplest explanation of why the virtuous person does not contradict the praise and thanks is that not just that he has made or is about to make a sacrifice of the same order of magnitude, but that he has made the very sacrifice he is being thanked for.

But millions of Christians have praised and thanked God for saving them from sin at the cost of death on the cross, and have not found God to contradict them. And many of these Christians have been quite rational and psychologically healthy. Assuming that God exists—this argument needs to assume that—this gives significant evidence that God did what he is thanked for doing. So, likely, what they thank God for doing is just what God has done.

This argument cannot be used against Christianity since no other religion praises God for a good of a higher order of magnitude. Indeed, it seems unlikely that God could do anything of a higher order of magnitude for us.

Might "animal" be a stage term?

Consider this argument:

1. It is possible for me to exist disembodied.
2. It is not possible for an animal to exist disembodied.
3. So, I am not an animal.

While I accept (1), I am not convinced of (2). However I want to try a somewhat different tack in this post. Compare:

4. It is possible for Tom Brady to exist disembodied.
5. It is not possible for a football player to exist disembodied.
6. So, Tom Brady is not a football player.

But (6) is false (or so I understand from one website). And even if (4) were false, we shouldn't be able to derive its falsity simply from (5) and the fact that Tom Brady is a football player. So there has to be something wrong with the second argument. And the diagnosis is very simple: "football player" is a stage term. An entity can exist at one time as not a football player and at another time as a football player. Thus, (5) is ambiguous between two claims:

• It is not possible for someone who is presently a football player to exist disembodied at any time.
• It is not possible for someone to exist disembodied while being a football player.

The second of these may be true[note 1] but it is insufficient for deriving (6) from (4)—it only implies that Tom Brady can't be football player when disembodied, not that he can't exist when disembodied. And the first reading simply begs the question.

Why not draw the same conclusion from the first argument? Granted (I am not sure of this) one can't be disembodied while being an animal. But why can't someone who is an animal at one time be disembodied at another time, ceasing to be an animal then? Then "animal" would be a stage term. (It could even be the case that "animal" is a stage term while "person" isn't.)

If animalism is the claim that we are animals, then this would be compatible with animalism. One couldn't, however, straightforwardly say that we are essentially animals. But one could say that it is an essential property of beings like us that they begin their existence as animals, or at least (maybe God could create someone already in the disembodied stage?) that they normally do so.

One could say that these are claims about all animals or just about rational ones. Maybe only some animals—say, the rational ones—have the capability of becoming disembodied souls.

An argument for expected utility maximization

Until very recently, I thought there was only one argument for the idea that, barring deontic concerns and the like, rationality is connected to the maximization of expected utilities, namely the long-run advantage argument based on the Law of Large Numbers. But there is another: an argument from a plausible set of axioms for rational preferability. Fix a probability space Ω. Say that a gamble is a bounded real-valued random variable on Ω. Suppose that there is a rational preferability ordering < on gambles, where we write A<B if B is preferable to A. Here are some plausible axioms for <:

1. Transitivity: < is transitive
2. Domination: If A(ω)≤B(ω) for all ω∈Ω, then for all C, if C<A, then C<B, and if B<C, then A<C.
3. Sure Thing: If A and B are gambles that have certainty of getting payoffs a and b respectively, with a<b, then A<B.
4. Additivity: If A<C and B<D, then A+B<C+D.
5. Equivalence: If A and B are probabilistically equivalent (i.e., P(A∈U)=P(B∈U) for every measurable U), then for all C we have A<C if and only if B<C, and C<A if and only if C<B.

The most controversial will be, I suppose, Additivity. But there is a very simple argument for it: If you should choose C over A, and D over B, then you should choose the combination of C and D over the combination of A and B.

Add a handy technical assumption:

6. Continuity: There is a collection of events E_a, for 0<a<1, such that P(E_a)=a and E_a is a subset of E_b when a<b.

To get Continuity, all we need to do is suppose we've got some irrelevant continuous random process going on in our probability space, like the decay of an radioactive sample, or else suppose that we've got an infinite sequence of independent identically distributed coin flips, etc. Even if our world doesn't contain such a process, surely the same preferences would be rational in a world where some irrelevant-to-us such process takes place. So we can assume Continuity.

