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.)

Scepticism and the Principle of Sufficient Reason

The following argument is inspired by one that Rob Koons gives here:

1. If the Principle of Sufficient Reason is not true, then it is possible for there to be only one finite conscious being, a completely unexplained brain in a vat with perceptual states just like mine.
2. Completely unexplained contingent states of affairs have no objective probability.
3. So, if the Principle of Sufficient Reason is not true, it is not objectively unlikely that the only finite conscious being is a brain in a vat with perceptual states like mine.
4. If it is not objectively unlikely that the only finite conscious being is a brain in a vat with perceptual states like mine, then I do not know that I have two hands.
5. I know I have two hands.
6. So, the Principle of Sufficient Reason is true.

The a priori and the a posteriori

I've been thinking about Chalmers' Constructing the World. It is absolutely crucial for Chalmers to have a distinction between what is a priori knowable and what is a posteriori knowable.

Now, imagine that we have evolved to believe that the number one has a successor (call this proposition "Two") as well as that many snakes are dangerous ("Snake"). In both cases, let us suppose, we evolved to believe the true claim because believing it conduced to our survival. (We may add that the reason belief in the claim conduced to our survival was explained by the truth of the claim, if we are worried about debunking arguments.) It now looks like Two and Snake are on par: either both beliefs are a priori or both beliefs are a posteriori. However, Chalmers cannot afford to say that Snake could be a priori—that will destroy much of his story.

So it seems that if Chalmers' story is to work, we will have to say that Snake and Two are a posteriori. However, it is also important for Chalmers to say that fundamental principles like Two are knowable a priori. The story doesn't destroy this: after all, something can be both a posteriori known and a priori knowable (say, a result of a calculation done with a calculator). Nonetheless, there is a problem here. It may well be that all fundamental principles like Two are known by us through something like this evolutionary mechanism (and "something like" includes the theistic variant where God instills this belief in us on account of its truth). And if so, then what reason do we have to think that they are a priori knowable, given that we don't know them a priori? One might have some conviction that some hypothetical or actual non-human reasoner knows them a priori, but it is difficult to see that conviction as justified.

However, I think there is a speculative non-naturalist story one could give that would help make the distinction. Suppose that all our knowledge is at base perceptual. However, sometimes what we perceive are abstracta and sometimes what we perceive are concreta. Knowledge that is grounded only in perception of abstracta is a priori, while knowledge that is grounded at least in part in perception of concreta is a posteriori. It may be that we can just see the number two as the successor of the number one. There is some phenomenological plausibility to this.

This would be really nice for Chalmers. For it is plausible that if any abstracta and their abstract relations are observable, they all are. If so, then all facts about the realm of abstracta are a priori knowable.

Granted, on this story knowledge of abstracta is observational and hence empirical. But while this does mean that the above use of "a priori" and "a posteriori" is idiosyncratic, the above use nonetheless may help recover at least some of Chalmers' story.

Some, but perhaps not the two-dimensionalism? For the account above is apt to make it a priori that wateriness is H₂O-ness, since that seems to just be a fact about abstracta.

Inequality

1. There is nothing intrinsically bad in heaven.
2. There is inequality in heaven.
3. So, inequality is not intrinsically bad.

Some reasons for accepting (2):

• God is in heaven and humans are in heaven, and there is infinite inequality there.
• There are angels in heaven and humans are in heaven, and there is inequality there.
• Heavenly bliss is proportional to merit (though of course the merit is the work of God's grace), and merit is unequal.

Even more on simplicity and theism

Some naturalists say that theism needlessly complicates our view of the world by positing that

• in addition to the material concrete contingent things, there is something immaterial, necessary and concrete.

But the naturalist needs to say that the naturalist needlessly complicates our view of the world by being committed to the claim that

• in addition to the dependent concrete contingent things, there is something independent, concrete and contingent.

(Say, the Big Bang or the universe as a whole.)

Does talking of needless complication get us ahead here?

Simplicity and theism

I have argued elsewhere, as my colleague Trent Dougherty also has and earlier, that when we understand simplicity rightly, theism makes for a simpler theory than naturalism. However, suppose I am wrong, and naturalism is the simpler theory. Is that a reason to think naturalism true? I suspect not. For it is theism that explains how simplicity can be a guide to truth (say, because of God's beauty and God's desire to produce an elegant universe), while on naturalism we should not think of simplicity as a guide to truth, but at most as a pragmatic benefit of a theory. Thus to accept naturalism for the sake of simplicity is to cut the branch one is sitting on.

