Determining camera pose from three landmarks and accelerometer: initial notes

Suppose we have three landmarks at known locations in three dimensional space, not all on one line, and a camera that is not located at any landmark but sees all three landmarks, with known optical parameters such as focal length. The problem of determining the camera pose—the camera position and angles—from the positions of the landmarks in the camera image is known as the P3P problem. It is known that in general there will be at most four solutions for camera pose. And indeed sometimes there will be four solutions.

What if we have some additional information, namely we know how the camera is oriented with respect to gravity (e.g., because the camera is held horizontally or it’s equipped with an accelerometer)? Call the problem of reconstructing the camera image from n landmarks and gravity data PnPA. I recently showed that with just two landmarks, i.e., P2PA, there will be either one, two or infinitely many solutions, and geometrically characterized exactly which case occurs when.

Question: What can we say about the number of solutions to P3PA?

In this post I will make some slight progress on this question.

First we characterize when there are more than two solutions. Note that once we know the camera position, we can calculate the direction it’s pointing from the camera image (Lemma 4 in my paper). So we only need to look at the number of solutions for camera position.

Fact 1: There are at most two solutions for P3PA, except in the case where the three landmarks and camera all lie on one horizontal circle, in which case there are infinitely many solutions.

Proof: For P2PA, we have more than two solutions in precisely the following cases: (a) the two landmarks are on a single vertical line; (b) the two landmarks are in the same horizontal plane and so is the camera; and (c) the two landmarks and the camera are all on one line. To have more that two solutions for P3PA, each pair of landmarks must satisfy at least one of (a)–(c). Suppose this is so.

Suppose first that two landmarks, say m₁ and m₂, satisfy (a). Next suppose that no two landmarks lie on the same horizontal plane, so (b) is satisfied for no pair of landmarks. Then the third landmark m₃ does not lie on the same line as both m₁ and m₂, and hence neither the pair m₁ and m₃ nor th epair m₂ and m₃ satisfies (a), and at least one of these pairs fails to satisfy (c). Hence we have a pair that fails to satisfy any of (a)–(c), and we have at most two solutions by my P2PA result.

Now, continuing to suppose m₁ and m₂ satisfy (a), suppose that some pair of landmarks lies on the same horizontal plane. It can’t be m₁ and m₂ (as then they will be at the same point, and hence all three landmarks will be on one line). Without loss of generality, suppose m₁ and m₃ lie on the same horizontal plane H. The pair m₁ and m₃ cannot satisfy (a) (or else m₃ is at the same point as m₁). If it satisfies either (b) or (c), the camera is on the plane P, and hence in any case we have (b).

Furthermore, if no two landmarks satisfy (a), then since it can’t be that every pair of landmarks satisfies (c) as that would put all the landmarks on one line, at least one pair of landmarks must satisfy (b).

We thus have reduced to the case where a pair of landmarks satisfies (b): they are on the same horizontal plane H as the camera C. Let’s say that these landmarks are m₁ and m₂. Let m′₃ be the projection of m₃ to this plane. From the camera’s optical parameters and the camera image, we can calculate the angles m₁Cm₂, m₁Cm′₃ and m₂Cm′₃. It is known that the locus of points in a plane that subtend the same angle to two fixed points is an arc through these points. Thus, if we are to have more than two solutions, the arcs respectively through m₁Cm₂, m₁Cm′₃ and m₂Cm′₃ must intersect in at least three points. This would require m₁, m₂, m′₃ and C to all lie on the same circle T.

Now, suppose m₃ lies off the plane H. Then given the camera image and the gravity vector, we can measure the angle between m₃, the camera and the plane H, and given the position of m₃ we can compute the distance from m′₃ to the camera. This constrains the camera to lie on the circle T as well as on a second circle T′ around m′₃. Since m′₃ lies on T, these two circles intersect in at most two points. Thus, we have at most two solutions.

On the other hand, when m₃ is in the same plane, so m₁, m₂, m₃ and the camera lie on the same horizontal circle, we will have infinitely many solutions. For if A and B are two fixed points on a circle, and C is a third point on the same side as A and B, the angle ACB will be constant regardless of the choice of C. The landmarks m₁, m₂ and m₃ split T into three arcs, and the camera could be anywhere in the arc it’s in as far as the image goes.

Fact 2: There are cases where there are exactly two solutions.

Proof by picture: Suppose m₁ and m₂ lie on the same plane and m₃ lies off the plane. Suppose the camera is horizontally oriented, and pointed at m₁. Wherever the camera is on the blue arc, it sees m₁ and m₂ the same way (because the angle indicated by the dotted lines does not change, as discussed above). The image of m₃ in the camera is always directly above or below the image of m₁, and as long as the distance from camera to m₁ is the same, the image of m₃ does not move. Thus, the image of m₃ does not change as the camera moves on the red circle. Hence, at the two points where the red and blue circles intersect, the camera sees the same thing, and hence we have two solutions.

Against Arianism

1. The Christian’s deepest desire is for direct union with God.

2. Our union with the Father must always be mediated by the Son.

3. So, if only the Father is God, then the Christian’s deepest desire cannot be fulfilled.

4. The Christian’s deepest desire can be fulfilled.

5. So, it is false that only the Father is God.

An Aristotelian argument for Causal Finitism

The Aristotelian account of individuation for beings in the same species proceeds by matter. But to fill out this account, we may need to talk about the times at which the matter is different.

One reason for talking about this is the possibility on some versions of Aristotelianism that a chunk of matter can survive substantial change, and hence can compose different substances—albeit at different times. If one wolf kills and eats another, there is in principle no reason why eventually the cannibal wolf’s matter might not entirely be the same as the matter of the victim, then.

So, perhaps, we may want to say that what distinguishes members x and y of the same material species is that they have different matter at the same time. But that’s not quite right. For it could be that x and y don’t ever exist at the same time! We might thus want to say that what grounds the distinction between x and y is:

1. at some time t, either x exists and y doesn’t or x doesn’t exist and y does or x and y both exist and have different matter.

This has at least two problems. First, if I travel backwards in time to a time when I already existed, then I will have different matter from my past self at the same time (since I’ve changed some cells in the meanwhile).

But the second is probably the more serious. According to the story in (1), past, present and future differences in matter all ground the distinction between x and y. But it is implausible that future differences in matter be relevant to my present distinction from you. So we should minimally restrict (1) to talking of past and present times t. But if we do that, then what grounds our difference changes over time, and does so pointlessly, because as soon as our distinction has once been established, that’s surely all we need. In fact, the last point is particularly important given the soul’s survival past death: what makes your soul past death different from mine is presumably the past difference in matter. This suggests that if we want to get at what truly grounds our distinction, we have to go back in time as far as we can, to the very beginnings of our existence, and say:

2. either x and y differ in the time at which they began to exist, or x and y began at the same time with different matter.

There is another reason to prefer (2) to (1). We not only want an account of the distinction of substances within a world, but between possible worlds. But there are many possible worlds where I exist but where my current matter is different from the matter I now have in the actual world (namely, because my diet was different). Thus, (1) has no hope of being extended to an account of transworld distinction. But (2) has some hope of such an extension. At the very least, it gives a sufficient condition for distinctness in the transworld case, while (1) does not even do that.

But now here is an interesting thing. If (2) is the true story about the distinction between material substances of the same species, it follows that:

3. It is impossible for there to be two material substances of the same species that have no beginning in time.

But if Causal Finitism (the doctrine that one effect cannot have infinitely many causes) is false, it seems like it should be possible to have an organism that has lived for an infinite amount of time, and likewise possible to have two such organisms in one species. So (2) gives us a new reason to accept Causal Finitism.

However, personally, I don’t like the Aristotelian account of distinctions between substances.

A theological argument against Causal Finitism

Some people are attracted to this idea:

1. God gives infinitely many chances for everyone to be saved.

At the same time, by and large (with the main exception being Origen) the Christian tradition accepts:

2. Whether you are saved or not is determined at the time of death.

It’s interesting to note that if you deny Causal Finitism, you can have both (1) and (2). You can suppose, for instance, that God miraculously sets up a supertask for you in the millisecond prior to death, miraculously accelerating the functioning of your soul in such a way that in the first half millisecond you have one choice whether to be saved or not, in the next quarter millisecond you have another choice, in the next eighth you have another, and so on. And then you’re saved if any of these infinitely many choices was made for God, and damned only if all them were against God.