Theorem: Assume (1)-(6). If E(A)<E(B) for gambles A and B, then A<B.

The proof is given in this footnote: [note 1].

Personally, I am suspicious of transitivity in general, but I am less suspicious of it in the case of real-valued bounded gambles.

Conscience and intending the impossible

One of the toughest problems is what to do about cases of mistaken conscience. Let's say Samantha has a justified false belief that it is right to kill one innocent that ten might live (whether anyone can be really justified in thinking this is a question to bracket), and acts on it. Then it seems: Samantha did wrong and she did right. She did wrong in killing the innocent but she did right in following her conscience. But that shouldn't be the whole story. For suppose that Samantha had done the opposite—refrained from killing the innocent. Then she would have done wrong in disobeying conscience and right in refraining from killing the innocent. So whatever she does, she does wrong and she does right. And yet the two cases are not analogous. For when she kills, then she is inculpable of the murder by reason of her justified false belief. And when she refrains, she is culpable for violating her conscience.

We could leave it at this. But that would leave unexplained why it is that the duties of conscience are what culpability aligns with.

For years I've been trying to explore a story here, and I am never quite happy with it, and I am still not happy with it, but let me give it one more try. Intention, permissibility and impermissibility applies to action types, and not just to action tokens. And while there are no impossible action tokens, there are impossible action types that can be the objects of one's intentions. Many people, some sane and some not, have intended to trisect an angle (with ruler and straightedge). In so doing, their intentions had an object, the action type trisecting an angle. Moreover, their intended action type was permissible, albeit also impossible. We might say that per impossibile had they succeeded, they would have done something permissible and worthwhile.

Now go back to Samantha. Samantha intends a consequentially justified killing of an innocent. This action type, just like the trisecting of an angle, is impossible. It is impossible for consequences to justify the killing of an innocent. But if per impossibile she succeeded, she would have done something permissible and worthwhile. Plausibly, Samantha's intended action type while impossible is permissible, just as trisecting an angle is.

So on this story Samantha intended to do something permissible, but failed. She ended up doing something other than she intended. Samantha's action was an attempt at a consequentially justified killing of an innocent, and at least if she was justified in thinking that the attempt would succeed, she did right to make the attempt. And had she refrained from the attempt, she would have done wrong. Compare the case of someone who is ignorant of the impossibility of trisecting an angle and is told that an innocent will die if he does not trisect an angle. He acts well by trying to trisect and would be we doing wrong by refraining from trying.

On this story, if she kills, Samantha doesn't do both right and wrong. She simply does the right thing. But this right thing is an attempt at the impossible, and hence fails. And its failure, tragically, results in the death of an innocent (though if indeed the ten are saved, there is a silver lining, though not a justification). And if she believes that the killing would be consequentially justified, then in refraining from trying to kill, she simply does wrong.

But don't we want to say that Samantha unjustifiedly killed an innocent, and that's wrong? We need to be cautious here. The experienced surgeon who does her very best but who nonetheless kills the innocent patient does not do wrong. Her performance of the surgery is, indeed, a killing. And it's not a justified killing. But the surgeon's action is not intentional under the description killing the patient, and to say that the surgeon did wrong or right in killing the patient jars in the same way that it jars to ask whether my stumbling over a bump while walking to work was right or wrong. The stumbling was a part of my attempt to get to work, and hence was a part of a right action. But it was an accidental part as far as my intentions go. The same goes for the surgeon. It is harder to say this in Samantha's case, but perhaps not impossible. She did not intend a killing simpliciter, but a justified one. I would be inclined to say that both the surgeon's and Samantha's killings are non-justified, rather than unjustified.

The case of Samantha is particularly striking because it is impossible for a killing of an innocent to be consequentially justified. But one can also have similar cases where what is intended is possible. Suppose I reveal a secret that I promised to keep silent because I justifiedly but falsely believe that I ought to. Then I intend to break confidence as I ought. I fail—my breaking confidence is not justified. But what I intend is in fact a possible action type—there are times when one ought to break confidence. I do the right thing simpliciter: I attempt to do what I ought.

