Tuesday, January 31, 2023

Scoring rules and publication thresholds

One of the most problematic aspects of some science practice is a cut-off, say at 95%, for the evidence-based confidence needed for publication.

I just realized, with the help of a mention of p-based biases and improper scoring rules somewhere on the web, that what is going on here is precisely a problem of a reward structure that does not result in a proper scoring rule, where a proper scoring rule is one where your current probability assignment is guaranteed to have an optimal expected score according to that very probability assignment. Given an improper scoring rule, one has a perverse incentive to change one’s probabilities without evidence.

To a first approximation, the problem is really, really bad. Insofar as publication is the relevant reward, it is a reward independent of the truth of the matter! In other words, the scoring rule has a reward for gaining probability 0.95 (say) in the hypothesis, regardless of whether the hypothesis is true or false.

Fortunately, it’s not quite so bad. Publication is the short-term reward. But there are long-term rewards and punishments. If one publishes, and later it turns out that one was right, one may get significant social recognition as the discoverer of the truth of the hypothesis. And if one publishes, and later it turns out one is wrong, one gets some negative reputation.

However, notice this. Fame for having been right is basically independent of the exact probability of the hypothesis one established in the original paper. As long as the probability was sufficient for publication, one is rewarded for fame. Thus if it turns out that one was right, one’s long-term reward is fame if and only if one’s probability met the threshold for publication and one was right. And one’s penalty is some negative reputation if and only if one’s probability met the threshold for publication and yet one was wrong. But note that scientists are actually extremely forgiving of people putting forward evidenced hypotheses that turn out to be false. Unlike in history, where some people live on in infamy, scientists who turn out to be wrong do not suffer infamy. At worst, some condescension. And it barely varies with your level of confidence.

The long-term reward structure is approximately this:

  • If your probability is insufficient for publication, nothing.

  • If your probability meets the threshold for publication and you’re right, big positive.

  • If your probability meets the threshold for publication and you’re wrong, at worst small negative.

This is not a proper scoring rule. It’s not even close. To make it into a proper scoring rule, the penalty for being wrong at the threshold would need to be way higher than the reward for being right. Specifically, if the threshold is p (say 0.95), then the ratio of reward to penalty needs to be (1−p) : p. If p = 0.95, the reward to penalty ratio would need to be 1:19. If p = 0.99, it would need to be a staggering 1:99, and if p = 0.9, it would need to be a still large 1:9. We are very, very far from that. And when we add the truth-independent reward for publication, things become even worse.

We can see that something is problematic if we think about cases like this. Suppose your current level of confidence is just slightly above the threshold, and a graduate student in your lab proposes to do one last experiment in her spare time, using equipment and supplies that would otherwise go to waste. Given the reward structure, it will likely make sense for you to refuse this free offer of additional information. If the experiment favors your hypothesis, you get nothing out of it—you could have published without it, and you’d still have the same longer term rewards available. But if the experiment disfavors your hypothesis, it will likely make your paper unpublishable (since you were at the threshold), but since it’s just one experiment, it is unlikely to put you into the position of yet being able to publish a paper against the hypothesis. At best it loses you the risk of the small negative reputation for having been wrong, and since that’s a small penalty, and an unlikely one (since most likely your hypothesis is true by your data), so that’s not worth it. In other words, the the structure rewards you for ignoring free information.

How can we fix this? We simply cannot realistically fix it if we have a high probability threshold for publication. The only way to fix it while keeping a high probability threshold would be by having a ridiculously high penalty for being wrong. But we should’t do stuff like sentencing scientists to jail for being wrong (which has happened). Increasing the probability threshold for publication would only require the penalty for being wrong to be increased. Decreasing probability thresholds for publication helps a little. But as long as there is a larger reputational benefit from getting things right than the reputational harm from getting things wrong, we are going to have perverse incentives from a probability threshold for publication bigger than 1/2, no matter where that threshold lies. (This follows from Fact 2 in my recent post, together with the observation that Schervish’s characterization of scoring rules shows implies that any reward function corresponds to a unique up to additive constant penalty function.)

What’s the solution? Maybe it’s this: reward people for publishing lots of data, rather than for the data showing anything interestingly, and do so sufficiently that it’s always worth publishing more data?

Monday, January 30, 2023

Epistemic goods

We think highly morally of teachers who put an enormous effort into getting their students to know and understand the material. Moreover, we think highly of these teachers regardless of whether they are in a discipline, like some branches of engineering, where the knowledge and understanding exists primarily for the sake of non-epistemic goods, as when they are in a discipline, like cosmology, where the knowledge and understanding is primarily aimed at epistemic goods.

The virtues and vices in disseminating epistemic goods are just as much moral virtues and vices as those in disseminating other goods, such as food, shelter, friendship, or play, and there need be little difference in kind. The person who is jealous of another’s knowledge has essentially the same kind of vice as the one who is jealous of another’s physical strength. The person generous with their time in teaching exhibits essentially the same virtue as the one generous with their time in feeding others.

There is, thus, no significant difference in kind between the pursuit of epistemic goods and the norms of the pursuit of other goods. We not infrequently have to weigh one against the other, and it is a mark of the virtuous person that they do this well.