While I buy Causal Finitism, and hence probably have to deny the conjunction of (1) and (2) since on (1) and (2) your state post-death appears to be determined by infinitely many causes (the choices), I think this is a somewhat attractive reason to deny Causal Finitism.

A tweak to the Aristotelian account of individuation

The standard Aristotelian story about individuation of finite substances is two-fold:

1. What makes x and y distinct when x and y are members of different species is their difference in form

2. What makes x and y distinct when x and y are members of the same species is their difference in matter.

But I think a slight tweak is in order. Suppose Alice and Spot are both members of different species and have a difference of matter. By the standard story, what accounts for their distinction is just their having different form. But it seems right to say that their difference in matter is also sufficient to distinguish them. The distinction between Alice and Spot is overdetermined by a difference in species and a difference in matter.

If so, then intead of (1) and (2), we should have an inclusively disjunctive account:

3. What makes x and y distinct is that they have a difference in form or have a difference in matter or both.

To make this account work even better, I will stipulate that the difference between an immaterial substance (say, an angel) and a material substance (say, a human) also counts as a difference in matter. Thus, there are two sorts of differences of matter: two things having different matter, or one thing having matter and the other not.

More on infinite chances at salvation

Yesterday, I offered a Toy Model of how God could give infinite chances to someone and they could still reject him. The crucial assumption of the model was that it might take multiple good choices to dig oneself out of the moral hole that one’s wickedness got one into.

But one can also have a model without that assumption. Suppose that every day I make a choice for or against God. If I choose for God, I am saved and God makes me virtuous. If I choose against God, I become habituated into a bit more vice. How much more? Well, we can measure a person’s virtue by how likely they are to choose for God. So let’s suppose that there is some number β strictly between 0 and 1 such that each time I choose against God, my goodness—as measured by the probability of my next choice being for God—is multiplied by β. We can think of β as a kind of wicked habituation constant.

What is the probability that I will always reject God? Suppose my initial level of goodness is α. Then the probability that I will reject God on day 1 is 1 − α. If I do that, then on day 2 my level of goodness will be αβ, and the probability that I will reject God will be 1 − αβ. And if I do that, then on day 3 my level of goodness will be αβ², and the probability that I will reject God will be 1 − αβ². So the probability that I will always reject God is:

• P(R_∞) = (1−α)(1−βα)(1−β²α)....

Easy mathematical fact: as long as α < 1 (i.e., I am not maximally good on day 1) and β < 1 (the bad choice does decrease my goodness), P(R_∞) is bigger than zero. In other words, on this model, I have infinite chances, and each choice has the possibility of resulting in my complete redemption, and yet I have a non-zero probability of choosing badly for eternity.

Let’s make up some conservative numbers. Suppose the infinite chances regimen starts with death, and that β = 0.99: one gets 1% worse each time one chooses against God—and that’s 1% of my previous level of goodness, rather than an absolute 1%. Presumably people die with various levels of goodness between 0 and 1. Suppose one in a thousand people dies with a goodness level α ≤ 0.1. Now if one has goodness level α = 0.1 when one starts the infinite chances regimen at death, and β = 0.99, it turns out (or so Mathematica calculates QPochhammer[.1,.99]) that the probability of damnation is about 0.000035. So with 8 billion people, there would be 8 million dying with goodness level at most 0.1, and so we might well expect at least 8, 000, 000 × 0.000035 = 280 of them to be damned even if they got the infinite chances regimen. So on this toy model, we would expect some people now alive to be damned.

Now it seems plausible that it is compatible with God’s goodness that he would allow one in a thousand people to die with a goodness level of 0.1, and that he would allow a character deterioration of at least 1% per posthumous choice against God—this would give people a real ability to affect their character, while yet being very generous with redemptiveness. In other words, even on an infinite chances model, it is compatible with God’s goodness that hundreds of people be forever separated from God. But if hundreds of people damned is compatible with God’s goodness on an infinite chances model, why wouldn’t hundreds of people damned also be compatible with God’s goodness apart from an infinite chances model, as long as it was also a model where God is very generous in redemptiveness and gives people a real ability to affect their character?

A variant of Molinism

The biggest problem with Molinism is the grounding problem: there is nothing in reality wherein the subjunctive conditionals of free will (“middle truths”) could be grounded. As Reuben perceptively noted in a recent comment, the problem is especially big if God is simple, as then the middle truths can’t even be grounded in God.

Anyway, it seems to me that there might be some benefit to Molinists to think about what one might call Individually Grounded Weak Molinism (IGWM). On IGWM, there are no middle truths about the free actions of non-actual persons, but there are middle truths about the free actions of actual persons, which are grounded in properties of these very persons. Thus, “when” (in the explanatory and not temporal sense) God is deciding whether to create Curley, there is no fact of the matter whether Curley would accept a $5000 bribe in circumstances C. But “as soon as” Curley is actual (on eternalism, this will be the case even before Curley temporally exists—maybe as soon as God decides to create Curley), there will be a fact of the matter whether he would accept that bribe in C.

On IGWM, there is an “ontological home” for the middle truths. Moreover, IGWM ameliorates another problem for standard Molinism: the problem of evil. On standard Molinism, God presumably has a vast amount of knowledge that he can use in optimizing the world. E.g., God knows that Adam and Eve would sin, but there is a vast number of couples almost exactly like Adam and Eve, and it seems very unlikely that they would all have sinned in Adam and Eve’s circumstances, so why didn’t God create them? But on IGWM, God only knows that Adam and Eve would sin in the Garden of Even once he decides to create them. One might objet that are presumably a vast number of gardens almost like Eden that God could put Adam and Eve in, and on IGWM God middle-knows how they would behave in each garden, and it would seem surprising if Adam and Eve would have sinned in all of them. Not so. For it might be that there are close correlations between middle-truths about how Eve (say) would act in circumstance C₁ and how she would act in C₂ if C₁ and C₂ are very similar.

But IGWM still has enough bite to yield some of the advantages of Molinism. First, we have the biblical cases. Granted, these could be interpreted in a non-Molinist way, but a Molinist way seems particularly neat. For instance, in the parable of Lazarus and the rich man, God tells the rich man that his brothers wouldn’t repent even if someone came back from the dead to talk to them. On IGWM, this conditional makes sense, and is grounded in the rich man’s brothers. (Granted, this is a parable, so it’s not a prooftext for Molinism.) Second, we still have the nice story about how God can work in our lives and achieve his gracious goals without undercutting our freedom: God knows what we would choose in which circumstances, and can control the circumstances we are in. Third, the Molinist account of efficacious grace works on IGWM with no changes. Fourth, and this is the immediate motivation for my post, my Molinist defense of hell works just fine on this story.

IGWM does not solve all the problems with Molinism. The Adams circularity argument continues to be problematic. Suppose I freely choose A in circumstances C, and God put me in C because he middle knew that I would freely choose A in C. It’s surely note a coincidence that both (a) I freely choose A in C and (b) I would freely choose A in C. There needs to be some kind of explanatory connection between (a) and (b). If (b) is explanatorily prior to (a), then it seems that the middle facts about me explain my actions, and that seems to undercut my freedom. But if (a) is explanatorily prior to (b), then we have a circularity. For (b) is prior to C, and C is prior to (a).

And there may be a way in which the freedom worry about Molinism feels bigger on IGWM. If it’s a property of me that I would freely choose A in C, rather than just being an ungrounded truth, then it looks a bit more like that property constrains me and takes away my freedom.

Infinite chances and a Molinist defense of damnation

Let’s start with a line of thought that doesn’t require Molinism. Let Infinite Chances be the thesis that God never definitively damns someone: even if on some day you have chosen against God, you will still have choices on future days that can get you back to the path of virtue and hence to God.