In the cases of Samantha and of confidence breaking, mistaken conscience enters into the story by making it possible for the agent to intend what otherwise the agent could not intend. Thus mistaken conscience functions much as the attempted trisector's false belief that one can trisect an angle.

There are probably some really serious problems with the above as a general proposal of what happens in conflicts of conscience. Here is one that particularly bothers me. I had breakfast today. Suppose, however, that I had promised someone not to have breakfast today (say, in order to experience solidarity with the less fortunate) but I completely innocently forgot the promise (imagine someone slipped me a forgetfulness pill). The analogue to the Samantha story would be that I intended to have a breakfast that I did not promise not to have. But of course I am exceedingly unlikely to have intended this while eating breakfast (wouldn't thinking about promises have brought my promise to mind?). Do you ever have such intentions when eating breakfast? (I suppose if one was in a habit of making promises to skip breakfast, one might. But one shouldn't make a habit of skipping breakfast—it's not healthy.)

Perhaps, though, whenever we do anything, we should be intending to do it rightly, or to glorify God through it, or the like. And if I tried to, say, rightly eat breakfast, while bound by promise not to eat, my action was a failure. So we do have the same pattern as in the Samantha story.

But is it really the case that whenever we act we need to have some such intention? Personally, I find this a plausible proposal. After all, we are to love God with our whole heart, soul, might and mind, and St Paul tells us to pray always and to take captive every thought (noema) for Christ. Thus every action of ours should be at least implicitly directed (perhaps in a way that even an atheist can) at the glory of God. When we fail to have it so directed, we do wrong.

This sounds right, but I don't know that it solves the breakfast problem. For suppose that I eat breakfast with no such intention, and eat contrary to my innocently forgotten promise. Then indeed I do wrong by not having the right God-glorifying intention in eating breakfast. But my innocent ignorance of my promise is still relevant. I am culpable for not intending to glorify God, but I am not culpable for breaking my promise, it seems. So something has yet to be explained.

But that we can handle a number of cases using the above method suggests that we may be able to do even more if we put our minds to it. Maybe there is something special about the promise case, for instance.

Consequentialism and doing what is very likely wrong

Consider a version of consequentialism on which the right thing to do is the one that has the best consequences. Now suppose you're captured by an eccentric evil dictator who always tells the truth. She informs you there are ten innocent prisoners and there is a game you can play.

• If you refuse to play, the prisoners will all be released.
• If you play, the number of hairs on your head will be quickly counted by a machine, and if that number is divisible by 50, all the prisoners will be tortured to death. If that number is not divisible by 50, they will be released and one of them will be given a tasty and nutritious muffin as well, which muffin will otherwise go to waste.

Now it is very probable that the number of hairs on year head is not divisible by 50. And if it's not divisible by 50, then by the above consequentialism, you should play the game—saving ten lives and providing one with a muffin is a better consequence than saving ten lives. So if you subscribe to the above consequentialism, you will think that very likely playing is right and refusing to play is wrong. But still you clearly shouldn't play—the risk is too high (and you can just put that in expected utility terms: a 1/50 probability of 10 being tortured to death is much worse than a 49/50 probability of an extra muffin for somebody). So it seems that you should do what is very likely wrong.

So the consequentialist had better not say that the right thing to do is the one that has the best consequences. She would do better to say that the right thing to do is the one that has the best expected consequences. But I think that is a significant concession to make. The claim that you should act so as to produce the best consequences has a very pleasing simplicity to it. In its simplicity, it is a lovely philosophical theory (even though it leads to morally abhorrent conclusions). But once we say that you should maximize expected utility, we lose that elegant simplicity. We wonder why maximize expected utility instead of doing something more risk averse.

But even putting risk to one side, we should wonder why expected utility matters so much morally speaking. The best story about why expected utility matters have to do with long-run consequences and the law of large numbers. But that story, first, tells us nothing about intrinsically one-shot situations. And, second, that justification of expected utility maximization is essentially a rule utilitarian style of argument—it is the policy, not the particular act, that is being evaluated. Thus, anyone impressed by this line of thought should rather be a rule than an act consequentialist. And rule consequentialism has really serious theoretical problems.

