Reinstall Microsoft Store on Windows 10

This is really just a note to self for future reference. I had uninstalled Microsoft Store very completely on my Win10 laptop, but then wanted it back. None of the solutions I found online worked out of the box. Finally, here is what worked (a variant of something I found online):

1. Go to https://store.rg-adguard.net/
2. Enter in the URL: https://www.microsoft.com/store/productId/9WZDNCRFJBMP and click on the checkmark
3. Download the latest WindowsStore*.AppxBundle and WindowsStore*.BlockMap files
4. Start powershell in admin mode
5. cd [your download directory]
6. get-item *WindowsStore*.appx* | Add-AppxPackage

Another KN95 mask mod

As an experiment, I cut the ear loops of one of my KN95 masks, and then made them join very snugly behind the head with rubber bands. I found that both comfort and fit improved greatly: I tested this while rock climbing. I expect this might work for surgical-style masks as well.

Important note: If you do this, strengthen the ear loop joints with Shoe Goo or some other goopy adhesive first, or they will likely not withstand the added pressure.

An optional improvement I made later is to add a clasp to the lower strap to make it easier to put on despite the loop being tight. Here are my 3D printing files. If you don't have a 3D printer, you could probably make a clasp out of a paperclip.

This is a lot of work for a disposable mask. But I reuse them.

The inhumanity problem for morality

When a state legislates, it often carves out very specific exceptions to the legislation. Sometimes, of course, one is worried that the exceptions are a sign that the legislators are pursuing special interests rather than the common good, but sometimes the exceptions are quite reasonable. For instance, you shouldn’t possess child pornography… except, say, if you are involved in the law enforcement process and need it as evidence to get the child pornographers. There is something ugly about carving out exceptions, but the point is to make society work well rather than make the laws elegant. Special-case clauses seem to be unavoidable in practice, given the messiness and complexity of human life. Elegant exceptionless legislation—with some important exceptions!—is apt to be inhuman.

I kind of wonder if an analogous thing might not be true in the case of morality, and for the same reason, the messiness and complexity of human life. Could it be that elegant exceptionless moral laws would necessarily have to be inhuman?

What solutions are available to this problem?

Well, we might just dig in our heels, either optimistically or pessimistically.

The optimistic version says: yes, we have elegant exceptionless moral laws, and they do work well for us. One way of running the optimistic variant is to make the moral laws leave a lot to human positive law. Thus, there are going to be exceptions to any prohibition of theft, but perhaps morality leaves the specification of this to the state. Or perhaps one could be really optimistic and have moral laws that do not leave a lot to positive law, but nonetheless they work. Act utilitarianism could be thought to provide this kind of solution, having a simple rule “Maximize utility!”, but its problem is that this rule is just wrong. Rule utilitarianism provides a nicer solution by having the elegant meta-rule “Do those things that fall under a utility-maximizing rule”, but I think the technical details here are insuperable.

The pessimistic variant says: yes, we have elegant exceptionless moral laws, and we’re stuck with that, even though it doesn’t work that great for us. That might be a better way to take act utilitarianism, but such pessimism is not a very attractive approach.

But what if we don’t want to dig in our heels? One could think that there are just brute (perhaps metaphysically necessary) facts about the moral rules, and many of these brute facts have specific exceptions: “Don’t lie, except to save a life or to prevent torture.” I think bruteness, and especially inelegant bruteness, is a last resort.

One might think that moral particularism is a solution: there are general elegant moral laws, but they all have unspecified exceptions. They say things like: “Don’t torture, other things being equal.” There is still a fact of the matter as to what to do in a particular situation, a fact that a virtuous agent may be able to discern, but these facts cannot be formulated in a general way, because any finite description of the particular situation will leave out factors that could in some other case trump the described considerations. There are exceptionless moral rules on such a view, but they are infinite in length. Unless some story is given as to where these infinite rules come from, this seems like it might be just an even worse version of the brute fact story.