But if this is all correct, then by parallel we should not make a significant distinction in kind between the pursuit of epistemic goods for oneself and the pursuit of non-epistemic goods for oneself. Hence, norms governing the pursuit of knowledge and understanding seem to be just a species of prudential norms.

Does this mean that epistemic norms are just a species of prudential norms?

I don’t think so. Consider that prudentially we also pursue goods of physical health. However, norms of physical health are not a species of prudential norms. It is the medical professional who is the expert on the norms of physical health, not the prudent person as such. Prudential norms apply to voluntary behavior as such, while the norms of physical health apply to the body’s state and function. We might say that norms of the voluntary pursuit of the fulfillment of the norms of physical health are prudential norms, but the norms of physical health themselves are not prudential norms. Similarly, the norms of the voluntary pursuit of the fulfillment of epistemic norms are prudential norms, but the epistemic norms themselves are no more prudential norms than the health norms are.

Saturday, January 28, 2023

Making single-proposition scoring rules

Accuracy scoring rules measure the value of your probability assignment’s closeness to the truth. A scoring rule for a single proposition p can be thought of as a pair of functions, T and F on the interval [0,1] where T(x) tells us the score for assigning x to p when p is true and F(x) tells us the score for assigning x to p when p is false. The scoring rule is proper provided that:

  • xT(x) + (1−x)F(x) ≥ xT(y) + (1−x)F(y)

for all x and y. If you assign probability x to p, then xT(y) + (1−x)F(y) measures your expected value of the score for someone who assigns y. The propriety condition thus says that by your lights there isn’t a better probability to assign. After all, if there were, wouldn’t you assign it?

I’ve been playing with how to construct proper scoring rules for a single proposition, and I found two nice ways that are probably in the literature but I haven’t seen explicitly. First, let F be any monotone (not necessarily strictly) decreasing function on [0,1] that is finite except perhaps at 1. Then let:

  • TF(x) = F(1/2) − ((1−x)/x)F(x) − ∫1/2xu−2F(u)du.

I think we then have the following:

Fact 1: The pair (TF,F) is a proper scoring rule.

Second, let T be any monotone increasing function on [0,1] that is finite except perhaps at 0. Let:

  • FT(x) = T(1/2) − (x/(1−x))T(x) + ∫1/2x(1/(1−u)2)T(u)du.

I think then we have the following:

Fact 2: The pair (T,FT) a proper scoring rule.

In other words, to generate a proper scoring rule, we just need to choose one of the two functions making up the scoring rule, make sure it is monotone in the right direction, and then we can generate the other function.

Here’s a sketch of the proof of Fact 1. Note first that if F = c is constant, then TF(x) = c − ((1−x)/x)c + c(x−1−(1/2)−1) = c + c − 2c = 0 for all x. Since the map F ↦ TF is linear, it follows that if F and H differ by a constant, then TF and TH are the same. Thus subtracting a constant from F, we can assume without loss of generality that F is non-positive.

We can then approximate F by functions of the form ici1[ai,1] with ci non-positive (here I have to confess to not having checked all the details of the approximation) and by linearity we only need to check propriety for F =  − 1[a,0]. If a = 0, then F is constant and TF will be zero, and we will trivially have propriety. So suppose a > 0. Let T(x) =  − ((1−x)/x)F(x) − ∫0xu−2F(u)du. This differs by a constant from TF, so (TF,F) will be proper if and only if (T,F) is. Note that T(x) = 0 for x < a and for x ≥ a we have:

  • T(x) = ((1−x)/x) + ∫axu−2du = ((1−x)/x) − (x−1a−1) = a−1 − 1.

Thus, T = ((1−a)/a) ⋅ 1[a,0]. Now let’s check if we have the propriety condition:

  1. xT(y) + (1−x)F(y) ≤ xT(x) + (1−x)F(x).

Suppose first that x ≥ a. Then the right-hand-side is x(1−a)/a − (1−x). This is non-negative for x ≥ a, and the left-hand-side of (1) is zero if y < a, so we are done if y < a. Since T and F are constant on [a,1], the two sides of (1) are equal for y ≥ a.

Now suppose that x < a. Then the right-hand-side is zero. And the left-hand-side is zero unless y ≥ a. So suppose y ≥ a. Since T and F are constant on [a,1], we only need to check (1) at y = 1. At y = 1, the left-hand-side of (1) is x(1−a)/a − (1−x) ≤ 0 if x < a.

Fact 2 follows from Fact 1 together with the observation that (T,F) is proper if and only if (F*,T*) is proper, where T * (x) = T(1−x) and F * (x) = F(1−x).

Thursday, January 26, 2023

A cure for some cases of TMI

Sometimes we know things we wish we didn’t. In some cases, without any brainwashing, forgetting or other irrational processes, there is a fairly reliable way to make that wish come true.