Another model of hell worth thinking about?

Suppose that objectively, hell lasts forever. But while the first objective year of hell is experienced subjectively as a year long, the second objective year "goes by faster" as we say, and only takes half a year, the the third objective year "goes by even faster" and only takes a quarter of a year, and so on. Thus, while the damned will always exist and always be suffering, they will only experience two years' worth of suffering over that objectively eternal suffering.

Now the difficult question is whether this is an orthodox view of hell. When Jesus talks about the suffering being everlasting, is he talking of subjective or objective time? We certainly wouldn't find the analogous view of heaven satisfactory. But heaven and hell aren't exact parallels: in heaven one is with God, and the absence of God is not much of a parallel to God.

Now, without affirming the model, it can still be of some use in apologetics. For suppose a non-Christian objects that nobody deserves an everlasting hell. One answer is Anselm's: an infinite crime deserves infinite punishment and some crimes against an infinite being are infinite. But given the above model or the alternate model here, one can say that an everlasting hell could involve only a finite amount of suffering. So one can say: if someone is damned, then either she committed a crime that deserves infinite punishment or her total suffering is finite. Since both options are compatible with everlasting hell, in neither case does the objection to an everlasting hell go through. And one can give this disjunctive answer while strongly inclined to think that the Anselmian infinite crime model of hell is superior, as long as the alternate model is not a heresy (if it is, I will of course withdraw it).

wincloudprint.py: Google Cloud Print Windows service

We're thinking of replacing my wife's netbook with a Chromebook, but printing is an issue. Our laser printer is over a decade old and while it works fine on our network (with a network adapter) it certainly doesn't support Google Cloud Print. Google really should have added support for local network printers. Their standard solution is a proxy that runs on some computer on the network via Chrome. But the standard way of doing that has two problems: (1) Chrome presumably takes up a lot of memory (I haven't checked just how much) and I don't want it running in the background all the time, and (2) it runs as a user application, not as a service, so a user for whom this has been configured needs to be actually logged in on the computer. Google has a solution to (2), but I didn't manage to get it working.

Fortunately, I managed to adapt the Linux python scripts from cloudprint to make a Windows solution, available here as "wincloudprint" (GPL3). Alas, installation is a bit of a bear due to license issues (I can't just include everything in one self-contained download): you need to install python, pywin32, SumatraPDF and wincloudprint. (SumatraPDF is used for handling the actual printing.) All the instructions are at the link. I don't know know that anybody other than myself will be interested in this, but I thought I'd share it.

Knowledge last: An argument

1. If p partly grounds q and p is explanatorily irrelevant to r, then q does not strictly explain r.
2. Something's being ungettiered is explanatorily irrelevant to all philosophically important facts, except for knowledge facts.
3. That one's belief that p is ungettiered partly grounds that one knows p.
4. So facts about what one knows do not strictly explain anything philosophically important.

The concept of strict explanation here is a kind of purification of explanation where irrelevant elements are removed. For instance, that George was late for work because George was mugged by a Polish-Canadian is not a case of strict explanation, because that the mugger was Polish-Canadian is irrelevant to explaining George's lateness.[note 1] But maybe it is a case of explanation. (Wes Salmon says that irrelevancy destroys explanation, but maybe sometimes it just renders it unstrict.)

So, in the order of explanation among philosophically important facts, knowledge facts come last, if at all.

Ethics for deer: A question

Suppose a species of deer who are persons evolved. And suppose one of the deer discovered that meat tastes good. Would it be permissible for the deer to kill animals from non-person species for meat?

Conditional commitments

Compare:

1. Assuming you pass at least one of your classes this spring, we will hire you in May.
2. We will hire you in May.

To a literalist it sounds like 2 makes the stronger commitment than 1.

But suppose that you get the lowest passing grade in one of your classes, and Fs in all the others. Then if I said 2, I could say: "Well, of course, but I was assuming half-decent performance." But if I said 1, I can't say that!