Divine command theory, on the other hand, could provide a very nice solution to the problem, exactly analogous to the legislative solution. If God is the author of moral laws, he can legislate: “Thou shalt not kill, except in cases of types A, B and C.”

Natural law could also provide such a solution, at least given theism: God could select for instantiation a nature that has a complex teleology with various specific exceptions.

Where do I fall? I think I want to hold out for a two-level theistic natural law story. On one level, there is a simple, single and elegant moral rule embedded in our nature: “Love everything!” However, the content of that love is specified in a very complex way by our nature and by the circumstances (love needs to be appropriate to the specifics of the relationships). This specification is embedded in our nature by much more complex rules. And God chose this nature for instantiation because it works so well.

Making KN95 masks better

I have found (using a small single blind test) that disposable masks have better audio quality for teaching: cloth muffles the voice, especially I think an already somewhat muddy male voice like mine. Surgical-style masks have a lot of leaks around the edges, so I went with cheap ebay KN95 masks.

I reuse them, "disinfected" by airing for seven days (as recommended for N95 masks by the N95's inventor), leaving them hanging on a little wooden rack with nails in it. (I had some fun with my CNC router on it as you can see.)

The cheap KN95s have some air leaking around the edges and top, and occasionally I've had an earloop break off. So, here are three mods I've made. You can do the first two without special equipment, but the third requires a 3D printer. (But if you're someone in my social circle at Baylor, I could 3D print some for you.)

First: add a bit of Shoe Goo under, around and over where the earloops meet the mask. This seems to greatly increase the strength of the connection. No broken off earloops since.

Second: I added some rubber bands joining the earloops. The main point was to make the fit more snug, reducing air leaking. Sadly, it puts more pressure on the ears. (There is probably a sweet spot in rubber band length where it reduces pressure on the ears, but I'm putting safety over comfort.)

Third: I replaced the flimsy metal nosepiece with a hefty 3D printed one. It took me five prototypes until I got the size and shape right, but then it worked great in teaching. It's 2mm thick, printed in PLA, and glued on with Shoe Goo. (The metal strips came off very easily from at least one of the brands of cheap KN95s.) No more fiddling with fitting the nosepiece, and no more feeling of air going up and out along the nose bridge, so I expect it increased the protection for others from exhalation. I don't normally need to wear glasses with a mask, but I tested with my sunglasses while walking home from class and found no fogging. I still fiddle with and adjust the mask, but a nice bonus is that I mainly need to touch the plastic strip, which is probably cleaner than the filtering surface.

I don't know that the strip increased the protection for me as significantly, because the KN95s already would seal around the face when breathing in. (There have been a lot of claims made that masks protect others more than they protect the wearer. I am somewhat skeptical of this in the case of KN95s and well-fitted cloth masks, because my experience is that when you breathe in, fitted masks pull to the face and seal much more tightly than when breathing out.)

My 3D printing files are here. Unless I have an identical twin I don't know about, you'll need to edit the OpenSCAD files and tweak the Bezier parameters to make them work for you. Mine I did mainly by trial and error with five prototypes, but when I made one for my son, I had him press a wire around the bridge of his nose, and then scanned the wire along with a ruler for size, and traced Beziers over the wire (if you're in my Baylor social circle, I can do this for you, on the basis of a good photo of a bent wire and a ruler or other calibrating object).

The coincidence between the right and the beneficial

One of the earliest and most important discoveries in Western philosophy was:

1. Doing the right thing is sometimes bad for you (in non-moral ways).

This precludes any easy reduction of morality to self-interest. But at the same time, the philosophers could see that:

2. Doing the right thing is usually good for you (even in non-moral ways).

This leads to an interesting problem that has occasionally been discussed, but not as much as one might hope:

3. What explains why acting morally well tends to be good for you (even in non-moral ways).

Living in accordance with one’s conscience, while having that conscience be well-formed, tends to lead to a kind attractive happiness that we can often see in people. There is that smile which reflects both a kindliness of nature and an inner joy. Why is there this harmony between the right and the beneficial?