Suppose that a necessary condition for knowing is that my evidence yields a credence of 0.9900, and that I know p with evidence yielding a credence of 0.9910. Then here is how I can rid myself of the knowledge fairly reliably. I find someone completely trustworthy who would know for sure whether p is true, and I pay them to do the following:

  1. Toss three fair coins.

  2. Inform me whether the following conjunction is true: all coins landed heads and p is true.

Then at least 7/8 of the time, they will inform me that the conjunction is false. That’s a little bit of evidence against p. I do a Bayesian update on this evidence, and my posterior credence will be 0.9897, which is not enough for knowledge. Thus, with at least 7/8 reliability, I can lose my knowledge.

This method only works if my credence is slightly above what’s needed for knowledge. If what’s needed for knowledge is 0.990, then as soon as my credence rises to 0.995, there is no rational method with reliability better than 1/2 for making me lose the credence needed for knowledge (this follows from Proposition 1 here). So if you find yourself coming to know something that you don’t want to know, you should act fast, or you’ll have so much credence you will be beyond rational help. :-)

More seriously, we think of knowledge as something stable. But since evidence comes in degrees, there have got to be cases of knowledge that are quite unstable—cases where one “just barely knows”. It makes sense to think that if knowledge has some special value, these cases have rather less of it. Maybe it’s because knowledge comes in degrees, and these cases have less knowledge.

Or maybe we should just get rid of the concept of knowledge and theorize in terms of credence, justification and truth.

Wednesday, January 25, 2023

The special value of knowledge

Suppose there is a distinctive and significant value to knowledge. What I mean by that is that if two epistemic are very similar in terms of truth, the level and type of justification, the subject matter and its relevant to life, the degree of belief, etc., but one is knowledge and the other is not, then the one that is knowledge has a significantly higher value because it is knowledge.

Plausibly, then, if we imagine Alice has some evidence for a truth p that is insufficient for knowledge, and slowly and continuously her evidence for p mounts up, when the evidence has crossed the threshold needed for knowledge, the value of Alice’s state with respect to p will have suddenly and discontinuously increased.

This hypothesis initially seemed to me to have an unfortunate consequence. Suppose Alice has just barely exceeded the threshold for knowledge of p, and she is offered a cost-free piece of information that may turn out to slightly increase or slightly decrease her overall evidence with respect to p, where the decrease would be sufficient to lose her knowledge of p (since she has only “barely” exceeded the evidential threshold for knowledge). It seems that Alice should refuse to look at the information, since the benefit of a slight improvement in credence if the evidence is non-misleading is outweighed by the danger of a significant and discontinuous loss of value due to loss of knowledge.

But that’s not quite right. For from Alice’s point of view, because the threshold for knowledge is not 1, there is a real possibility that p is false. But it may be that just as there is a discontinuous gain in epistemic value when your (rational) credence becomes sufficient for knowledge of something that is in fact true, it may be that there is a discontinuous loss of epistemic value when your credence becomes sufficient for knowledge of something false. (Of course, you can’t know anything false, but you can have evidence-sufficient-for-knowledge with respect to something false.) This is not implausible, and given this, by looking at the information, by her lights Alice also has a chance of a significant gain in value due to losing the illusion of knowledge in something false.

If we think that it’s never rational for a rational agent to refuse free information, then the above argument can be made rigorous to establish that any discontinuous rise in the epistemic value of credence at the point at which knowledge of a truth is reached is exactly mirrored by a discontinuous fall in the epistemic value of a state of credence where seeming-knowledge of a falsehood is reached. Moreover, the rise and the fall must be in the ratio 1 − r : r where r is the knowledge threshold. Note that for knowledge, r is plausibly pretty large, around 0.95 at least, and so the ratio between the special value of knowledge of a truth and the special disvalue of evidence-sufficient-for-knowledge for a falsehood will need to be at most 1:19. This kind of a ratio seems intuitively implausible to me. It seems unlikely that the special disvalue of evidence-sufficient-for-knowledge of a falsehood is an order of magnitude greater than the special value of knowledge. This contributes to my scepticism that there is a special value of knowledge.

Can we rigorously model this kind of an epistemic value assignment? I think so. Consider the following discontinuous accuracy scoring rule s1(x,t), where x is a probability and t is a 0 or 1 truth value:

  • s1(x,t) = 0 if 1 − r ≤ x ≤ r

  • s1(x,t) = a if r < x and t = 1 or if x < 1 − r and t = 0

  • s1(x,t) =  − b if r < x and t = 0 or if x < 1 − r and t = 1.

Suppose that a and b are positive and a/b = (1−r)/r. Then if my scribbled notes are correct, it is straightforward but annoying to check that s1 is proper, and it has a discontinuous reward a for meeting threshold r with respect to a truth and a discontinuous penalty  − a for meeting threshold r with respect to a falsehood. To get a strictly proper scoring rule, just add to it any strictly proper continous accuracy scoring rule (e.g., Brier).