One might think that Infinite Chances would all but guarantee that everyone is saved in the end. There is no such guarantee, however. Consider this Toy Model. On each future day n, I have level V_n of virtue, negative if I am on balance vicious and positive if I am on balance virtuous. On each day, I make a free choice. The choice will either be good or bad, and will respectively increase or decrease your virtue by one unit. The more virtuous I am, the more likely the choice will be for good, and the more vicious I am, the more likely that it will be for bad, but no matter how vicious I am, there is a non-zero probability that I will choose for good. I also assume that if I am on-balance vicious, the probability that I will choose for good is less than 1/2. Suppose for simplicity that there is some level S > 0 of virtue at which I become definitively saved.

This story is consistent with Infinite Chances. For no matter how vicious I am, there is also a non-zero probability that I will choose the good, thereby decreasing my level of vice by one unit, and the next day choose the good again, and so on, until I reach level S. So I will always have a chance to dig myself out of my vice. But at the same time, it is a mathematical fact that under the above circumstances if I am in fact vicious on day n (i.e., V_n < 0), then there is a non-zero probability that even over an infinite future I will never be better than I am on day n. My next day’s choice might be bad, and then I might bounce up and down, but never dig myself out. Remember that once my level of vice is  − m units, it would take m steps to get back to neutral.

Next, note that the Toy Model seems consistent with God’s goodness. It allows our actions to have genuine effects on our character, and it allows us genuine freedom to mold that character. Yet on this toy model there is a non-zero probability of my being damned. Thus we get a first conclusion:

1. It is consistent with God’s goodness that not everyone is saved.

This isn’t a defense of hell or damnation as it stands, though it is a defense of non-univeralism. I take it (perhaps pace C. S. Lewis) that it is a part of the doctrine of damnation that once one is damned, one doesn’t have a chance to get out of hell. In other words, the doctrine of hell implies a denial of Infinite Chances, why my toy model implies Infinite Chances.

However, given Molinism (I wish Molinism were true, but alas I think it’s not) one can turn this into a defense of damnation. Let’s imagine that God has created a bunch of people in a situation where the Toy Model is true, while with his middle knowledge God knew that given the infinite chances that the Toy Model posits, all but one—let’s call him “Judas”—would eventually be saved. This is compatible with God’s goodness by something like the line of thought that led to (1).

But now imagine that God decided to create this bunch of people, while knowing with his middle knowledge that given the infinite chances of the Toy Model they would be saved except for Judas, but hadn’t yet decided whether to give the people the infinite chances of the Toy Model. Among the things that God middle-knew is that, if he made the Toy Model true, there would a day n on which Judas will have a level of vice which he would never do better than—the next day he would choose badly, and while he might bounce back up a little, he would never be better than V_n. And so God decided that he would freeze Judas morally at the level V_n achieved on day n, taking away Judas’ freedom of choice on subsequent days. As a result, Judas was irredeemably damned on day n, as the doctrine of hell holds.

Would this be compatible with God’s goodness? Yes! Here is why. We have granted that giving Judas the infinite chances of the Toy Model would be compatible with God’s goodness, even if Judas is never saved. But freezing Judas’ moral level seems a better thing to do. For God knows that if Judas’ moral level is frozen at V_n, then Judas won’t get worse, while if Judas had the infinite choices, then he would still never be better than level V_n, and he would be worse than that on at least day n + 1 (since we have supposed that after day n, Judas is never better than V_n, so V_n + 1 is not greater than V_n, and hence it must be less, as it was assumed that all choices either improve or make worse). Since one’s level of virtue or vice is a primary constituent of human wellbeing, this freezing seems a good thing.

I wonder if there is any way to rescue this argument in the absence of Molinism? Maybe. For there may come a time when the probability of Judas ever digging himself out of his vice has gotten so small, while the probability of his digging himself deeper is pretty high, and so it even absent Molinism it might be a mercy not to give Judas more chances.

Libertarian universalism and open futurism

Consider an interesting pair of thesis. First, consider libertarian universalism, which holds that everyone eventually freely—and with the ability to do otherwise—definitively chooses for God. On the grounds of Christian revelation, I think libertarian universalism is false, but I also think it’s clearly logically possible, so it’s only contingently false.

Second, consider open future views of time on which there either are no facts about future contingents or the facts are all-false. Both open future views are logically compatible with the truth of libertarian universalism, but only in a funny sort of way. For it could be that “everyone” (in the sense of the libertarian universalist doctrine, which includes both past, present and future people) has already freely definitively chosen for God. This would require that the end of the world is predestined to come soon, because otherwise the “everyone” includes—or at least possibly includes!—people who have yet to be conceived, and they clearly have yet to definitively choose for God. Moreover, it would require that all the people now alive have already definitively chosen for God, which seems highly implausible.

Let’s suppose that, as seems plausible, that we all agree that some people have yet to already definitively choose for God. Then open futurists cannot affirm libertarian universalism. Can they deny it? Yes, in at least two ways. They might say that God predestined some to non-freely choose for God. Or they might hold that some who have died have already definitively rejected God and that God has predestined them to have no further chances after that rejection.

So open futurists can deny libertarian universalism but cannot affirm it. Can they be open to it? Well, it initially seems hard to say what it means that one is open to a view that one believes is not true. But open futurists can say that although they believe libertarian universalism is not true, it can become true. And that’s a way of being open to it. But a kind of weakish way.

I don’t accept either open future or libertarian universalism, of course.

Hedonic adaptation

Hedonic adaptation is the theory that individual humans have different basic levels of subjective life satisfaction (SLS). When major changes in life circumstances make life change for the better (e.g., marriage) or worse (e.g., divorce or unemployment), life satisfaction goes up and down, but then over five years it creeps back to roughly where it was even if the circumstances do not change back.

Suppose this theory is true. Can we draw any philosophical conclusions from it?

Here is one thought. Suppose the right theory of human well-being is hedonism. It seems plausible to think that SLS measures one’s overall hedonic level averaged over a medium term. If so, then hedonic adaptation makes means we have less reason than we might have thought to seek out gains and to avoid losses in objective life circumstances for ourselves and others. Not that we have none. Short-term pains and pleasures are still real, and major changes in life circumstances do have a genuine effect on the life-time total hedonic level even if after five years much of the change in SLS disappears—and in the case of negative changes, even if much of the change disappears, it probably doesn’t disappear entirely.

Still, it seems plausible that if hedonism is the right theory of human well-being, then the discovery of hedonic adaptation (assuming it is real) should make us, both individually and societally, more passive than is plausible. One’s hedonic level, according to hedonic adaptation, is just not very responsive in the long-term to costly interventions. E.g., is it really worth a society’s putting a lot of effort into preventing unemployment if the unemployed would anyway return to a basic hedonic level, and hedonic level is what matters?

This suggests that hedonic adaptation provides an argument against hedonic theories of well-being, by yielding an implausible consequence of those theories. Now while this argument initially seemed to depend on hedonic adaptation being empirically true, perhaps it does not even require that. All we need, perhaps, is that hedonic adaptation is possibly true, and then we can argue:

1. If well-being is hedonic in nature, then if hedonic adaptation is true, we would have much less reason than we thought for major societal improvements.

But I think we may also find ourselves with the intuition:

2. Even if hedonic adaptation is true, we would not have much less reason than we thought for major societal improvements.

And it arguably follows from 1 and 2, together with

3. It is possible that hedonic adaptation is true

that:

4. Well-being is not hedonic in nature.

(The move from 1–3 to 4 uses modus tollens and two principles in subjunctive conditional logic, namely (a) if p → q and p → r then p → (q&r), and (b) if p is possible and p → q, then q is possible. I think (a) is true, but I am dubious of (b) in general, even though it is true on Lewis’s theory of counterfactuals. However, even if (b) is not true in general, I think it is true in typical cases, and seems likely to be true in the context of my argument agaisnt hedonism.)

In other words, thinking about hedonic adaptation might well shift us to thinking that a major portion of our well-being is more objective than hedonism would make it be. The reason for fighting unemployment isn’t primarily its deleterious effects on subjective life satisfaction, but its (typical) deleterious effects on objective human flourishing.