If we were non-realists about morality, we might give an evolutionary explanation: our moral beliefs evolved to benefit us. But if we are realists about morality, then that only makes it puzzling: why is it that the true moral beliefs are the ones that tend to benefit us?

Divine command ethics has a plausible story grounded in God’s loving desire that we live under moral rules that are good for us. Natural law ethics has a different story: our natures are harmonious, and hence the many ends we have are mutually supportive. That story, of course, only shifts the problem to the more general question of the mutual support of our ends, and I suspect that this more general question cannot be answered without bringing in theism, either by positing that God is more likely to instantiate harmonious natures or that because created natures are way of participating in the God whose inner life is a harmony of love, they tend to be (or maybe even all are) harmonious.

Sofas and uncaused events

Consider three worlds where a sofa rises and in each of which Alice and Bob have the same lifting powers, and nothing beyond Alice and Bob influences the sofa’s rise.

• w₁: the sofa is lifted by Alice and Bob, with neither of them sufficiently power to lift the sofa on their own

• w₂: the sofa rises without Alice or Bob exerting their lifting abilities

• w₃: the sofa rises with Alice exerting her lifting abilities.

If we think w₁ and w₂ are both possible, we should think that w₃ is also possible. It would be too weird if eliminating both Alice and Bob’s exertions were compatible with the sofa rising, but somehow keeping Alice’s exertions precluded the sofa from rising.

But in w₃, Alice can’t be the cause of the sofa’s rising. But she seems to have the same influence on it as in w₁. So it seems she is a merely partial cause of the sofa’s rising.

However, Alice can’t be a merely partial cause of the sofa’s rising without being a part of a full cause of the sofa rising. But nothing else influences the sofa’s rise. So, there is no full cause, and yet Alice can’t be a merely partial cause.

Thus, w₃ is impossible. But if w₁ and w₂ are possible, so is w₃. So, w₂ is impossible. So, uncaused events are impossible.

When can we have exact symmetries of hyperreal probabilities?

In many interesting cases, there is no way to define a regular hyperreal-valued probability that is invariant under symmetries, where “regular” means that every non-empty set has non-zero probability. For instance, there is no such measure for all subsets of the circle with respect to rotations: the best we can do is approximate invariance, where P(A)−P(rA) is infinitesimal for every rotation. On the other hand, I have recently shown that there is such a measure for infinite sequences of fair coin tosses where the symmetries are reversals at a set of locations.

So, here’s an interesting question: Given a space Ω and a group G of symmetries acting on Ω, under what exact conditions is there a hyperreal finitely-additive probability measure P defined for all subsets of Ω that satisfies the regularity condition P(A)>0 for all non-empty A and yet is fully (and not merely approximately) invariant under G, so that P(gA)=P(A) for all g ∈ G and A ⊆ Ω?

Theorem: Such a measure exists if and only if the action of G on Ω is locally finite. (Assuming the Axiom of Choice.)

The action of G on Ω is locally finite iff for every x ∈ Ω and every finitely-generated subgroup H of G, the orbit Hx = {hx : h ∈ H} of x under H is finite. In other words, we have such a measure provided that applying the symmetries to any point of the space only generates finitely many points.

This mathematical fact leads to a philosophical question: Is there anything philosophically interesting about those symmetries whose action is locally finite? But I’ve spent so much of the day thinking about the mathematical question that I am too tired to think very hard about the philosophical question.

Sketch of Proof of Theorem: If some subset A of Ω is equidecomposable with a proper subset A′, then a G-invariant measure P will assign equal measure to both A and A′, and hence will assign zero measure to the non-empty set A − A′, violating the regularity condition. So, if the requisite measure exists, no subset is equidecomposable with a proper subset of itself, which by a theorem of Scarparo implies that the action of G is locally finite.