Tuesday, January 24, 2023

Thresholds and precision

In a recent post, I noted that it is possible to cook up a Bayesian setup where you don’t meet some threshold, say for belief or knowledge, with respect to some proposition, but you do meet the same threshold with respect to the claim that after you examine a piece of evidence, then you will meet the threshold. This is counterintuitive: it seems to imply that you can know that you will have enough evidence to know something even though you don’t yet. In a comment, Ian noted that one way out of this is to say that beliefs do not correspond to sharp credences. It then occurred to me that one could use the setup to probe the question of how sharp our credences are and what the thresholds for things like belief and knowledge are, perhaps complementarily to the considerations in this paper.

For suppose we have a credence threshold r and that our intuitions agree that we can’t have a situation where:

  1. we have transparency as to our credences,

  2. we don’t meet r with respect to some proposition p, but

  3. we meet r with respect to the proposition that we will meet the threshold with respect to p after we examine evidence E.

Let α > 0 be the “squishiness” of our credences. Let’s say that for one credence to be definitely bigger than another, their difference has to be at least α, and that to definitely meet (fail to meet) a threshold, we must be at least α above (below) it. We assume that our threshold r is definitely less than one: r + α ≤ 1.

We now want this constraint on r and α:

  1. We cannot have a case where (a), (b) and (c) definitely hold.

What does this tell us about r and α? We can actually figure this out. Consider a test for p that have no false negatives, but has a false positive rate of β. Let E be a positive test result. Our best bet to generating a counterexample to (a)–(c) will be if the priors for p are as close to r as possible while yet definitely below, i.e., if the priors for p are r − α. For making the priors be that makes (c) easier to definitely satisfy while keeping (b) definitely satisfied. Since there are no false negatives, the posterior for p will be:

  1. P(p|E) = P(p)/P(E) = (rα)/(rα+β(1−(rα))).

Let z = r − α + β(1−(rα)) = (1−β)(rα) + β. This is the prior probability of a positive test result. We will definitely meet r on a positive test result just in case we have (rα)/z = P(p|E) ≥ r + α, i.e., just in case

  1. z ≤ (rα)/(r+α).

(We definitely won’t meet r on a negative test result.) Thus to get (c) definitely true, we need (3) to hold as well as the probability of a positive test result to be at least r + α:

  1. z ≥ r + α.

Note that by appropriate choice of β, we can make z be anything between r − α and 1, and the right-hand-side of (3) is at least r − α since r + α ≤ 1. Thus we can make (c) definitely hold as long as the right-hand-side of (3) is bigger than or equal to the right-hand-side of (4), i.e., if and only if:

  1. (r+α)2 ≤ r − α

or, equivalently:

  1. α ≤ (1/2)((1+6r−3r2)1/2−1−r).

It’s in fact not hard to see that (6) is necessary and sufficient for the existence of a case where (a)–(c) definitely hold.

We thus have our joint constraint on the squishiness of our credences: bad things happen if our credences are so precise as to make (6) true with respect to a threshold r for which we don’t want (a)–(c) to definitely hold. The easiest scenario for making (a)–(c) definitely hold will be a binary test with no false negatives.

We thus have our joint constraint on the squishiness of our credences: bad things happen if our credences have a level of precision equal to the right-hand-side of (6). What exactly that says about α depends on where the relevant threshold lies. If the threshold r is 1/2, the squishiness α is 0.15. That’s surely higher than the actual squishiness of our credences. So if we are concerned merely with the threshold being more-likely-than-not, then we can’t avoid the paradox, because there will be cases where our credence is definitely below the threshold and it’s definitely above the threshold that examination of the evidence will push us about the threshold.

But what’s a reasonable threshold for belief? Maybe something like 0.9 or 0.95. At r = 0.9, the squishiness needed for paradox is α = 0.046. I suspect our credences are more precise than that. If we agree that the squishiness of our credences is less than 4.6%, then we have an argument that the threshold for belief is more than 0.9. On the other hand, at r = 0.95, the squishiness needed for paradox is 2.4%. At this point, it becomes more plausible that our credences lack that kind of precision, but it’s not clear. At r = 0.98, the squishiness needed for paradox dips below 1%. Depending on how precise we think our credences are, we get an argument that the threshold for belief is something like 0.95 or 0.98.

Here's a graph of the squishiness-for-paradox α against the threshold r:

Note that the squishiness of our credences likely varies with where the credences lie on the line from 0 to 1, i.e., varies with respect to the relevant threshold. For we can tell the difference between 0.999 and 1.000, but we probably can’t tell the difference between 0.700 and 0.701. So the squishiness should probably be counted relative to the threshold. Or perhaps it should be correlated to log-odds. But I need to get to looking at grad admissions files now.

Monday, January 23, 2023

Respecting conscience

One of the central insights of Western philosophy, beginning with Socrates, has been that few if any things are as bad for an individual as culpably doing wrong. It is better, we are told through much of the Western philosophical tradition, that it is better to suffer than do injustice.

Now, acting against one’s conscience is always wrong, and is almost always culpably wrong. For the most common case when doing something wrong isn’t culpable is that one is ignorant of the wrongness, but when one acts against one’s conscience one surely isn’t ignorant that one is acting against conscience, and that we ought follow our conscience is obvious.

That said, I think a qualification is plausible. Some wrongdoings are minor, and in those cases the harm to the wrongdoer may be minor as well. But in any case, to get someone to act against their conscience in a matter that according to their conscience is major is to do them grave harm, a harm not that different from death.