Illocutionary force and propositions

Suppose I say to Bill: "Make all of your papers be between two and four pages." Bill hands in an eight page paper for his first assignment. I rebuke him and he apologizes. He then hands in another eight page paper for his second. When I rebuke him, he says: "You told me to bring it about that all my papers be between two and four pages. With my first paper I ensured that the proposition that all my papers are between two and four pages is false. Sorry! By the time of my second paper, it was too late to undo this: no matter what length of paper I wrote, that proposition would still be false. So I might as well write the length that I like."

Bill's mistake was thinking that the content of my command was the proposition that all his papers be between two and four pages. I didn't command that proposition. Rather, I commanded distributively of each of his papers that it be between two and four pages.

This means that we should not analyze my speech act as having a propositional content plus an illocutionary force. The content of the speech act wasn't a proposition, but something else. Perhaps the content of the speech act was an ordered pair of properties, the property P of being one of Bill's papers, and the property L of being between two and four pages in length. And the illocutionary force was of something one might call distributive command. Successful distributive command in respect of a pair of properties P and L creates for each instance x of P a reason to make x have L.

There are, I think, assertion-like speech acts that also have such a non-propositional content. For instance, assertoric endorsement. A paradigm case: I endorse what you are about to assert. The content of assertoric endorsement is a property which is supposed to be had by one or more propositions—say, the property of being soon asserted by you—and when successful, the assertoric endorsement makes you stand behind each of these propositions as if you asserted it. This kind of assertoric endorsement is distributive.

I wish I knew what kinds of entities can be contents of speech acts. The above suggests that some speech acts have propositions as contents, some have pairs of properties, some have single properties. There must be many other options.

Simplicity as a sign of design

It seems hard to deny that simplicity is a guide to truth in science. But the best account of the simplicity of a theory is brevity of expression in a language whose terms cleave nature at the joints. But that brevity of expression in a language is a guide to truth is a sign that a rational being is behind our universe.

A theory of contingency and an argument for a causal Principle of Sufficient Reason

Consider this theory, a modification of my causal power account of possibility:

• A proposition p is contingent provided that something has a power for p and something has a power for not-p.

Here, I say that x has a power for p if and only if x has the power to bring p about, or x has the power to bring it about that something has the power to bring p about, or ....

It follows from this theory that every contingent true propositions has a causal explanation.

For suppose for reductio ad absurdum that p is contingently true and has no causal explanation. Let q be the conjunction of p with the claim that p has no causal explanation. Then q is true, and it is not necessarily true since p is not necessarily true, so q is contingent. It follows from our account of contingency that something has the power to ... bring q about (where the "..." is a possible chain of causal power claims). But that's absurd, since something that brings q about thereby also brings p about, and then p isn't bereft of causal explanation!

Love of truth

Let's say I am grading final exams and am very curious how a student who had been struggling all semester will do in the course. So I forthwith submit a B+ for her to our grading system, without bothering with any more calculations, and my curiosity is satisfied.

There is something perverse here. Of course, there is a perversion of justice—that's clear. But I think there may also be a perversion of the intellectual life. Genuine love of truth is not satisfied by making a proposition true or false. Genuine love of truth, at least as proper to creatures, seeks to make the mind reflect the world, not to make the world reflect the mind. If this line of thought is wrong, then the counterexample to evidentialism in my previous post fails.

The issue comes also comes up in third-person cases. My friend thinks that I will be wearing a long-sleeved shirt today. Does a loving desire to promote his intellectual goods give me any reason to wear such a shirt? I doubt it. But if not, then this is very puzzling. For surely my friend is intrinsically the better off for getting right what I will wear.

Maybe the case is a bit like throwing a game. My daughter wants to beat me at chess. But she wants to beat me by her own powers, rather than because I didn't try hard. Is there any value to beating me if I don't try at all?

This example suggests that when I wear a long-sleeved shirt to make my friend be right, his being right is not an achievement of his, and hence it's not much of a victory. But maybe this makes the epistemic life sound too proud. Maybe we should rather see it humbly as a comformation of our minds to reality.

Maybe the direction-of-fit issue here is parallel to one with desires. I bought my friend a trinket for his birthday. I then slip him a pill that induces a desire for the trinket. Surely that's perverse—it gets things the wrong way around. I should make the world conform to my friend's (reasonable) desires, not the other way around. And to make my friend's beliefs conform to the world, not the other way around.

It might be different in the case of God. Aquinas says that God knows creation by creating. Maybe here is a crucial difference between God and creatures.

A counterexample to evidentialism?

Consider Williamson-style beliefs that obviously have the property that they have to be correct if they are believed. For instance, if I believe that I have a belief, then that belief is guaranteed to be correct. Call beliefs like this obviously self-guaranteeing.