Now for the converse. If we could show the result for all finitely-generated groups G, by using ultraproduct along an ultrafilter on the partially ordered set of all finitely generated subgroups of G we could show this for a general G.

So, suppose that G is finitely generated and the orbit of x under G is finite for all x ∈ Ω. A subset A of G is said to be G-invariant provided that gA = A for all g ∈ G. The orbit of x under G is always G-invariant, and hence every finite subset of A is contained in a finite G-invariant subset, namely the union of the orbits of all the points in A.

Consider the set F of all finite G-invariant subsets of Ω. It’s worth noting that every finite subset of G is contained in a finite G-closed subset: just take the union of the orbits under G. For A ∈ F, let P_A be uniform measure on A. Let F^* = {{B ∈ F : A ⊆ B}:A ∈ F}. This is a non-empty set with the finite intersection property. Let U be an ultrafilter extending F^*. Let ^*R be the ultraproduct of the reals over F with respect to U, and let P(C) be the equivalence class of the function A ↦ P_A(A ∩ C) on F. Note that C ↦ P_A(A ∩ C) is G-invariant for any G-invariant set A, so P is G-invariant. Moreover, P(C)>0 if C ≠ ∅. For let C′ be the orbit of some element of C. Then {B ∈ F : C′⊆B} is in F^*, and P_A(A ∩ C′) > 0 for all A such that C′⊆A, so the set of all A such that P_A(A ∩ C′) > 0 is in U. It follows that P(C′) > 0. But C′ is the orbit of some element x of C, so every singleton subset of C′ has the same P-measure as {x} by the G-invariance of P. So P({x}) = P(C′)/|C′| > 0, and hence P(C)≥P({x}) > 0.

Invariance under independently chosen random transformations

Often, a probabilistic situation is invariant under some set of transformations, in the sense that the complete probabilistic facts about the situation are unchanged by the transformation. For instance, in my previous post I suggested that a sequence of fair coin flips should be invariant under the transformation of giving a pre-specified subset of the coins an extra turn-over at the end and I proved that we can have this invariance in a hyperreal model of the situation.

Now, a very plausible thesis is this:

Randomized Invariance: If a probabilistic situation S is invariant under each member of some set T of transformations, then it is also invariant under the process where one chooses a random member of T independently of S and applies that member to S.

For instance, in the coin flip case, I could choose a random reversing transformation as follows: I line up (physically or mentally) the infinite set of coins with an independent second infinite set of coins, flip the second set of coins, and wherever that flip results in heads, I reverse the corresponding coin in the first set.

By Randomized Invariance, doing this should not change any of the probabilities. But insisting on this case of Randomized Invariance forces us to abandon the idea that we should assign such things as an infinite sequence of heads a non-zero but infinitesimal probability. Here is why. Consider a countably infinite sequence of fair coins arranged equidistantly in a line going to the left and to the right. Fix a point r midway between two successive coins. Now, use the coins to the left of r to define the random reversing transformation for the coins to the right of r: if after all the coins are flipped, the nth coin to the left of r is heads, then I give an extra turn-over to the nth coin to the right of r.

According to Randomized Invariance, the probability that all the coins to the right of r will be tails after the random reversing transformations will be the same as the probability that they were all tails before it. Let p be that probability. Observe that after the transformations, the coins to the right of r are all tails if and only if before the transformations the nth coin to the right and the nth coin to the left showed the same thing (for we only get tails on the nth coin on the right at the end if we had tails there at the beginning and the nth coin on the left was tails, or if we had heads there at the beginning, but the heads on the nth coin to the left forced us to reverse it). Hence, p is also the probability that the corresponding coins to the left and right of r showed the same thing before the transformation.

Thus, we have shown that the probability that all the paired coins on the left and right equidistant to r are the same (i.e., we have a palindrome centered at r) is the same as the probability that we have only tails to the right of r. Now, apply the exact same argument with “right” and “left” reversed. We conclude that the probability that the coins on the right and left equidistant to r are always the same is the same as the probability that we have only tails to the left of r. Hence, the probability of all-tails to the left of r is the same as the probability of all-tails to the right of r.