Now, the state, just like individuals, should ceteris paribus avoid causing grave harm. Hence, the state should generally avoid getting people to do things that violate their conscience in major matters.

The difficult case, however, is when people’s consciences are mistaken to such a degree that conscience requires them to do something that unjustly harms others. (A less problematic mistake is when conscience is mistaken to such a degree that conscience requires them to do something that’s permissible, but not wrong. In those cases, tolerance is clearly called for. We shouldn’t pressure vegetarians to eat animals even if their conscientious objection to eating animals happens to be mistaken.)

One might think that what I said earlier implies that in this difficult case the state should always allow people to follow their conscience, because after all it is worse to do wrong—and violating conscience is wrong—than to have wrong done to one. But that would be absurd and horrible—think of a racist murderer whose faulty conscience requires them to kill.

A number of considerations, however, keep one from reaching this absurd conclusion.

  1. The harm of violating one’s conscience only happens to one if one willingly violates one’s conscience. If law enforcement physically prevents me from doing something that conscience requires from me, then I haven’t suffered the harm. Thus, interestingly, the consideration I sketched against violating one’s conscience does not apply when one is literally forced (fear of punishment, unless it is severe enough to suspend one’s freedom of will, does not actually force, but only incentives).

  2. In cases where doing wrong and suffering wrong are of roughly the same order of magnitude, it is very intuitive that we should prevent the suffering of wrong rather than the doing of wrong. Imagine that Alice is drowning while at the same time Bob is getting ready to assassinate a politician, but we know for sure that Bob’s bullets have all been replaced with blanks. If our choice is whether to try to dissuade Bob from attempting murder or keep Alice from drowning, we should keep Alice from drowning, evne if on the Socratic view the harm to Bob from attempting murder will be greater than that to Alice from drowning. (I am assuming that in this case the two harms are nonetheless of something like the same order of magnitude.)

  3. A reasonable optimism says that in most cases most people’s consciences are correct. Thus typically we would expect that most violators of a legitimate law will not be acting out of conscience—for a necessarily condition for the legitimacy of a law is that it does not conflict with a correct conscience. Thus, even if there is the rare murderer acting from mistaken conscience, most murderers act against conscience, and by incentivizing abstention from murder, in most cases the law helps people follow their conscience, and the small number of other cases can be tolerated as a side effect. Thus the considerations of conscience favor intolerant laws in such cases. Nonetheless, there are cases where most violators of a law would likely be acting from conscience. Thus, if we had a law requiring eating meat, we would expect that most of the violators would be conscientious. Similarly, a law against something—say, the wearing of certain clothes or symbols—that is rarely done except as a religious practice would likely be a law most violators of which would be conscientious.

  4. When someone’s conscience mistakenly requires something that violates an objective moral rule, there is a two-fold benefit to that person from a law incentivizing following the moral rule. The law is a teacher, and the state’s disapproval may change one’s mind about the matter. And even if it a harm to one to violate conscience, it is also a harm to one to do something wrong even inculpably. Thus, the harm of violating conscience is somewhat offset by the benefit from not doing something else that is wrong.

  5. In some cases the person of mistaken conscience will still do the wrong deed despite the law’s contrary incentive. In such a case, both the perpetrator and the victim may be slightly better off for the law. The victim has a dignitary benefit from the very fact that the state says that the harm was unlawful. That dignitary benefit may be a cold comfort if the victim suffered a grave harm, but it is still a benefit. And the perpetrator is slightly better off, because following one’s conscience against external pressure has an element of admirability even when the conscience is mistaken.

Nonetheless, there will be cases where these considerations do not suffice, and the law should be tolerant of mistaken conscience.

In a just defensive war, to refuse to fight to defend one’s fellow citizens without special reason (perhaps priests and doctors should not kill) is wrong. But a grave harm is done to a conscientious objector who is gotten to fight by legal incentives. Let’s think through the five considerations above. The first mainly applies to laws prohibiting a behavior rather than ones requiring a behavior. Short of brainwashing, it is impossible to make someone fight. (We could superglue their hands to a gun, and then administer electric shocks causing their fingers to spasm and fire a bullet, but that wouldn’t count as fighting.) The second applies somewhat: we do need to weigh the harms to innocent citizens from enemy invaders, harms that might be prevented if our conscientious objector fought. But note that there is something rather speculative about these harms. Someone who fights contrary to conscience is unlikely to be a very effective fighter, and it is far from clear that their military activity would actually prevent any actual harm to innocents. Now, regarding the third consideration, one can design a conscription law with an exemption that few who aren’t conscientious objectors would take advantage of. One way to do this is to require evidence of one’s conscience’s objection to fighting (e.g., prior membership in a pacifist organization). Another way is to impose non-combat duties on conscientious objectors that are as onerous and maybe as dangerous as combat would be. Regarding the fourth consideration, it seems unlikely that a typical conscientious objector’s objections to war would be changed by legal penalties. And the fifth seems a weak consideration in general. Putting all these together, we do not outweigh the prima facie considerations against pressuring conscientious objectors to act against their (mistaken) conscience from the harms in going against conscience.