But there is also another even more speculative consequence of hedonic adaptation. Even if we reject hedonic level or SLS as the measure of human flourishing, it is clear that that we all want SLS, and indeed want long-term SLS. Hedonic adaptation tells us that improving our life-circumstances won’t get us long-term SLS: after half a decade, we’ll go back to where we were. Plausibly, our desire for long-term SLS is a normal and central human desire. Plausibly, normal and central human desires are fulfillable. Yet this desire for long-term SLS is not fulfillable in this life. Hence there is life after death.

In other words, the phenomenon of hedonic adaptation would empirically bolster the argument from desire considered as an argument for an afterlife.

Some Fitch-style arguments

Let strong physicalist supervenience (SPS) be the view that necessarily all truths supervene on first-order truth about the spatiotemporal arrangement of physical stuff.

Here is a plausible thesis:

1. Every first-order truth about the spatiotemporal arrangement of physical stuff is possibly known.

Add this premise:

2. If a conjunction is known, so are its conjuncts.

It follows from SPS, 1 and 2 that:

3. Every actual first-order truth about the spatiotemporal arrangement of physical stuff is actually known.

For suppose that p is any actual first-order truth about the spatiotemporal arrangement of physical stuff. Let q be the conjunction of all other actual first-order truths about the spatiotemporal arrangement of physical stuff. Let r be the conjunction of p and q. Then r is a first-order truth about the spatiotemporal arrangement of physical stuff. By 1, there is a possible world w where r is known. Since r contains all actual first-order truths about the spatiotemporal arrangement of physical stuff, w and the actual world have the same first-order truths about the spatiotemporal arrangement of physical stuff. By SPS, it follows that w and the actual world have the same truths. Thus, since it’s true at w that r is known, it’s actually true that r is known. And since r is actually known, so is p by 2. Hence, we have 3.

In exactly the same way, given:

4. Every first-order truth about the spatiotemporal arrangement of physical stuff is possibly explained

5. If a conjunction is explained, so are its conjuncts

and SPS, we get:

6. Every actual first-order truth about the spatiotemporal arrangement of physical stuff actually has an explanation.

But given physicalism, it is clear that 3 and 6 are false. Regarding 3, it is clear that if physicalism is true, nobody knowns how many hairs I have on my head. Regarding 6, given physicalism, there is no explanation of the sum total of physical reality. Thus:

7. Given either 1 and 2, or 4 and 5, at least one of physicalism and SPS is false.

It is tempting to turn this into an argument against physicalism by holding that:

8. If SPS is true, physicalism is true.

But actually I can imagine someone denying 8. Imagine a classical theist who thinks that everything other than God has to be physical. Such a classical theist will actually believe SPS. For on classical theism, facts about God supervene on facts about everything other than God, since classical theism includes divine simplicity according to which there are no contingent facts intrinsically about God, so any two worlds that differ in their facts about God will have to differ in something other than God. And according to our classical theist, everything other than God has to be physical. So any two worlds that differ must differ in facts about the arrangement of physical stuff, which is what SPS says. However, the classical theist is not a physicalist, because God is not physical on classical theism.

Still, I think an argument against the conjunction of physicalism and SPS is still interesting.

My first (and last?) computer science paper

Some years back I was searching for a better method to play lightgun games like Duck Hunt under emulation by using a Wii remote as a light gun.

The standard Wii remote way of pointing at a TV with a sensor bar underneath is not precise enough. The Wii remote has an infrared camera that tracks the LEDs in the sensor bar, and uses that to estimate where the remote is pointing. But since all the LEDs in the sensor bars lie in a single line, there is no way to determine from the camera image exactly how the Wii remote points. (It’s easy to visualize how rotating the Wii remote about the axis defined by the sensor bar won’t change the camera image in it.) The Wii just guesses, but that’s not good enough for lightgun games.

I already had a way that involved putting four infrared LEDs around my TV, but I was wondering if one could do this with a standard sensor bar by leveraging the additional information from the accelerometer in the Wii remote. A web search came up with a paper by d’Alfonso et al. on determining camera pose using two landmarks and accelerometer. The paper claimed a theorem that, except in certain singular cases, one could determine the pose up to a two-option ambiguity—i.e., there will be two camera orientation/position setups that would result in the same landmark images and accelerometer data.

After thinking about the paper, I realized that the claimed theorem was incorrect: there were singular cases the authors didn’t know about, but more interestingly I discovered cases where one could determine camera pose from two landmark images and accelerometer data without any ambiguity. Moreover, one of these cases was precisely the one relevant to me: the two landmarks are on the same horizontal line (the Wii sensor bar supplies precisely that) and the camera is above or below the plane of that line (easy to arrange). Sadly, experiments showed that Wii remote accelerometer precision was insufficient to determine camera pose in this way.

Anyway, I don’t like the idea of there being a published “theorem” that is incorrect. At some point I worked out all the details with trigonometry, and found a precise characterization of the exact conditions under which in theory (i.e., given a sufficiently precise accelerometer, unlike the one in the Wii remote) one could determine camera pose from two landmark images and gravity data with no ambiguity, with an ambiguity of two, or with an ambiguity of infinity (in some singular cases).

I just got a paper with all the details published.

Sadly, as noted, the Wii remote application didn’t pan out. However, I did write some proof of concept code for my Pixel 7 Pro, and found that it worked pretty decently with its accelerometer. Here’s a screenshot where the accelerometer data plus the screen positions of the (centers of) the bottom two Aruco markers, together with data about the camera and the dimensions of my display board, is used to calculate the screen positions of the upper two Aruco markers (the calculated positions are marked in red).

I kind of suspect that the accelerometer in the right-hand Nintendo Switch joycon might be sufficiently precise for use as a lightgun with two infrared LEDs for positioning, but I haven’t had time to implement this. (Part of the problem is that the infrared LEDs that I have at home are the wrong wavelength for the joycon.)

While this may be my last computer science paper, if some computer science grad student or the like is interested in working with me, we could do some empirical work on seeing how applicable the algorithm is.

From one divine person to two or more

1. Personhood is teleologically relational in the sense that a person is the kind of being that necessarily can only be fulfilled in relationships with other persons.

2. Necessarily, any person other than God is a creature.

3. So, if God is a single person, this person necessarily can only be fulfilled in relationships with creatures.

4. If a being only finds its fulfillment in relationships with Fs, then the being’s fulfillment depends on Fs.

5. So, if God is a single person, God’s fulfillment depends on creatures.

6. God’s fulfillment does not depend on creatures.

7. Thus, God is not a single person.

8. thus, if God is a person, God is two or more persons.

(I think most non-Christian monotheists these days will say that God is a single person. But I wonder if it wouldn’t be a better view for them to deny that God is a person at all, and to insist that the centrality of personhood to the understanding of God is a mistaken Christian approach.)

Here is another variant on the argument:

9. Necessarily, a person can only flourish in relationship to another equal person.

10. It is possible that God flourishes.

11. If God is a single person, then it is impossible for there to be another person equal to God.

12. So, God is a single person, it is impossible that God flourishes. (9 and 11)

13. So, God is not a single person. (10 and 12)

14. So, if God is a person, God is two or more persons.

Now, granted, it is a standard view in Christianity that one cannot prove the doctrine of the Trinity by reason. If the arguments above are sound, have I contradicted that view? No, for several reasons. First, the arguments don’t establish that God is three persons. Second, the arguments only establish that if God is a person at all, then God is two or more persons—but a further argument is needed that God is a person at all. (Perhaps perfect being theology can yield that.)

Value supervenes on being

Consider this thesis, which both Augustine and Aquinas would accept:

1. Value supervenes on being (VSB): How valuable reality is is determined by what entities exist.

However:

2. It is good to have a true belief and bad to have a false belief.

We now have an argument against presentism (only present things exist) and growing block (only present and past things exist). Among our beliefs, we have beliefs about contingent future events. Some of these beliefs are contingently true and others are contingently false. (I am assuming a bivalent temporal logic.) The value of these beliefs is determined by what entities exist. But present and past entities are insufficient to determine whether the beliefs are true, and hence what the value of the beliefs is. Thus, contrary to presentism and growing block, there must be future things.

Broadening harms considered in research ethics

In typical lab research ethics, the following types of ethical considerations count against an experiment:

1. risks of harm to the world outside the lab

2. risks of harm to persons in the lab

3. risks of pain to conscious entities in the lab.