And this argument does not depend on the choice of the midpoint r between two coins. But as we move r one coin to the right, the probability of all-tails to the right of r is multiplied by two (there is one less coin that needs to be tails) and the probability of all-tails to the left of r is multiplied by a half. And yet these numbers have to be equal as well by the above argument. Thus, 2p = p/2. The only way this can be true is if p = 0.

Therefore, Randomized Invariance, plus the thesis that all the non-random reversing transformations leave unchanged the probabilistic situation (a thesis made plausible by the fact that even with infinitesimal probabilities, we provably can have a model of the probabilities that is invariant under these transformation), shows that we must assign probability zero to all-tails, and infinitesimal probabilities are mistaken.

This is, of course, a highly convoluted version of Timothy Williamson’s coin toss argument. The reason for the added complexity is to avoid any use of shift-based transformations that may be thought to beg the question against advocates of non-Archimedean probabilities. Instead, we simply use randomized reversal symmetry.

Hyperreal modeling of infinitely many coin flips

A lot of my work in philosophy of probability theory has been devoted to showing that one cannot use technical means to get rid of certain paradoxes of infinite situations. As such, most of the work has been negative. But here is a positive result. (Though admittedly it was arrived at in the service of a negative result which I hope to give in a future post.)

Consider the case of a (finite or infinite, countable or not) sequence of independent fair coin flips. Here is an invariance feature we would like to have for our coin flips. Suppose that ahead of time, I designate a (finite or infinite) set of locations in the infinite sequence. You then generate the sequence of independent fair coin flips, and I go through my pre-designated set of locations, and turn over each of the coins corresponding to that location. (For instance, if you will make a sequence of four coin flips, and I predesignate the locations 1 and 3, and you get HTTH, then after my extra flipping set the sequence of coin flips becomes TTHH: I turned over the first and third coins.) The invariance feature we want is that no matter what set of locations I predesignate, it won’t affect the probabilistic facts about the sequence of independent fair coin flips.

This invariance feature is clearly present in finite cases. It is also present if “probabilistic facts” are understood according to classical countably-additive real-valued probability theory. But what if we have infinitely many coins, and we want to be able to do things like comparing the probability of all the coins being heads to all the even-numbered coins being heads, and say that the latter is more likely than the former, with both probabilities being infinitesimal? Can we still have our reversal-invariance property for all predesignated sets of locations?

There are analogous questions for other probabilistic situations. For instance, for a spinner, the analogous property is adding an extra predesignated rotation to the spinner once the spinner stops, and it is well-known that one cannot have such invariance in a context that gives us “enough” infinitesimal probabilities (e.g., see here for a strong and simple result).

But the answer is positive for the coin flip case: there is a hyperreal-valued probability defined for all subsets of the set of sequences (with fixed index set) of heads and tails that has the reversal-invariance property for every set of locations.

This follows from the following theorem.

Theorem: Assume the Axiom of Choice. Let G be a locally finite group (i.e., every finite subset generates a finite subgroup) and suppose that G acts on some set X. Then there is a hyperreal finitely additive probability measure P defined for all subsets of X such that P(gA)=P(A) for every A ⊆ X and g ∈ G and P(A)>0 for all non-empty A.

To apply this theorem to the coin-flip case, let G be the abelian group whose elements are sets of locations with the exclusive-or operation (i.e., A ⊕ B = (A − B)∪(B − A) is the set of all locations that are in exactly one of A and B). The identity is the empty set, and every element has order two (i.e., A ⊕ A = ∅). But for abelian groups, the condition that every finite subset generates a finite subgroup is equivalent to the condition that every element has finite order (i.e., some finite multiple of it is zero).