What's going on? Normally when I say what I will do, there are some unstated conditions. But when I get into the business of stating conditions, I had better list all of them, or at least all the ones that are likely to be as relevant as the ones I list.

Knowledge last (if at all) epistemology

The hot thing in epistemology these days is "knowledge first" epistemology. What I think about epistemology and knowledge is best summed up as: "knowledge last (if at all)". I don't know of anything philosophically interesting, besides others facts about what one knows, that is explained by facts about what one knows—as opposed to by facts about what one believes, what one is justified in, what is true, what one understands, etc. Facts about knowledge come last in the order of explanation, if at all. But I'm not an epistemologist—just a formal epistemologist—so nobody should care much about what I think about epistemology.

Pascal's Wager rescued

In its classical formulation, Pascal's Wager contends that we have something like the following payoff matrix:

	God exists	No God
Believe	+∞	−a
Don't believe	-b	c

where a,b,c are finite. Alan Hajek, however, observes that it is incorrect to say that if you don't choose to believe, then the payoff is finite. For even if you don't now choose to believe, there is a non-zero chance that you will later come to believe, so the expected payoff whether you choose to believe or not is +∞.

Hajek's criticism has the following unhappy upshot. Suppose that there is a lottery ticket that costs a dollar and has a 9/10 chance of getting you an infinite payoff. That's a really good deal intuitively: you should rush out and buy the ticket. But the analogue to Hajek's criticism will say that since there is a non-zero chance that you will obtain the ticket without buying it—maybe a friend will give it to you as a gift—the expected payoff is +∞ whether you buy or don't buy. So there is no point to buying. So Hajek's criticism leads to something counterintuitive here, though that won't surprise Hajek. The point of this post is to develop a rigorous principled response to Hajek's criticism embodying the intuition that you should go for the higher probability of an infinite outcome over a lower probability of it.

A gamble is a random variable on a probability space. We will consider gambles that take their values in R*=R∪{−∞,+∞}, where R is the real numbers. Say that gambles X and Y are disjoint provided that at no point in the probability space are they both non-zero. We will consider an ordering ≤ on gambles, where X≤Y means that Y is at least as good a deal as X. Write X<Y if X≤Y but not Y≤X. Then we can say Y is a strictly better deal than X. Say that gambles X and Y are probabilistically equivalent provided that for any (Borel measurable) set of values A, P(X∈A)=P(Y∈A). Here are some very reasonable axioms:

1. ≤ is a partial preorder, i.e., transitive and reflexive.
2. If X and Y are real valued and have finite expected values, then X≤Y if and only if E(X)≤E(Y).
3. If X and Y are defined on the same probability space and X(ω)≤Y(ω) for every point ω, then X≤Y.
4. If X and Y are disjoint, and so are W and Z, and if X≤W and Y≤Z, then X+Y≤W+Z. If further X<W, then X+Y<W+Z.
5. If X and Y are probabilistically equivalent, then X≤Y and Y≤X.

For any random variable X, let X* be the random variable that has the same value as X where X is finite and has value zero where X is infinite (positively or negatively).

The point of the above axioms is to avoid having to take expected values where there are infinite payoffs in view.

Theorem. Assume Axioms 1-5. Suppose that X and Y are gambles with the following properties:

6. P(X=+∞)<P(Y=+∞)
7. P(X=−∞)≥P(Y=−∞)
8. X* and Y* have finite expected values

Then: X<Y.

It follows that in the lottery case, as long as the probability of getting a winning ticket without buying is smaller than the probability of getting a winning ticket when buying, you should buy. Likewise, if choosing to believe has a greater probability of the infinite payoff than not choosing to believe, and has no greater probability of a negative infinite payoff, and all the finite outcomes are bounded, you should choose to believe.

Proof of Theorem: Say that an event E is continuous provided that for any 0≤x≤P(E), there is an event F⊆E with P(F)=x. By Axiom 5, without loss of generality {X∈A} and {Y∈A} are continuous for any (Borel measurable) A. (Proof: If necessary, enrich the probability space that X is defined on to introduce a random variable U uniformly distributed on [0,1] and independent of X. The enrichment will not change any gamble orderings by Axiom 5. Then if 0≤x≤P(X∈A), just choose a∈[0,1] such that aP(X∈A)=x and let F={X∈A&U≤a}. Ditto for Y.)