(The categories overlap: the persons in the lab are conscious entities, and pain harms them. Likewise, the persons in the lab have lives outside the lab, and harm to them in the lab can imply harm to what is outside the lab.)

Now, note that the harms to the world outside the lab can include things that neither harm persons nor cause pain to conscious entities, such as damage to a non-conscious ecosystem. Thus, an experiment that risks the escape of microbes that could cause the world-wide extinction of a plant species would be ethically problematic, even if there were only negligible risks of harm to persons or pain to conscious critters. Further, note that non-pain harms to persons in or out of the lab certainly do count against an experiment—that is why consent is so important.

But I think the above categories of harms are too narrow, as they leave out:

4. risks of non-painful harms to conscious non-persons in the lab

5. risks of harms to non-conscious non-persons in the lab.

As the extinction point shows, we do worry about such risks when they concern what is outside the lab. We should, I think, likewise worry about them when they are in the lab.

The premature death of a plant or insect is a bad thing, even if plants or insects are not conscious (I have no idea whether insects are): a good thing has been erased from the world before its time. Likewise, disabling an insect in an experiment is a bad thing, even if it does not cause pain to the insect. Indeed, depriving an organism, of whatever sort, of the kind of environment where it thrives is a a bad thing.

I am not saying that we shouldn’t experiment on plants. But the fact that a living thing is harmed, regardless of whether it is a person or it feels pain, is an ethical consideration against the experiment. This ethical consideration may be outweighed by the epistemic or pragmatic value of the information gained from the experiment. But for the experiment to be worthwhile, it needs to be so outweighed. It’s not automatic that a non-pain harm to a non-person is outweighed by information gains. Some information might be too trivial.

Aristotelian abstraction

On the Aristotelian account of abstraction, we abstract the forms of things via perception. But here is a problem. First of all, we sometimes have better knowledge of things not perceived than of things perceived. Let’s say you read a detailed scientific book about pangolins, though you’ve never encountered one, while I once saw a koala in a zoo, but don’t much know anything more about them than that they are cute marsupials. It seems quite problematic to say I have the form of the koala in my mind, but you do not have the form of the pangolin. For that would make the mental possession of the form of the thing an unimportant flourish to what really epistemically matters, since clearly you are better off epistemically for reading the book about pangolins than I am for seeing the koala.

Of course, presumably, the author of the pangolin book presumably saw the pangolin. But that doesn’t seem to matter much, either. There are many dinosaur species where the paleontologist knows more about the species than I know about koalas, and yet no human ever perceptually interacted with a dinosaur, but typically only with minerals that have displaced their carcasses.

What should someone committed to the Aristotelian story about abstraction say? Two options come to mind, a modest and an expansive one.

Modest option: It really doesn’t really matter much epistemically whether one has the form of a pangolin or a koala in one’s mind. What matters is the propositional knowledge. However, it is essential to our possession of the propositional knowledge that our thoughts have intentionality, that they refer to the world. And it is crucial to securing intentionality that we have Aristotelian abstraction at the interface between the world and the mind. The triceratops expert’s intentionality with respect to triceratopses is derivative from the causal chain from triceratopses to their corpses to their fossils to the light reflecting from the fossils into the expert’s eyes to (eventually somehow) the expert’s thoughts. Where the causal chain crosses from the world to the mind, we need a transmission of form to ensure intentionality for the whole chain. Maybe the transmission happens when the minerals that have replaced the bones transmit their mineral forms to the expert’s mind, in some mysterious way mediated by the light.

Expansive option: Everything in the world carries the forms of everything nondivine that is causally upstream from itself. The triceratops corpse somehow has the form of the triceratops in it, and so do the bones of the triceratops, and so do the minerals that replace the bones, and so does the light modulated by reflection from the minerals, and so do the electrical impulses in the nerves from the retinal receptors to the brain. It is easier to extract the form when it comes from seeing a live koala than when it comes from seeing fossils, so that a layman can do the former while the scientist is needed to do the latter. Similarly, when you read a book about pangolins, the ink on the page, being causally modulated by the author’s knowledge, carries the form of the pangolin (in addition to any forms naturally contained in the ink, say those of the particles constituting the ink).

Medievals already said something like this about the modulated light, so it’s just a matter of extending the story. And while the problem I am concerned about is one that does not require any science or technology beyond what the medievals had—the medievals knew about the gaining of expertise from books (if anything, they might have overestimated it!)—in a modern setting we it feels like an Aristotelian has to do this. After all, it is hard to deny that one can gain the same kind of knowledge of koalas by looking at them through high-end augmented reality goggles as from directly seeing them with our eyes. Thus, on a form-transmission story, the electrical potentials of the capacitors in the computer memory in the goggles have to somehow carry the form of the koala. But given the many ways that computer memory can be realized (think of magnetized rings versus capacitors versus markings on a CD), it seems plausible that any effect will have to have to be admitted to carry the form of its cause. It is tempting to say the form is only found in substances where there is sufficient data to reconstruct significant information about the causally-upstream object the form is of, but I think is not tenable. Presumably each bacterium is a substance, and we could have a type of biological memory in augmented reality glasses where each bit of information is stored in a different bacterium.

This story coheres very nicely with the essentiality of origins. It is a kind of a reversal of the principle of proportionality of causation on which the reality of the effect is actually or eminently found in the cause: the formal reality of a cause is found in the effect. I think it’s a defensible story. But I also find it hard to believe.

From chickens and eggs to a cosmological argument

Suppose we have an infinite regress of chickens and eggs, with no first item, and suppose that if there is any last item, it’s not a chicken.

Plausibly, then:

1. The plurality of the chickens explains the eggs.

But also plausibly:

2. If x has an explanation, then x has a fundamental explanation.

However:

3. No plurality of chickens and eggs fundamentally explains the eggs.

4. So, something that is other than a plurality of chickens and eggs fundamentally explains the eggs.

Why is (3) true? Well, there are two kinds of chicken and egg pluralities.

1. Pluralities with an earliest item.
2. Pluralities with no earliest item.

A Type I plurality does not explain the chickens and eggs, because it does not explain any chickens and eggs earlier than the earliest item in the plurality.

A Type II plurality at best non-fundamentally explains the eggs. For given a Type II plurality, choose any item x in the plurality. Item x is a chicken or egg, and there is an earlier chicken or egg y in the plurality. Since y explains x, by a plausible case of transitivity, the subplurality where we drop x also explains the eggs. So the initial Type II plurality did not fundamentally explain the eggs.

Since every plurality of chickens and eggs is of Type I or Type II, we have (3).

The argument above applies beyond chickens and eggs. Suppose:

5. Every physical item in the world is caused.

Then:

6. Either there is a physical item with a non-physical cause or every physical item has a physical cause.

Add:

7. There are no loops of physical causes.

8. Physical causation is transitive.

As I show in my 1999 paper using Zorn’s Lemma, if every physical item has a physical cause, it is then possible to partition the physical items into an “even” and an “odd” subset, where every item in the even subset causes an item in the odd subset, and every item in the odd subset causes an item in the even subset. Call the odd items “eggs”. Call the non-final even items “chickens” (a non-final item is one that causes a further physical item). Then, plausibly, the chickens explain the eggs. By the argument above, something non-physical thus explains the eggs. Anything non-physical that explains the eggs explains all of physical reality (because for any physical item, it is either an egg or there is an egg prior to it). So:

9. There is a non-physical item that explains all of physical reality.

I think the thing that needs the most defense is premise (2). It’s somewhat close to causal finitism.

Virtues in heaven

Here are two Aristotelian theses:

1. Complete human flourishing requires the exercise of all the human virtues.

2. Courage is a human virtue.

Add a highly plausible thesis that likely everyone will accept:

3. The exercise of courage requires apparent or real danger.

Now add some theses that a Christian will surely accept:

4. In heaven, there are no apparent danger.

5. In heaven, there is no real danger.

6. In heaven, there is complete human flourishing.

Contradiction!

I find it nearly impossible to deny 3-6. That leaves 1 and 2.