Mathematical notes: The subgroup condition on G in the Theorem entails that every element of G has finite order, but is stronger than that in the non-abelian case (due to the non-trivial fact that there are infinite finitely generated torsion groups). In the special case where X = G, the condition that every element of G have finite order is necessary for the theorem. For if g has infinite order, let A = {gⁿ : n ≥ 0}, and note that gA is a proper subset of A, so the condition that non-empty sets get non-zero measure and finite additivity would imply that P(gA)<P(A), which would violate invariance. It is an interesting question whether the condition that every finite subset generates a finite subgroup is also necessary for the Theorem if X = G.

Proof of Theorem: Let F be the partially ordered set whose elements are pairs (H, V) where H is a finite subgroup of G and V is a finite algebra of subsets of X closed under the action of H, with the partial ordering (H₁, V₁)≼(H₂, V₂) if and only if H₁ ⊆ H₂ and V₁ ⊆ V₂.

Given (H, V) in F, let B_V be the basis of V, i.e., a subset of pairwise disjoint non-empty elements of V such that every element of V is a union of (finitely many) elements of B_V. For A ∈ B_V and g ∈ H, note that gA is a member of V since V is closed under the action of H. Thus, gA = B₁ ∪ ... ∪ B_n for distinct elements B₁, ..., B_n in B_V. I claim that n = 1. For suppose n ≥ 2. Then g⁻¹B₁ ⊆ A and g⁻¹B₂ ⊆ A, and yet both g⁻¹B₁ and g⁻¹B₂ are members of V by H-closure. But since A is a basis element it follows that g⁻¹B₁ = A = g⁻¹B₂, and hence B₁ = B₂, a contradiction. Thus, n = 1 and hence gA ∈ B_V. Moreover, if gA = gB then A = B, so each member g of H induces a bijection of B_V onto itself.

Now let P_(H, V) be the probability measure on V that assigns equal probability to each member of B_V. Since each member of H induces a bijection of B_V onto itself, it’s easy to see that P_(H, V) is an H-invariant probability measure on V. And, for convenience, if A ∉ V, write P_(H, V)(A)=0.

Let F^* = {{B ∈ F : A ≼ B}:A ∈ F}. This is a nonempty set with the finite intersection property (it is here that we will use the fact that every finite subset of G generates a finite subgroup). Hence it can be extended to an ultrafilter U. This ultrafilter will be fine: {B ∈ F : A ≼ B}∈U for every A ∈ F. Let ^*R be the ultraproduct of the reals R over F with respect to U, i.e., the set of functions from F to R modulo U-equivalence. Given a subset A of X, let P(A) be the equivalence class of (H, V)↦P_(H, V)(A).

It is now easy to verify that P has all the requisite properties of a finitely-additive hyperreal probability that is invariant under G and assigns non-zero probability to every non-empty set.

Complete Probabilistic Characterizations

Consider the concept of a complete probabilistic characterization (CPC) of an experiment. It’s a bit of a fuzzy concept, but we can get some idea about it. For instance, if I have a coin loaded in favor of heads, then saying that heads is more likely than tails is not a CPC. Minimally, the CPC will give exact numbers where the probabilities have exact numbers. But the CPC may go beyond giving numerical probabilities. For instance, if you toss infinitely main fair coins, the numerical probability that they are all heads is zero as is the probability that all the even numbered ones are heads. But intuitively it is more likely that the even numbered ones are heads than that all of them are heads. If there is something to this intuition, the CPC will include the relevant information: it may do that by assigning different infinitesimal probabilities to the two events, or by giving conditional probabilities conditioned on various zero-probability events.

A deep question that has sometimes been discussed by philosophers of probability is what CPCs are like. Here are three prominent candidates:

1. classical real-valued probabilities

2. hyperreal probabilities assigning non-zero (but perhaps infinitesimal) probability to every possible event

3. primitive conditional probabilities allowing conditioning on every possible event.