Now, given an event A and a random variable X, let AX be the random variable equal to X on A and equal to zero outside of A. Let A={X=−∞} and B={Y=−∞}. Define the random variables X₁ and Y₁ on [0,1] with uniform distribution by X₁(x)=−∞ if x≤P(A) and X₁(x)=0 otherwise, and Y₁(x)=−∞ if x≤P(B) and Y₁(x)=0 otherwise. Since P(A)≥P(B) by (7), it follows that X₁(x)≤Y₁(x) everywhere and so X₁≤Y₁ by Axiom 3. But AX and BY are probabilistically equivalent to X₁ and Y₁ respectively, so by Axiom 5 we have AX≤BY. If we can show that A^cX<B^cY then the conclusion of our Theorem will follow from the second part of Axiom 4.

Let X₂=A^cX and Y₂=B^cY. Then P(X₂=+∞)<P(Y₂=+∞), X₂* and Y₂* have finite expected values and X₂ and Y₂ never have the value −∞. We must show that X₂≤Y₂. Let C={X₂=+∞}. By subdivisibility, let D be a subset of {Y₂=+∞} with P(D)=P(C). Then CX₂ and DY₂ are probabilistically equivalent, so CX₂≤DY₂ by Axiom 5. Let X₃=C^cX₂ and Y₃=D^cY₃. Observe that X₃ is everywhere finite. Furthermore P(Y₃=+∞)=P(Y₂=+∞)−P(X₂=+∞)>0.

Choose a finite N sufficiently large that NP(Y₃=+∞)>E(X₃)−E(Y₃*) (the finiteness of the right hand side follows from our integrability assumptions). Let Y₄ be a random variable that agrees with Y₃ everywhere where Y₃ is finite, but equals N where Y₃ is infinite. Then E(Y₄)=NP(Y₃=+∞)+E(Y₃*)>E(X₃). Thus, Y₄>X₃ by Axiom 2. But Y₃ is greater than or equal to Y₄ everywhere, so Y₃≥Y₄. By Axiom 1 it follows that Y₃>X₃. but DY₂≥CX₂ and X₂=CX₂+X₃ and Y₂=DY₂+Y₃, so by Axiom 4 we have Y₂>X₂, which was what we wanted to prove.

Vague baldness with no hair

You might think that someone who has no hair is definitely bald. Not so. For only someone who has a normally hirsute head can be bald, and it can be vague whether a particular hairless person has a head. For instance, we can imagine aliens that have a part that resembles our heads to some degree--maybe they have eyes on it but all their other sensory organs and mouth are on their hand--and it will be vague whether they have heads. So, if Bob is such an alien and has no hair anywhere, then it is definitely true that either Bob is maximally bald (if he has a head) or Bob is not bald at all (if he has no head).

This is inspired by a remark by Kenny Pearce that, in the case he was writing about, something was a paradigm case of F without being at all an intense case of F. The lesson is that it can be important to keep degrees of being F apart from whether one is definitely F or not. This remark may damage this argument.

Choosing a zero probability of an infinite good over a certainty of a finite good

Sometimes it is rational to choose a zero probability of an infinite good over a certainty of a finite good.

Suppose there are uncountably many benevolent people, each of whom is assigned a number in (0,1), the interval from zero to one non-inclusive.  A random number Y is chosen in (0,1) with a continuous distribution (say, a uniform one, or a cut-off Gaussian).  The people aren't informed of its value, but they know the setup of the story.

Person number x is now given this choice:  

• wager: if Y=x, then everyone gets $1; else, nothing happens.  
• don't wager: the person with number x/2 gets $1.

Then:

• If everybody wagers, then everybody gets $1.
• If nobody wagers, then all and only the people with numbers in (0,1/2) get $1.

So surely at least some, and probably all, should wager.  But if you wager, you're choosing a zero probability of an infinite good (since the probability that your number matches Y is zero) over the certainty of a finite good.  (The goods are to others, but since you're benevolent, that doesn't matter.)