One move available to the Christian Aristotelian is to weaken 1. Perhaps complete human flourishing only requires the exercise of fundamental virtues, and courage is a non-central virtue. But traditionally fortitude also counts as a cardinal virtue, cardinal seems to imply fundamental, but fortitude is a virtue of dealing with evil—whether in the form of real or apparent danger, or of suffering, or of hardship, while there is no evil in heaven. So the problem comes back for fortitude. One solution is to deny the traditional cardinal virtues as being fundamental, and insist that the only fundamental virtue is love. Fortitude is just a central situational way for love to manifest.

Another move is simply to deny 1 altogether, of course, denying the whole link between virtues and flourishing. This is probably too radical.

And of course one might just deny 2. The most plausible move would be to say that only love is a virtue.

The thesis of the uniqueness of our self-consciousness

I am toying with the thesis that each person’s self-consciousness is qualitatively different. Perhaps all persons have uniquely individuating properties, something like haecceities, and for each such individuating property there is a different quale of seeing oneself as an instance of it. One might even think the quale is the unique individuating property. Or, alternately, it might be that each person’s qualia are different: my quale of red is a quale of me-seeing-red, my quale of middle-C is a quale of me-hearing-middle-C and your quale of red is a quale of you-seeing-red while your quale of middle-C is a quale of your-hearing-middle-C. There is, presumably, something that all of my qualia have in common, and all your qualia have in common, and so on.

What consequences might there be of such a view?

First, some people have the intuition that there is something mysterious about self-consciousness, something that goes significantly beyond mere consciousness (pace my thoughts here). The qualitative conscious difference thesis could be a way of capturing that mysterious extra in self-consciousness.

Second, this would be a way to support and explain intuition of the non-fungibility and irrepeatability of persons. There is something irrevocably unique about all our perceptions of the world.

Third, we might get a neat account of second-person knowledge. What happens in second-person knowledge—very mysteriously!—is that we come to see that special qualitative feature of the other, albeit in a different mode, from the outside.

Fourth, the thesis might make one even more cautious about some thorny epistemological matters. There cannot be such a thing as two people with the same qualitative evidence: each person’s evidence must be qualitatively different from that of every other person. The concept of being an epistemic peer becomes dubious: you unavoidably see the world differently from me. The application of Bayesianism to fundamental matters becomes even more fishy. Plausibly, we have essentiality of origins, so that if there is a God, you and I couldn’t have existed in a world without God. On the qualitative uniqueness thesis, this means that there are qualitative features of our mental lives that cannot be exemplified in a world without God. We can then expect that we will have some trickiness when we ask questions like: “How likely is it that the evidence I have would obtain if God didn’t exist?” Similar issues come up for fundamental laws of nature. In particular, we cannot really do Bayesian reasoning on our total evidence, and we should probably see Bayesianism as a mere toy model.

A false trilemma

Kant writes:

only three [forms of sovereignty] are possible: namely, either only one, or some in association, or all those together who constitute the civil society possess sovereign power (autocracy, aristocracy, and democracy …).

The division of the sovereigns into “one”, “some” or “all” members of a society sure sounds like an exhaustive division, at least assuming that there is a sovereign rather than anarchy, which is a fair assumption, since an anarchy probably doesn’t count as a “civil society”.

But not quite! For what if there is a sovereign, but the sovereign is not a member of the civil society? I can think of at least two possibilities like that.

For, a civil society is a mutually interacting body of persons. Thus, first, we could have a personal sovereign with a one-way relationship with the civil society, where the sovereign rules but is in no way affected by what happens in the society. Since causal interaction among embodied beings is always bidirectional, the sovereign would need to be a supernatural being, such as God.

Second, we could have a sovereign that is not a person—a robot overlord. Since only persons can be members of society, such a sovereign would not be a member of society.

One might object that the robot overlord isn’t really a sovereign, since it doesn’t have a will of its own. But it is evident that one could have a fairly functional civil society run by the dictates of a robot overlord, enforced by human or robotic minions, or simply by people’s confidence that the robot overlord’s plan is a good one. If so, and if a robot overlord is not a sovereign, we have a counterexample to the thesis that anarchy is not a civil society.

Making the happiness of others our end

Kant says that a virtuous persons has the happiness of others as an end.

Note that this is not an end in the sense of a goal which we try to achieve. For the goal can’t be that all others are happy. If that were the goal, we might as well not even try: we’re not going to achieve it. Nor can the goal be that some others are happy. For that’s already so, and so nothing need be done. Could the goal be that many others are happy? I suppose that my actions could make a difference between not-many and many being happy. But it seems rather unlikely that they could. And if a careful examination of the world were to show that independently of my actions many others are happy, again such a goal would lead to my not having to do anything.

Maybe the Kantian end of happiness is distributive. For each person distinct from yourself, you aim at this person’s being happy. So you have a large number of goals, one for each person distinct from yourself. But that’s not right, either. For suppose that Alice is miserable, and you know you can’t make her happy. But you can make her happier. Surely that’s a part of what the virtuous person aims at. So is it, perhaps, that in the case of each person distinct from yourself, you aim to make them happier than they currently are? However, imagine that Alice is miserable, but you know that her condition will at least slightly improve over time, no matter what you do. Then you don’t need to do anything to make her happier, and so a goal of making others happier doesn’t make you do anything for Alice—even if you could make Alice much happier.

Perhaps, though, the end of happiness is doubly distributive: it distributes over persons and over what you might call “possible pieces of happiness”. For each person x and each possible piece H of happiness, you aim at x having H. But now the virtuous person’s aims are vast, since there are many, many possible pieces of happiness—maybe even infinitely many. Maybe this is right, but it seems implausible.

All this makes me think that when Kant talks of the happiness of others as our end, he is not talking of an end as a set of specific goals to be achieved. Maybe he is saying that in general a virtuous person has a tendency to perform actions with specific goals of the form: Make Alice happier with respect to her toothache. The “end” of happiness isn’t an end, or a multiplicity of ends even, but a kind of architectonic pattern in our goals.

Self-locating evidence and bearers of epistemic good

In the case of non-epistemic goods, it’s an obvious feature of life that someone there is a choice to be made by an individual between their own first-order good and the first-order good of the community—each requires the sacrifice of the other. In the case of epistemic goods, this is less obvious.

In the pragmatic case, the typical reason for such competition between goods is due to limited resources. This, of course, also happens in the epistemic sphere. Suppose Alice is much more intellectually talented than Bob, but only Bob has the money to go to university. If Bob spends the money on himself, he will gain private epistemic goods, but will contribute little epistemically to society as a whole. But if he gives the money to Alice, she may become a brilliant scholar or scientist, significantly contributing to society’s knowledge.

More interesting than these, however, are cases of competition between private and communal epistemic goods that are not due to epistemic resources. I find it interesting that some cases of self-locating evidence appear to be such.

Suppose there are ten billion people in the world, currently isolated from one another. A device produced by a mad scientist has a 99.9% chance at noon today of triggering a death ray that randomly kills 99.9999% of the world population. Noon has just passed. You are still alive. Should you think the device worked? Sleeping Beauty style arguments say “No”. This time I want to think about this in terms of individual epistemic goods. In N runs of the device, 0.001N runs will have you survive because the device doesn’t trigger and 0.999 ⋅ 0.000001N runs will have you survive despite the device triggering. Thus, the vast majority of the runs where you survive are runs where the device didn’t trigger. Hence, it’s best for you individually to adopt the epistemic policy of thinking the device didn’t trigger.

But on the other hand, suppose we all adopt the epistemic policy of thinking the device didn’t trigger. Then 99.9% of the time, we are unanimously collectively wrong. And if we all adopt the epistemic policy of thinking the device did trigger, then 99.9% of the time, we are unanimously collectively right. It seems thus that if we look at the epistemic goods of society, then a policy of thinking the device did trigger is best.

If this is right, it points to a potential diagnosis of why the problems about self-locating evidence (doomsday, multiverses, Sleeping Beauty, etc.) are so difficult. For there may be different bearers of epistemic goods at play—say, society vs. the individual—and it could be that different answers are appropriate depending on whose goods we are pursuing. Maybe.