The argument against (1) and for (2) and (3) is that (1) doesn’t distinguish things that should be distinguished—like the heads case above. I want to offer an argument against (2) and (3), however.

Here is a plausible principle:

4. If X and Y are measurements of two causally independent experiments, then the CPC of the pair (X, Y) is determined by the CPCs of X and Y together with the fact of independence.

If (4) is true, then a challenge for a defender of a particular candidate for CPC is to explain how the CPC of the pair is determined by the individual CPCs of the independent experiments.

In the case of (1), the challenge is easily met: the pair (X, Y) has as its probability measure the product of the probability measures for X and Y.

In the cases of (2) and (3), the challenge has yet to be met, and there is some reason to think it cannot be met. In this post, I will argue for this in the case of (2): the case of (3) follows from the details of the argument in the case of (2) plus the correspondence between Popper functions and hyperreal probabilities.

Consider the case where X and Y are uniformly distributed over the interval [0, 1]. By independence, we want the pair (X, Y) to have a hyperreal finitely additive probability measure P such that P(X ∈ A, Y ∈ B)=P(X ∈ A)P(Y ∈ B) for all events A and B. But it turns out that this requirement on P highly underdetermines P. In particular, it seems to be that for any positive real number r, we can find a hyperreal measure P such that P(X ∈ A, Y ∈ B)=P(X ∈ A)P(Y ∈ B) for all A and B, and such that P(X = Y)=rP(Y = 0). Hence, independence highly underdetermines what value P assigns to the diagonal X = Y as compared to the value it assigns to Y = 0.

Maybe some other conditions can be added that would determine the CPC of the pair. But I think we don’t know what these would be. As it stands, we don’t know how to determine the CPC of the pair in light of the CPC of the members of the pair, if CPCs are of type (2).

Product spaces for hyperreal and full conditional probabilities

I think the following is a consequence of a hyperreal variant of the Horn-Tarski extension theorem for measures on boolean algebras:

Claim: Suppose that <Ω_i, F_i, P_i> for i ∈ I is a finitely additive probability space with values in some field R^* of hyperreals. Then, assuming the Axiom of Choice, there is a hyperreal-valued finitely additive probability space <Ω, 2^Ω, P> where Ω = ∏_i ∈ IΩ_i and where the Ω_i-valued random variables π_i given by the natural projections of Ω to Ω_i are independent and have the distributions given by the P_i.

Note that the values of P might be in a hyperreal field larger than R^*.

Given the Claim, and given the well-known correspondences between hyperreal-valued probabilities and full conditional real-valued probabilities, it follows that we can define meaningful product-space conditional real-valued probabilities.

It would be really nice if the product-space conditional probabilities were unique in the special case where F_i is the power set of Ω_i, or at least if they were close enough to uniqueness to define the same real-valued conditional probabilities.

For a particularly interesting case, consider the case where X and Y are generated by uniform throws of a dart at the interval [0, 1], and we have a regular finitely additive hyperreal-valued probability on [0, 1] (regular meaning that all non-empty sets have positive measure). Let Z be the point (X, Y) in the unit square.

Looking at how the proof of the Horn-Tarski extension theorem works, it seems to me that for any positive real number r, and any non-trivial line segment L along the x = y diagonal in the square [0, 1]², there is a product measure P satisfying the conditions of the Claim (where P₁ and P₂ are the uniform measures on [0, 1]) such that P(L)=rP(H), where H is the horizontal line segment {(x, 0):x ∈ [0, 1]}. For instance, if L is the full diagonal, we would intuitively expect P(L)=2^1/2P(H), but in fact we can make P(L)=100000P(H) or P(L)=P(H)/100000 if we like. It is clear that such a discrepancy will generate different conditional probabilities.

I haven’t checked all the details yet, so this could be all wrong.

But if it is right, here is a philosophical upshot. We would expect there to be a unique canonical product probability for independent random variables. However, if we insist on probabilities that are so fine-grained as to tell infinitesimal differences apart, then we do not at present have any such unique canonical product probability. If we are to have one, we need some condition going beyond independence.

