Very new moon

It's over the Baylor Science Building, appropriately enough.

Another absurdity about infinite fair lotteries

It is easy to generate a method for choosing a natural number in such a way that the probability of choosing n is 2⁻ⁿ. For instance, toss a fair coin and let n be the number of the first toss on which you get heads. Thus, n=1 if you get heads on your first toss, n=2 if you get tails on the first and heads on the second, n=3 if you get tails on the first two and heads on the third, and so on. (And if you never get heads, count that as just another way of choosing 1—after all, the probability of never getting heads is zero.)

Could one have a way of choosing a natural number, guaranteed to return some natural number, but such that the probability of choosing n is between 4⁻ⁿ and 3⁻ⁿ? Surely not! That would be absurd: the probabilities would be too small.

But now suppose you can have an infinite fair lottery. Follow the following procedure. Toss a fair coin until you get heads. If it took an even number m of tosses to get to heads, let your chosen number be n=m/2. If it took an odd number of tosses to get to heads, or if you never got to heads, then choose the natural number n via an infinite fair lottery.

What's the probability of getting n in this new process? Well, the probability of getting heads for the first time on the 2nth toss is 2⁻²ⁿ=4⁻ⁿ. But if you didn't get heads for the first time on an even-numbered toss, you still have a chance of getting n, namely via the infinite fair lottery. The latter chance is either zero or infinitesimal. So, the probability of getting n is 4⁻ⁿ or 4⁻ⁿ plus an infinitesimal. But it's absurd to have a random choice of natural number where the probability of getting n is 4⁻ⁿ. And if it's 4⁻ⁿ plus an infinitesimal, then it's between 4⁻ⁿ and 3⁻ⁿ, which we agreed was absurd.

So infinite fair lotteries lead to absurdity.

Not caring what other people think

I used to think I was the sort of person who didn't care what other people think. I was wrong. I was the sort of person who wanted other people to think he was the sort of person who didn't care what other people think. A very different critter.

Deep Thoughts XXXVII

Everybody is somebody.

Deep Thoughts XXXVI

The necessary is always possible.

Another quick argument for the possibility of incommensurability

Suppose Curley is deciding whether to accept a $1,000,000 bribe and lose his soul, or remain honest. It seems quite possible to have situations like this where neither option is preferable to Curley. Of course, prima facie that need not be a case of incommensurability--it might be a case of equal preference. But we cannot say that all similar cases like this are cases of equal preference. For in most cases like this, Curley also wouldn't have a preference between a $1,010,000 bribe and honesty. If that too has to be a case of equal preference rather than incommensurability, then by the transitivity of equal preference, we would have to say in such cases that Curley has equal preference as to a $1,000,000 bribe and a $1,010,000 bribe. But of course that's false: in most cases like this, he prefers the larger bribe.

Non-sentences

Consider:

1. Many a philosopher was celebrated during his life. His work was culturally influential. And then after he died, he was all but forgotten.

Notice that "His work was culturally influential" has the grammatical form of a sentence, but is not a sentence. It is an open formula with "His" being a free variable, bound ultimately by the "Many a philosopher" quantifier. But there seem to be other contexts in which "His work was culturally influential" seems to be a sentence:

2. Bergson was celebrated during his life. His work was culturally influential. And then after he died, he was all but forgotten.

So it seems whether something is a sentence is sometimes contextually determined.

Maybe we can say that "his" is actually two words: one word functions as a variable and the other as a name (with reference anaphorically coming from another name). But on grounds of Ockham's razor this seems a poor move.

There is another move one can make here that seems better: The third-person pronoun is always a variable. In (1), it is bound by the "Many a philosopher" quantifier. In (2), it is bound by the "Bergson" quantifier. (Here I am following Montague's insight that names can be seen as functioning as quantifiers.)

Note added later: I think the name-as-variable move doesn't get one out of the contextuality of what's a sentence. Suppose that after I said (2), you said: "He is still quite influential." What you said is clearly a genuine sentence.

Real dilemmas and conscience

Some people—notably, St. Thomas—think that you can have real dilemmas, ones where you are obligated to do each of two incompatible actions, but only when you've done something wrong. For instance, one might make incompatible promises to two different people, and then have a real dilemma, but of course it's wrong to make such incompatible promises.