Is there some sort of a probability problem with a humongous but finite universe?

It’s easy to generate probabilistic paradoxes in a universe (or multiverse) with infinitely many people (e.g., if infinitely many people roll a die, equal numbers of people get 1 as get more than 1, so why think it’s more likely to get more than 1?). But what about a very large but finite universe? I used to think: “The only relevant difference is between finite and infinite. Really big but finite—no problem.” Now I am not so sure.

Paul Heyl measured the gravitational constant G as 6.670 × 10⁻¹¹ m³ kg⁻¹ s⁻², and denote the latter quantity by G₀. Consider two theories:

• H₁: The gravitational constant is between 6.665 × 10⁻¹¹ m³ kg⁻¹ s⁻² and 6.675 × 10⁻¹¹ m³ kg⁻¹ s⁻².

• H₂: The gravitational constant is between 7.676 × 10⁻¹¹ m³ kg⁻¹ s⁻² and 7.686 × 10⁻¹¹ m³ kg⁻¹ s⁻².

It seems obvious that:

1. Heyl’s measurement strongly supports H₁ but does not completely rule out H₂.

But let’s think this through. Suppose Heyl’s evidence is the proposition E which he would express as “I measured G to be G₀.” But, very plausibly, it is an essential property of a human being that they exist in a world with such-and-such a gravitational constant. One way of getting to this conclusion is to say that the forces of gravity are part of our causal history, and then to apply the essentiality of origins. Another is to say that we couldn’t have been made of completely different matter, but the forces exerted by the matter in our bodies are an essential property of that matter.

Given this essentiality of gravitational constant assumption, it follows that at least one of H₁ and H₂ is incompatible with Heyl’s existence. Now, to get (1), we need prior probabilities on which P(H₁|E) > P(H₂|E) > 0. Such prior probabilities will assign a non-zero value to H₁E and to H₂E. But at least one of these two claims is impossible since E entails Heyl’s existence, and a probability assignment that assigns a non-zero value to something impossible is screwed up, and we should be quite suspicious of what we get from it.

We might try to avoid this by using self-locating evidence. But my colleague Yoaav Isaacs has this great paper that gives a pretty strong argument that there isn’t a good way to working with self-locating evidence. So suppose we put this option aside.

Or we might make a distinction between logical impossibility and metaphysical impossibility. I find that suspicious, too.

So, what’s left? Well, here’s one remaining suggestion. Heyl’s evidence is equivalent to the proposition that Heyl measured G to be G₀, a proposition that rigidly refers to Heyl, and hence won’t be compatible with both H₁ and H₂. But we can weaken Heyl’s evidence to something that is compatible with H₁ and H₂, something purely qualitative, like:

• E_Q: A physicist named “Paul Heyl”, who married someone named “Lucy Daugherty”, and who …, measured G to be G₀.

Here, “…” is all the other purely qualitative stuff we know about Paul Heyl, so that E_Q is compatible with both H₁ and H₂.

But now here is a problem. Suppose we live in a vast but finite universe with, say, 10¹⁰¹⁰ people. In such a universe, we might well expect large numbers of people named “Paul Heyl” who satisfy all the conditions in E_Q, including the measurement of G to be G₀, even if in fact G is in the range indicated in H₁ (measurement error!). Thus, P(E_Q|H₂) is close to 1 as is P(E_Q|H₁). Granted, we do have P(E_Q|H₁) > P(E_Q|H₂) > 0. But because the two probabilities are so close to each other, the support E_Q gives to H₁ over H₂ is very slight, and hence we no longer have (1).

It follows that unless we can find some other way of solving the problem that the essentiality of the laws of nature to humans poses for Bayesian reasoning, a fair amount of fundamental physics research would be undercut by a large enough—even if finite—universe.

Of course, maybe we can find some other way of solving it. But maybe we can’t. And if we can’t, then the E_Q solution might be our best bet—and it’ll work just fine in a universe that isn’t too vast.

Consent to deceitful experiments

1. In order to give valid consent, the subjects in an experiment need to be informed about all the harms that the experimenter plans to impose on them.

2. Being deceived is a harm.

3. Therefore, in order to give valid consent, the subjects in an experiment where the experimenter plans to use deceit need to be informed that deceit will be imposed.

Often, consent conditions are phrased in terms of risks, and it is stipulated that only non-minimal risks need to be disclosed, where minimality is measured relative to the risks in ordinary life. Being deceived about a minor matter might be thought to be a minimal risk even when the probability of deceit is nearly 100%, since in ordinary life people routinely suffer deception, and sometimes don’t mind much if at all (see here for discussion).

However, I think there may be a difference between disclosing risks and disclosing planned harms. For instance, minor pains are a daily occurrence for people. But deliberately imposing a minor pain on an experimental subject who did not consent to such imposition—apart from special cases such as pushing someone away from danger—would seem to be a morally impermissible assault.

Knowledge and induction

Assume that in fact all ravens are black. Suppose you are sequentiallly observing ravens, and noting each one to be black. After observing n ravens, your evidence that the next raven is black will typically be significantly better than the evidence that all ravens are black. Now at some point, say after observing n_A ravens, your evidence that all ravens are black will rise to the level of knowledge. Thus, plausibly, at an earlier point in the sequence, call it n_N, your evidence that the next raven is black will have risen to the level of knowledge.

Suppose now you have observed n_A − 1 ravens, and you have been handed a raven in an opaque box, which you are certain you are about to open. Since n_N < n_A, at this point you have reached n_N. Hence:

1. You do not know that all ravens are black.

2. You do know that the next raven is black.

3. You know that when you observe the next raven, you will have sufficient evidence for knowledge that all ravens are black.

But note that while you know you will have sufficient evidence for knowledge that all ravens are black, you don’t know that you will know that all ravens are black. There is nothing deeply surprising about this distinction. We might well say about someone who has been subjected to misleading or Gettiered evidence that they have sufficient evidence to know something but nonetheless they don’t know, though the case at hand feels different.

One interesting thing about this case, as I read it, is that it contradicts the thesis that K = E, i.e., that knowledge is evidence. For if knowledge is evidence, and you know that the next raven is black, then you already have the evidence you will gain by observing the next raven, and hence you are already in the position to know that all ravens are black.

Another interesting thing is that it shows that you can know something and nonetheless it be rational for you to investigate it. For you know that the next raven is black, but it’s worth investigating further, since it is only upon observation that your knowledge of the next raven’s blackness turns into the kind of evidence that gives you knowledge that all ravens are black.

All this might make one think that I have misconstrued the epistemic facts, and it is false that there can be a point n_N prior to n_A at which you know that the next raven is black. Here is one way to back up my intuition that there can be such a point n_N < n_A. Suppose that we know for sure we live in a world where the color distributions of birds are always uncorrelated between the males and the females of the species, so that information about the color of members of one sex are irrelevant to the color of members of the other sex. Also assume that you know for sure that ravens have equal numbers of each sex, that you are observing ravens in an alternative female-male-female-male-… sequence, and that your priors for the color distributions of the two sexes of ravens are the same. Then if p_M and p_F are the probabilities that all male ravens are black and all female ravens are black, and p_A is the probability that all ravens are black, then at any given point in the observation sequence p_A = p_Mp_F. Let n_M and n_F be the points in the sequence where you know that all male and all female ravens are black, respectively. Then, n_M < n_A and n_F < n_A, since p_A = p_Mp_F is significantly smaller than either p_M and p_F at all points in the sequence except when we’ve observed all the ravens of one sex, and since p_M and p_F rise fairly gradually as we go through the sequence. Thus, at the point n_A − 1, we will have already reached knowledge that all the male ravens are black and the knowledge that all the female ravens are black. In particular, then, we know that the next raven is black, since if n_A is even, the n_Ath raven is male and we know all male ravens are black, and if n_A is odd, then the n_Ath raven is female, and we know that all female ravens are black.

Epistemic rationality and Pascal's Wager

Pascal’s Wager is an argument that it is prudentially rational to engage in theistic belief promotion practices (TBPP), namely practices apt to promote one’s belief in God.