Suppose now that I am unable to introspect my beliefs and am not a sufficiently good observer to gain evidence as to what I believe on the basis of my behavior. Unsure whether I have any beliefs, but thinking that true beliefs are valuable to have although false ones are valuable to avoid, I try to will myself to believe that I have a belief, because it is clear to me that that claim will be true if I believe it. (You might ask: If I do that, don't I already believe something, namely that the belief will be true if I believe it? Maybe, but that's beside the point, since I am unable to tell that I believe it.) I don't know if I will succeed—and even if I do succeed, I won't know that I have succeeded—since willing myself to have a belief is a notoriously shaky thing. There seems to be nothing incompatible with the love of truth in willing myself to believe that I have a belief, indeed there seems to be nothing epistemically bad. But I am (a) willing myself to believe something I now do not have evidence for, and (b) if I do come to believe it, I will believe it without any evidence for it. If indeed there is nothing epistemically bad here, then (b) gives a counterexample to synchronic evidentialism and (a) gives a counterexample to diachronic evidentialism.

But perhaps there is something perverse here. See tomorrow's post.

Explaining the necessary with the contingent

It may seem initially slightly surprising, but there are necessary truths that are explained by contingent ones. For instance, it is a necessary truth that Obama is president or 2+2=4. And this necessary truth can be explained by the fact that the majority of the electoral college voted for Obama, or, perhaps even better, by facts about the way the Democrats and Republicans campaigned. Another necessary truth that can be explained in the same way is that it is or is not the case that Obama is president.

That the necessary can sometimes be explained with the contingent is, I think, a rather more trivial claim than that the contingent can be explained with the necessary.

An interesting epistemic scoring rule

A forecast p is an assignment of probabilities to events in some space Ω. A proper score is an assignment of a random variable s_p to each forecast on that space, with the property that E_ps_p≤E_ps_q whenever p is a consistent forecast (one that satisfies the axioms of probability) and q is any other forecast. Here, E_p is expectation with respect to the probability function p. Propriety basically says that if we have a consistent forecast, then by our own lights no other forecast is expected to have a better score. The scores are thought of as penalties or distances from truth—smaller is better.

One thing proper scoring rules have been used for is to argue that our credences should be consistent. For instance, under a simplifying assumption, Predd et al. have basically shown that the proper score for an inconsistent forecast is always dominated (from below) by a proper score for some consistent forecast. The simplifying assumption is that scores are computed for individual events and added.

Now, here is a curious proper score that does not satisfy this simplifying assumption. Suppose we're working with a finite space Ω with n points. Suppose p is consistent. Let m(p) be a point of Ω where p is maximized for a forecast p. (Use any tie-breaking method you like if that point isn't unique.) Then let s_p be 0 at m(p) and 1 everywhere else. Then if p and q are consistent, E_ps_q=1−p(m(q)) (where p(ω)=p({ω})). Since p(m(p))≥p(m(q)) by definition of m, it follows that E_ps_p≤E_ps_q. Observe that p(m(p))≥1/n. Thus, E_ps_p≤1−1/n. Finally, if p is inconsistent, let s_p be 1−1/n everywhere. Then s is a proper score.

For consistent forecasts, our s is a best guess score: a forecast's maximum point (with whatever tie breaker one likes) counts as the forecast's "best guess", and we get the perfect score 0 if we guessed right, and we get 1 otherwise. And for inconsistent forecasts, I just assigned a value that makes the score proper and, well, that makes what I am about to say true.

Namely: the above score s does not have the domination property that I talked about earlier. Let q be any inconsistent forecast. Then s_q is 1−1/n everywhere. If p is any consistent forecast, however, then s_p is 1 at all but one point, and so s_p does not dominate s_q from below.

Now, our score s is not a strictly proper score (Predd et al. actually work with strictly proper scores): for a strictly proper score s, whenever q differs from p and p is consistent, we will have E_ps_p<E_ps_q. But we can make our score strictly proper. Fix a small constant c. Then s+cb, where b is the standard Brier score, will be strictly proper. But if c is small enough, s+cb will also fail to have the domination property.

We should already have been suspicious of the argument for consistency based on proper scores and domination when proper scores were defined: the definition treated consistent forecasts in a special way (i.e., E_ps_p≤E_ps_q was only required when p is consistent—of course, it's hard to define E_p when p is inconsistent, so there is some excuse). But now we have even more reason to be suspicious: it is only some proper scores that have the property that scores of inconsistent forecasts are dominated by scores of consistent ones. Now, if we had some philosophical reason to think that the right way to score forecasts is by adding up scores for individual events, this would be better. But I don't know of such a philosophical reason.