Saturday, January 21, 2023

Knowing you will soon have enough evidence to know

Suppose I am just the slightest bit short of the evidence needed for belief that I have some condition C. I consider taking a test for C that has a zero false negative rate and a middling false positive rate—neither close to zero nor close to one. On reasonable numerical interpretations of the previous two sentences:

  1. I have enough evidence to believe that the test would come out positive.

  2. If the test comes out positive, it will be another piece of evidence for the hypothesis that I have C, and it will push me over the edge to belief that I have C.

To see that (1) is true, note that the test is certain to come out positive if I have C and has a significant probability of coming out positive even if I don’t have C. Hence, the probability of a positive test result will be significantly higher than the probability that I have C. But I am just the slightest bit short of the evidence needed for belief that I have C, so the evidence that the test would be positive (let’s suppose a deterministic setting, so we have no worries about the sense of the subjunctive conditional here) is sufficient for belief.

To see that (2) is true, note that given that the false negative rate is zero, and the false positive rate is not close to one, I will indeed have non-negligible evidence for C if the test is positive.

If I am rational, my beliefs will follow the evidence. So if I am rational, in a situation like the above, I will take myself to have a way of bringing it about that I believe, and do so rationally, that I have C. Moreover, this way of bringing it about that I believe that I have C will itself be perfectly rational if the test is free. For of course it’s rational to accept free information. So I will be in a position where I am rationally able to bring it about that I rationally believe C, while not yet believing it.

In fact, the same thing can be said about knowledge, assuming there is knowledge in lottery situations. For suppose that I am just the slightest bit short of the evidence needed for knowledge that I have C. Then I can set up the story such that:

  1. I have enough evidence to know that the test would come out positive,


  1. If the test comes out positive, I will have enough evidence to know that I have C.

In other words, oddly enough, just prior to getting the test results I can reasonably say:

  1. I don’t yet have enough evidence to know that I have C, but I know that in a moment I will.

This sounds like:

  1. I don’t know that I have C but I know that I will know.

But (6) is absurd: if I know that I will know something, then I am in a position to know that the matter is so, since that I will know p entails that p is true (assuming that p doesn’t concern an open future). However, there is no similar absurdity in (5). I may know that I will have enough evidence to know C, but that’s not the same as knowing that I will know C or even be in a position to know C. For it is possible to have enough evidence to know something without being in a position to know it (namely, when the thing isn’t true or when one is Gettiered).

Still, there is something odd about (5). It’s a bit like the line:

  1. After we have impartially reviewed the evidence, we will execute him.

Appendix: Suppose the threshold for belief or knowledge is r, where r < 1. Suppose that the false-positive rate for the test is 1/2 and the false-negative rate is zero. If E is a positive test result, then P(C|E) = P(C)P(E|C)/P(E) = P(C)/P(E) = 2P(C)/(1+P(C)). It follows by a bit of algebra that if my prior P(C) is more than r/(2−r), then P(C|E) is above the threshold r. Since r < 1, we have r/(2−r) < r, and so the story (either in the belief or knowledge form) works for the non-empty range of priors strictly between r/(2−r) and r.

Friday, January 20, 2023

Partial and complete explanations

  1. Any explanation for an event E that does not go all the way back to something self-explanatory is merely partial.

  2. A partial explanation is one that is a part of a complete explanation.

  3. So, if any event E has an explanation, it has an explanation going all the way back to something self-explanatory. (1,2)

  4. Some event has an explanation.

  5. An explanation going back to something self-explanatory involves the activity of a necessary being.

  6. So, there is an active necessary being. (4,5)

I am not sure I buy (1). But it sounds kind of right to me now. Additionally, (3) kind of sounds correct on its own. If A causes B and B causes C but there is no explanation of A, then it seems that B and C are really unexplained. Aristotle notes that there was a presocratic philosopher who explained why the earth doesn’t fall down by saying that it floats on water, and he notes that the philosopher failed to ask the same question about the water. I think one lesson of Aristotle’s critique is that if it is unexplained why the water doesn’t fall down it is unexplained why the earth falls down.

Thursday, January 19, 2023

What am I "really"?

What am I? A herring-eater, a husband, a badminton player, a philosopher, a father, a Canadian, a six-footer and a human are all correct answers. There is an ancient form of the “What is x?” question where we are looking for a “central” answer, and of course “a human” is usually taken to be that one. What makes for that answer being central?

Sometimes the word “essential” is thrown in: I am essentially human. But what does that mean? In contemporary analytic jargon, it just means that I cannot exist without being human. But the “central” answer to “What am I?” is not just an answer that states a property that I cannot lack. There are, after all, many such properties essential in such a way that do not answer the question. “Someone conceived in 1972” and “Someone in a world where 2+2=4” attribute properties I cannot lack, but are not the central answers.

So the sense of the “What am I centrally, really, deep-down, essentially?” question isn’t just modal. What is it?

Here is a start.