But given the binding force of conscience, this is not a stable position to hold. For consider a situation S where one would be in a real dilemma, say the situation where one made incompatible promises to two different people. But now imagine a situation S^* which is epistemically like S and where the same actions are open to one, but where one did nothing wrong to get into S^*. Maybe one has a justified false belief that one has made incompatible promises to two different people (say, a bad friend convinces you that you made such promises but forgot about them). Then while one does not have duties of promise to do the two incompatible actions, one does have duties of conscience to do them.

A complete denial of real dilemmas is a more stable position, as is the position that where there is a real dilemma, something has gone wrong morally or epistemically (but not necessarily through one's having done anything wrong, morally or epistemically).

Trying to do what one knows is impossible

I know that angles can't be trisected. But suppose that I know for sure that unless I manage to find a way to trisect angles in the next ten years, the human race will be destroyed. What should I do? Surely I should try to trisect, hoping that there are mistakes in the impossibility proofs. And surely here at least is a place where I can do what I should. So it is possible to try to do what one knows to be impossible.

Foam covered wooden swords

I made these for my big kids to spar with, as a birthday present to my son. Here are instructions on how to make them.

Thick and thin obligations

Suppose that all fundamental thick moral obligation terms can be put in the logical form: "required by virtue v". Then we have a plausible and neat account of thin obligation in terms of thick obligation:

1. A is obligatory if and only if there is a virtue v such that A is required by v.

Can we define the thick terms via thin obligation? A promising start is:

2. A is required by virtue v if and only if facts about v explain why A is obligatory.

But this is only a start, since there are obvious counterexamples. Suppose I promise you to give you ten dollars if courage is a virtue. Then a fact about courage, namely that it is a virtue, explains why it is obligatory for me to give you ten dollars, but I am not required by justice, and not by courage, to give you ten dollars.

To make something like (2) work, we would need to specify the way in which facts about v explain why A is obligatory. It is implausible to suppose that this can be done without recourse to something circular, like saying that facts about v explain in a requirement-inducing way why A is obligatory.

This suggests that it is easier to account for thin obligation in terms of thick obligation. And this, in turn, suggests that it is better to take as our fundamental concept of obligation not "is obligated to" but "is obligated by ... to".

Deflation of predicates

Some deflationary theories take some predicate, such as "is necessary" or "is true", and claim that there is really nothing in the predicate for philosophical investigation—the predicate is not in any way natural, but just attributes some messy, perhaps even infinite, combination of more natural properties.

But I know only three candidates for a way that we could come to grasp a meaningful predicate. One way is by ostension to a natural property. Here's a rough idea. The predicate "is circular" might be introduced as follows. We are shown a bunch of objects, A₁,...,A_n, and told that each "is circular", and a bunch of other objects, B₁,...,B_m, and told that each "is not circular." The predicate "is circular" is then grasped to indicate some property that all or almost all of the As have and all or almost all of the Bs lack. But there may be many (abundant) properties like that (for instance, being one of A₁,...,A_n). Which one do we mean by "is circular"? Answer: The most natural of the bunch.

The second way depends on a non-natural view of mind. It could be that our minds, unlike language, can directly be in contact with some properties. And it may be that a predicate tends to be used in circumstances in which both speaker and listener are directly contemplating a particular property, and that makes the predicate mean that property.

The third way is by stipulation. I just say: "Say that x is frozzly if and only if x is frozen and green."

The predicates, like "is true" and "is necessary", that are the subjects of these deflationary theories are not introduced in the first way if the theories are correct to hold that the predicates do not correspond to a natural property. Are they introduced in the third way? That is very unlikely. I doubt there was a first user of "is necessary" who stood up and said: "I say that p is necessary if and only if...." That leaves the second way, the non-naturalistic way. Therefore:

1. If these deflationary theories are correct, naturalism is false.

Which is interesting since the motivation for the theories is sometimes naturalistic (e.g., Hartry Field in the case of "is true"[note 1]).

But in any case, the following is very plausible. Any properties we are in direct non-natural cognitive contact with are either innately known or natural. So, the deflated predicates must refer to innately known predicates. I doubt, however, that we innately know any entirely non-natural predicates. And that leaves little room for these theories.