My interest this post is the standard epistemic rationality objection to the Wager, that engaging in TBPPs is irrational—a kind of brainwashing of oneself. Let’s think about the objection with a bit more care. Consider a specific TBPP Q, say Pascal’s example of going to Mass. If Q is indeed a TBPP, one expects engagement in TBPP to make it more likely that one believes in God. But how does one expect Q to achieve that goal? There are two possibilities. Either Q is expected to achieve that goal by providing one with evidence for theism or in some non-evidential way (or a combination of the two).

Suppose Q is expected to work evidentially. Then we already have the expectation of a higher credence given Q. This expectation is either rational or not. If it is not rational, then we don’t actually have good reason to engage in Q. If it is rational, however, then we should rationally raise our credence in theism right now, without having to bother engaging in Q, and for reasons having nothing to do with any wager. But if Q promotes belief in God non-rationally, then we should not engage in Q for the sake of such promotion of belief—we should not aim to non-rationally promote beliefs.

Let me make the first horn of the dilemma—namely, that the expectation of a higher credence is rational—a bit more precise. We can distinguish two (not mutually exclusive) ways in which a practice rationally increases one’s credence in a hypothesis H. One way is purely epistemic, by uncovering facts about reality. This is the usual way. But if that’s the way we expect to increase our credence in theism by engaging in Q, then we already have evidence that there are such theism-indicating facts to be discovered, and so we should already increase our credence in Q. The other way is practical, by promoting the hypothesis H in a way that shows up to us. The second way is a bit unusual, but here is an example: one way to increase your credence that you will not die of heart disease is to live a healthy life. For if you live a healthy life, you are less likely to die of heart disease, and since you will notice signs of improved cardiac health (e.g., lower resting heart rate, less huffing and puffing on stairs, etc.), your credence that you won’t die of heart diseases will also increase. But this practical way of increasing credence is utterly irrelevant in the case of theism, since nothing we can do can make God more or less likely to exist! So the only rational way that remains is the evidence-based way, and evidence-of-evidence is already evidence.

I used to be quite impressed by the worry that Pascal’s Wager leads to self-deception. I am less impressed. Here is why. There is a serious technical flaw in the argument for the first horn of the dilemma. A simple model for the relation of credence and belief is that you believe a proposition if and only if your credence is above some threshold β. This model might be false, but an analogue of what I will say should apply on more sophisticated models as well.

Here is the point. Consider a case where one is thinking about observing (and suppose this is a simple non-Newcombian observation that does not affect the hypothesis) whether some event E evidentially relevant to H has obtained. Then one’s expected posterior credence is:

1. C(E)C(H∣E) + C(∼E)C(H∣∼E),

where C is one’s credence function. But if one is a good Bayesian reasoner, then by total probability the value in (1) is simply equal to one’s prior C(H). Thus the value of one’s credence has no expectation of change upon observation when one is being rational. This seems to support the idea that if you expect your credence to go up, you should already raise it.

But in fact it’s not so simple. For even though the expected posterior credence equals one’s current credence, it could well be that it is more likely that the expected posterior credence exceeds the threshold β if you make the observation than if you don’t. Indeed, cases are obvious. Suppose the belief threshold β is 0.9, and you tossed a coin out of my sight. Suppose I have a prior credence 0.5 that this coin is fair and a prior credence 0.5 that it is double-headed. Then currently I don’t believe (or disbelieve) that the coin is fair. But if I look at the coin and I see tails, I will believe that it is fair—indeed, I will have posterior credence 1 in its fairness. But if I don’t look at the coin, I am not going to get any evidence, and I will continue not to believe that the coin is fair. If I look at the coin, the probability that I will see tails is 0.25 (I have credence 0.5 that it’s fair, and if it’s fair, the chance of tails is 0.5), and so the probability that I will believe if I look is 0.25 (since if I look and see tails, my posterior will be 1 which is bigger than β = 0.9), and the probability that I will believe if I don’t look is 0. Of course, if I don’t see tails, my credence that the coin is fair will go down. But while it will go down, it won’t affect whether I believe that the coin is fair—for I already don’t believe it (in the sense of not-believe, rather than in the sense of believe-not). And there is no irrationality of any sort in looking at the coin in this case.

In other words, the point is that while one’s rational credence has no positive or negative rational expectation of change upon observation, whether one’s rational credence is above a threshold certainly can have a positive or negative rational expection of change.

How could this work in a Pascal’s Wager situation? Let’s talk through one possibility. Take Pascal’s example of going regularly to Mass. Suppose, as Pascal says, your current credence in God is 0.5. You might think that if God exists, going to Mass has a decent chance, say 0.2, of resulting in an evident radical transformation E of your life, so evident and radical that updating your credence in theism on E will push your credence in God to above the threshold β. Of course, you might go to Mass it might not produce any such evident radical transformation (this is true even if God always improves the hearts of people who go to Mass, since he might do so more gradually or less evidently), and in that case your rational credence in God will go below 0.5. But going below 0.5 won’t affect whether you believe in God, since 0.5 is already, I assume, far below the belief threshold β. On the other hand, maybe if you don’t go to Mass, the probability that you will get evidence that will push your credence in theism above β is pretty small—smaller than 0.2. Very likely, your credence will just oscillate a little around 0.5 in the non-Mass-going case. Thus if there is a payoff you get for your credence exceeding the threshold β, it will be worth going to Mass, without there being any epistemic irrationality in the reasoning.

Thought of in this way, we get some practical guidance as to which TBPPs the agnostic or atheist should engage in. They should look for practices that, if God exists, have a decent chance of producing evidence for theism sufficient to push them above β.

I am a bit doubtful that Pascal meant us to think in the above way. He may well have been recommending TBPPs on the grounds of their non-rational effect on belief. My argument above does not defend that.

What's a properly epistemic difference?

There are different types of knowledge: mathematical, geological, biological, a priori, a posteriori, de se, interpersonal, hard-earned, esoteric, very well justified, professional, certain, firm, testimonial, propositional, etc.

When we divide up knowledge, some of the divisions are more properly epistemic than others. For instance, the difference between hard-earned knowledge and easy-come knowledge is not an epistemic difference at all—it is just a difference between how one came by it. If you and I live in villages on the opposite side of a mountain ridge, my knowledge of the waterfall outside our village is easy-come knowledge, while you had to work hard to earn that knowledge, climbing over the mountain ridge. But we are epistemically on par with respect to the knowledge of the waterfall. So, some types or classifications of knowledge do not divide up the knowledge epistemically at all. We might call these “wholly accidental” classifications of knowledge.

Other classifications of knowledge divide up the knowledge by means of the type of justification. For instance, the classic distinction between a priori and a posteriori is like this. Notice, however, that the very same thing can be known a priori and a posteriori. If you use a calculator to find out what 117 × 19 is, you learn it a posteriori, but if you use figure it out in your head, it’s a priori. And when the evidence is equal in strength and the content is the same, we do not have an epistemic difference to mark. (In the past, we might have said that a priori knowledge is better evidenced and maybe even infallible, but we now know that this is not in general true.) I am not sure whether the difference between a priori and a posteriori knowledge is a genuine epistemic difference. After all, the difference between knowledge gained by reading books in French and reading books in German doesn’t seem to be.

Yet other classifications of knowledge divide up the knowledge by the quality or degree of justification. Knowledge comes with better and worse justifications. This is a genuinely epistemic division.

The division of knowledge based on degree of certitude is ambiguous. By degree of certitude one could just mean a psychological feature of the agent, in which can this is not a properly epistemic division, or one could mean the degree of justification, in which case it is, or one could mean a combination of the two (“well-justified confidence”).

Another properly epistemic division of knowledge is with respect to content. Some differences of content do not seem to mark a genuine epistemic difference—say, the difference between biological and geological knowledge. But others do, like the difference between propositional and interpersonal knowledge do. At the same time, some of the content-based classifications include both an accidental and a properly epistemic element. Thus, it might be that interpersonal knowledge by definition must be gained through interpersonal interaction. But the epistemic benefits of interpersonal knowledge can be had without interpersonal interaction. So the concept of interpersonal knowledge may contain some epistemically accidental features about how the knowledge was in fact gained and some properly epistemic features as to the content.

This is all a bit of a mess.