  1. Necessarily, I am good and a human if and only if I am a good human.

But the same is not true for any other attribute besides “human” among those of the first paragraph. I can be good and a badminton player while not being a good badminton player, and I can be a good herring-eater without being good and a herring-eater. And I have no idea what it is to be a good six-footer, but perhaps the fact that I am a quarter of an inch short of six feet right now makes me not be one. (In some cases one direction may hold. It may be that if I am good and a human, then I am a good human.)

So our initial account of what is being asked for is that we are asking for an attribute F such that:

  1. Necessarily, x is a good F if and only if x is good.

But that’s not quite right. For:

  1. Necessarily, I am good and a virtuous human if and only if I am a good virtuous human.

And yet “a virtuous human” would not be the answer to the ancient “What am I centrally, really, deep-down, essentially?” question even if I were in fact a virtuous human.

But perhaps we can do better. The necessary biconditional (2) holds in the case “virtuous human”, but in a kind of trivial way: “a good virtuous human” is repetition. I think that, as often, we need to pass from a modal to a hyperintensional characterization. Consider that not only is (1) true, but also:

  1. Necessarily, if I am human, what it is for me to be good is for me to be a good human.

In other words, if I am a human, being a good human explains my being good. On the other hand, even if I were a virtuous human, my being a good virtuous human would not explain my being good. For redundancy is to be avoided in explanation, and “good virtuous human” is redundant.

Thus, I propose that:

  1. x is “centrally, really, deep-down, essentially” F just in case what it is for x to be good is for x to be a good F.

In other words, that which I am centrally, really, deep-down and essentially is that which sets the norms for me to be good simpliciter.

Objection 1: Some things are “centrally” electrons, but something’s being good at electronicity is absurd.

Response: I deny that it’s absurd. It’s just that all the electrons we meet are good at electronicity.

Objection 2: “Good” is attributive, and hence there is no such thing as being good simpliciter.

Response: “Good” is attributive in the sense that the schema

  1. x is good and x is F if and only if x is a good F

is not generally logically valid. But some instances of a schema can be logically valid even if the schema is not logically valid in general.

Wednesday, January 18, 2023

Probability, belief and open theism

Here are few plausible theses:

  1. A rational being believes anything that they take to have probability bigger than 1 − (1/10100) given their evidence.

  2. Necessarily, God is rational.

  3. Necessarily, none of God’s beliefs ever turn out false.

These three theses, together with some auxiliary assumptions, yield a serious problem for open theism.

Consider worlds created by God that contain four hundred people, each of whom has an independent 1/2 chance of freely choosing to eat an orange tomorrow (they love their oranges). Let p be the proposition that at least one of these 400 people will freely choose to eat an orange tomorrow. The chance of not-p in any such world will be (1/2)400 < 1/10100. Assuming open theism, so God doesn’t just directly know whether p is true or not, God will take the probability of p in any such world to be bigger than 1 − (1/10100) and by (1) God will believe p in these worlds. But in some of these worlds, that belief will turn out to be false—no one will freely eat the orange. And this violates (3).

I suppose the best way out is for the open theist to deny (1).

Tuesday, January 17, 2023


As far as I know, in all racquet sports players are told to follow-through: to continue the racquet swing after the ball or shuttle have left the racquet. But of course the ball or shuttle doesn’t care what the racquet is doing at that point. So what’s the point of follow-through? The usual story is this: by aiming to follow-through, one hits the ball or shuttle better. If one weren’t trying to follow-through, the swing’s direction would be wrong or the swing might slow down.

This is interesting action-theoretically. The follow-through appears pointless, because the agent’s interest is in what happens before the follow-through, the impact’s having the right physical properties, and yet there is surely no backwards causation here. But there not appear to be an effective way to reliably secure these physical properties of the impact except by trying for the follow-through. So the follow-through itself is pointless, but one’s aiming at or trying for the follow-through very much has a point. And here the order of causality is respected: one swings aiming at the follow-through, which causes an impact with the right physical properties, and the swing then continues on to the “pointless” follow-through.

Clearly the follow-through is intended—it’s consciously planned, aimed at, etc. But it need not be a means to anything one cares about in the game (though, of course, in some cases it can be a means to impressing the spectators or intimidating an opponent). But is it an end? It seems pointless as an end!

Yet it seems that whatever is intended is intended as a means or an end. One might reject this principle, taking follow-through to be a counterexample.

Another move is this. We actually have a normative power to make something be an end. And then it becomes genuinely worth pursuing, because we have adopted it as an end. So the player first exercises the normative power to make follow-through be an end, and then pursues that end as an end.

But there is a problem here. For even if there is a “success value” in accomplishing a self-set goal, the strength of the reasons for pursuing the follow-through is also proportioned to facts independent of this exercise of normative power. Rather, the reasons for pursuing the follow-through will include the internal and external goods of victory (winning as such, prizes, adulation, etc.), and these are independent of one’s setting follow-through as one’s goal.