More generally, the above considerations make it difficult to see how we could have any genuine non-natural, non-stipulative predicates. Thus, if we have good reason to think that P does not indicate a natural property, and is not stipulative, we have good reason to have an error theory about P.

Concepts of artifacts appear to be a counterexample. "Is a chair" is neither natural nor stipulated. My inclination is to say that it is not really a predicate ("Bob is chair" expresses some sentence about Bobbish reality being chairwise arranged, or something like that), which makes for a kind of error theory.

The human animal and the cerebrum

Suppose your cerebrum was removed from your skull and placed in a vat in such a way that its neural functioning continued. So then where are you: Are you in the vat, or where the cerebrum-less body with heartbeat and breathing is?

Most people say you're in the vat. So persons go with their cerebra. But the animal, it seems, stays behind—the cerebrum-less body is the same animal as before. So, persons aren't animals, goes the argument.

I think the animal goes with the cerebrum. Here's a heuristic.

• Typically, if an organism of kind K is divided into two parts A and B that retain much of their function, and the flourishing of an organism of kind K is to a significantly greater degree constituted by the functioning of A than that of B, then the organism survives as A rather than as B.

Uncontroversial case: If you divide me into a little toe and the rest of me, then since the little toe's contribution to my flourishing is quite insignificant compared to the rest, I survive as the rest. More controversially, the flourishing of the human animal is to a significantly greater degree constituted by the functioning of the cerebrum than of the cerebrum-less body, so we have reason to think the human animal goes with the cerebrum.

Another related heuristic:

• Typically, if an organism of kind K is divided into two parts A and B that retain much of their function, and B's functioning is significantly more teleologically directed to the support of A than the other way around, then the organism survives as A rather than as B.

My heart exists largely for the sake of the rest of my body, while it is false to say that the rest of my body exists largely for the sake of my heart. So if I am divided into a heart and the rest of me, as long as the rest of me continues to function (say, due to a mechanical pump circulating blood), I go with the rest of me, not the heart. But while the cerebrum does work for the survival of the rest of my body, it is much more the case that the rest of the body works for the survival of the cerebrum.

There may also be a control heuristic, but I don't know how to formulate it.

"God agrees with me" and "I have the truth"

Suppose I am convinced that God exists. Then if p is true, God believes p. So it seems that whenever I have the right to assert p, then as long I should also be willing to say: "And God agrees with me." But to say that God agrees with me sounds awfully arrogant!

I suppose some of the apparent arrogance comes from the implicature that I have independent evidence that God agrees with me—a special line to God. But of course in the typical case, my evidence that God agrees with me about p just is my evidence for p (plus my evidence that God exists and that I believe p).

Or maybe it's that one implicates certainty. (Why? Is it because there is a stereotype that when people make claims about God they are certain of them?)

There is a similar impression of arrogance one conveys when one says: "I have the truth about p." Yet, of course, if one is justified in believing p, one is typically justified in believing that p is true, and hence that one has the truth about p. Again, maybe the issue is that saying one has the truth implicates certainty?

There is, indeed, something odd about claiming that God agrees with one or that one has the truth on a subject where one has only a weak opinion. I am about to have random.org choose a number between 1 and 10, both inclusive. I think that the number will be smaller than 10. But it would be odd to say: "God agrees with me" or "I have the truth on that." Yet, my evidence that I have the truth on the number being smaller than 10 is almost as good as my evidenec that the number will be less than 10, and my evidence that God agrees with me is very good, given that I have very good evidence that God exists. (Oh, and I was right. The number turned out to be 1.)

A quick theological argument that we do not cease to exist at death

On materialist reconstitution views of the resurrection, we cease to exist at death, but then we are reconstituted at the resurrection. On Thomas Aquinas's view, we cease to exist at death, though our soul continues to exist, and then at the resurrection we come back to life.

On these views, we should view the badness of death as primarily constituted by a cessation of existence. But Christ did not cease to exist when he died. The Trinity did not become a Binity between Good Friday and Easter Sunday! So if the badness of death is primarily constituted by a cessation of existence, Christ either did not die or at least did not undergo the primary badness of death. And both options do serious damage to the doctrine of atonement.

(The view of death that seems right to me is that death is the destruction of the body. And Christ underwent that.)