TeXlipse

I've been editing LaTeX files using an old version of WinEdt. But at least in the old version I was using, I was having a terrible time ensuring things like matching \begin{...} and \end{...} for frame and itemize blocks when editing Beamer presentations. But I think I finally found a better way to handle LaTeX: the TeXlipse plugin for Eclipse. I can now have background building (at least when I save, and I press ctrl-s quite often instinctually), syntax highlighting and indenting, autocomplete, a handy hierarchical view, and very nice handling of error messages.

The downsides are that you need Eclipse (but I have it already installed for Android software development, and it is free after all) and Eclipse is bloated and doesn't start fast, setting up a new project takes a few more clicks than before, and the PDF viewer that comes with TeXlipse doesn't show all the graphical elements in a Beamer file. The last is a nuisance, but the nice way that the PDF file is linked with the LaTeX source compensates for it, as does the fact that I can just set up SumatraPDF as a secondary PDF viewer, and SumatraPDF (unlike Acrobat) will automatically reload the pdf file when it is regenerated. All in all, it seems worthwhile.

Proper scoring rules and gambling

There are two ways of evaluating a credence assignment. There is the decision-theoretic method: you consider how you are going to do given this credence assignment when presented with some gambles. And there is the scoring rule method: you consider how far you are from "the truth", i.e., the credence assignment that assigns 0 to the falsehoods and 1 to the truths, and you measure this with respect to a proper scoring rule.

There are various parallel results for the two methods.

It turns out that there is a good reason why there are parallel results. The two methods are equivalent. Assume an underlying probability space Ω. To avoid measurability issues, suppose Ω is finite. Denote a scoring rule by a function s(p,q) where p is a consistent credence assignment for some family of sentences and q is a consistent extreme credence assignment (0/1 valued) for the same family. By "the truth", I mean the extreme credence assignment that assigns 1 to each true sentence and 0 to each false one. A scoring rule is proper provided that E_p(−s(p',T))≤E_p(−s(p,T)) where T is the random variable that assigns to each point ω of Ω a function T(ω) that in turn assigns to each sentence its truth value at ω (i.e., 1 if true, 0 if false), and where E_p is expectation with respect to the credence assignment p.

Theorem. For any proper scoring rule s(p,q), there exists a family F of gambles such that for any consistent credence assignment there is a gamble that maximizes the expected payoff, and when you choose that maximizing gamble your payoff will be −s(p,T). Conversely, suppose that F is a family of gambles such that for any credence assignment there is a gamble that maximizes the expected payoff. Let V(p,q) be the payoff of such a gamble for credence assignment p when q is the truth. Then −V(p,q) is a proper scoring rule.

The proof is actually very simple. (I had very complicated proofs of special cases of this Theorem in the past, but now I see it is all very simple, even trivial.) For the left-to-right direction, for any possible credence assignment p, define the gamble G_p as follows: at ω, you get paid −s(p,T(ω)). Then the propriety of the scoring rule guarantees that G_p maximizes the expected payoff when p is your credence assigment. Conversely, let s(p,q)=−V(p,q). Propriety is easy to check—it just follows from maximization.

Explaining the contingent via the necessary

It has been claimed that contingent truths cannot be explained by necessary ones. Indeed, Peter van Inwagen has contended that this shows that the Principle of Sufficient Reason is false. But it seems that here is a case of a necessary truth explaining a contingent one: That it's extremely unlikely that 30 fair die throws would be all sixes explains why nobody has tossed 30 sixes in a row with a fair die.

Pantheism and omnipresence

If a view falls short with respect to the main doctrine it's organized around, that view is seriously flawed. For instance, if Calvinism fell short with regard to sovereignty, it would be seriously flawed. For pantheism, the relevant doctrine is omnipresence. On its face, pantheism is designed to make omnipresence work out perfectly: if God is everything, then he is where anything is.

But is that enough for omnipresence? First, perhaps omnipresence should also imply that God is in the places where nothing other than God exists—in otherwise empty space. Whether pantheism can account for that perhaps depends on whether it's deflationary (God is nothing but everything) or inflationary (everything is God, in addition to what it ordinarily is, and there may be more to God than ordinary things—and hence in particular God might be where there is nothing ordinary). That said, perhaps this is not so serious. If substantivalism about space is false, then maybe there are no empty places, except in a manner of speaking.

More seriously, by making God be everything, God comes to be only partly present everywhere. Only a part of God is in this room where I am—a very small part and, at least on the deflationary variant, a very insignificant part. Yes, God is in the stone and the butterfly and the galaxy—but all of these are very small bits of God. Classical theists, however, have the doctrine of divine simplicity and so we can say that where God is, all of God is.