Maybe we should say this. Even if all intentional action is end-directed, there are two kinds of reasons for an action: the reasons that come from the value of the end and the reasons that come from the value of the pursuit of that end. In the case of follow-through, there may be a fairly trivial success value in the follow-through—a success value that comes from one’s exercise of normative power in adopting the follow-through as one’s end—but that success value provides only fairly trivial reasons. However, there can be significantly non-trivial reasons for one’s pursuing that end, reasons independent of that end.

Anthropomorphism about God

Consider an anthropomorphic picture of God that some non-classical theists have:

  1. God is not simple, and in particular God’s beliefs are proper parts of God.

  2. God’s beliefs change as the reality they are about changes.

Putting these together, it follows that:

  1. I can bring about the destruction of a part of God.

How? Easy. I am now sitting, and God knows that. So, a part of God is the belief that I am sitting. But I can destroy that belief of God’s by standing up! For as soon as I stand up, the belief that I am sitting will no longer exist. But on the view in question, God’s beliefs are parts of him. So by standing up, I would bring it about that a part of God doesn’t exist.

But (3) is as absurd as can be.

And of course by standing up, I bring it about that a new divine belief exists. So:

  1. I can bring about the genesis of a part of God.

Which is really absurd, too.

Friday, January 13, 2023

Trivial and horrendous evils

Nobody seriously runs an argument against the existence of God from trivial evils, like hangnails or mosquito bites.

Why not? Here is a hypothesis. There are so very many possible much greater goods—goods qualitatively and not just quantitatively much greter—that it would be easy to suppose that God’s permitting the trivial evil could promote or enhance one of these goods to a degree sufficient to yield justification.

On the other hand, if we think of horrendous evils, like the torture of children, it is difficult to think of much greater goods. Maybe with difficulty we can come up with one or two possibilities, but not enough to make it easy to suppose a justification for God’s permission of the evil in terms of the goods.

However, if God exists, we would expect there to be an unbounded upward qualitative hierarchy of possible finite goods. God is infinitely good, and finite goods are participations in God, so we would expect a hierarchy of qualitatively greater and greater types of good that reflect God’s infinite goodness better and better.

So if God exists, we would expect there to be unknown possible finite goods that are related to the known horrendous evils in something like the proportion in which the known great finite goods are related to the known trivial evils. Thus, if God exists, there very likely is
an upward hierarchy of possible goods to which the horrendous evils of this life stand like a mosquito bite to the courage of a Socrates. If we believe in this hierarchy of goods, then it seems we should be no more impressed by the atheological evidential force of horrendous evils than the ordinary person is by the atheological evidential force of trivial evils.

There is, however, a difference between the cases. Many great ordinary goods that dwarf trivial evils, like the courage of a Socrates, are known to us. Few if any finite goods that dwarf horrendous evils are known to us. Nonetheless, if theism is true, it is very likely that such goods are possible. And since the argument from evil is addressed against the theist, it seems fair for the theist to invoke that hierarchy.

Moreover, we might ask whether our ignorance of goods higher up in the hierarchy of goods beyond the ordinary goods is not itself evidence against the existence of such goods. Here, I think the answer is that it is very little evidence. We would expect any particular finite being to be able to recognize only a finite number of types of good, and thus the fact that there are only a finite number of goods that we recognize is very little evidence against the hypothesis of the upward hierarchy of goods.

Humility and the existence of God

Here’s a valid argument:

  1. Humility is an appropriate attitude for everyone.

  2. For highly accomplished individuals, humility is only an appropriate attitude if God exists.

  3. Some individuals are highly accomplished.

  4. So, God exists.

Premise (1) is controversial. The ancient Greeks would have denied it. But I think the reason they denied it is that they didn’t have the examples that the Christian tradition does, highly attractive examples examples of accomplished lives of great humility.

Here’s the thought behind premise (2). If there is no God, then highly accomplished individuals have much to brag about. Many of their accomplishments are primarily theirs.

Premise (3) is obvious.

Here’s another way to think about it. Given (1) and (3), we need an explanation of how it is that humility is an appropriate attitude for a highly accomplished individual. Classical theism’s doctrine of participation provides such an explanation: all the efforts and all the accomplishments are not truly theirs but a participation in God’s perfection.

Monday, January 9, 2023

Means inappropriate to ends

Consider this thesis:

  1. You should adopt means appropriate to your ends.

The “should” here is that of instrumental rationality.

I am inclined to think (1) is false if by “end” is meant the end the agent actually adopts, as opposed to a natural end of the agent. If your ends are sufficiently irrational, adopting means appropriate to them may be less rational than adopting means inappropriate to them.

Suppose your end is irrationality. Is it really true that you should adopt the means to that, such as reasoning badly? Surely not! Instead, you should reject the end.

Instead of (1), what is likely true is:

  1. You should be such that you adopt means appropriate to your ends.

But what is wrong with being such that you adopt means inappropriate to your ends is not necessarily the means—it could be the ends.

Unjust laws have no normative force, and stupid ends have no normative force, either.

Three non-philosophy projects

Here are some non-philosophy hobby projects I’ve been doing over the break:

  • Measuring exercise bike power output

  • Dumping NES ROMs

  • Adapting Dance Dance Revolution and other mat controllers to work as NES Power Pad controllers for emulation.