This is part of a larger set of claims, namely that we do not at present have a clear notion of what “uniform probability” means once we make our probabilities more finegrained than classical real-valued probability.

Putative Sketch of Proof of Claim: Embedding R^* in a larger field if necessary, we may assume that R^* is |2^Ω|-saturated. Define a product measure on the cylinder subsets of Ω as usual. The proof of the Horn-Tarski extension theorem for measures on boolean algebras looks to me like it works for |B|-saturated hyperreal-valued probability measures where B is the boolean algebra, and completes the proof of our claim.

Real dilemmas, alas

I’ve been trying to avoid holding there are real moral dilemmas—ones where one is genuinely morally required to do something and to abstain from it. But here is a problem:

1. One is obligated to do what one believes to be obligatory.

2. Some people believe that ϕing is obligatory and that refraining from ϕing is obligatory.

3. So, some people are obligated to ϕ and not to ϕ.

Perhaps the most obvious case of (2) is killing in war. It seems to be not an uncommon view that (a) all killing is wrong, but (b) you should kill to defend the innocent in a just war.

Mistaken conscience and failure

Alice is a sniper tasked with stopping Bob the terrorist who is about to set up a bomb that will kill many. Let’s take it for granted that shooting Bob would have been permissible and even praiseworthy. Now, Alice takes all reasonable precautions but she misidentifies Carl the innocent as the terrorist and shoots Carl.

Among the infinitely many ways that we can describe Alice’s action, two are of particular moral relevance:

1. Trying to shoot Bob the terrorist.

2. Shooting an innocent person.

Alice is morally responsible for, and even praiseworthy, for performing (1). She is not responsible for (2), since she did (2) unintentionally and in non-culpable ignorance (remember that she took all reasonable precautions).

Did Alice do a morally impermissible action? It sounds like (2) is impermissible, and Alice indisputably did it. But perhaps this is too quick. For it is not clear to me that Alice’s shooting an innocent person is an action. Suppose that while Alice was sleeping, an evil tinkerer set up a pressure-sensitive switch connected to a gun pointed at David the innocent, and Alice rolled over onto it. Then Alice shot David, but we cannot say that she did anything: shooting David wasn’t an action, but a mere event. And if she didn’t do anything, she didn’t do anything impermissible.

Now, Alice’s trying to shoot Bob the terrorist identical with her shooting an innocent person. And since Alice’s trying to shoot Bob is an action, it follows that her shooting the innocent person is also an action. So it seems that Alice did do something impermissible.

But even this may not be quite right. For it may be that it is not quite right to say that (2) is impermissible. Rather, what are impermissible are actions that are non-accidental cases of shooting an innocent. And both the shooting of Carl and of David are accidental cases (and that of David isn’t even an action).

If this is right, then we can say that Alice did nothing wrong in either the case of Carl or of David.

Now, let’s switch to a harder case. Alice has a reasonable but false belief that she is pursuing a just war, but she is not. She shoots Ella the enemy combatant. Did Alice do anything morally wrong? It seems that she did: she shot Ella. But perhaps we can say something very similar to what we said above. There are two ways to describe Alice’s action;

3. Trying to shoot enemy combatant Ella in pursuit of a just war

4. Shooting enemy combatant Ella not in pursuit of a just war.

Action (1) is permissible, but unbeknownst to Alice was doomed to failure as the war was not just. Now, what is impermissible is non-accidentally shooting enemy combatants not in pursuit of a just war. But Alice did this accidentally: she reasonably thought it was a just war. So perhaps Alice is entirely off the hook for doing something morally wrong. Instead, she accidentally did something that it would be have been wrong to do non-accidentally.

Let’s switch to an even harder case. Alice has a reasonable (given her flawed upbringing and culture) but false belief that in order to save lives in the pursuit of a just war it is permissible to shoot innocent non-combatants, and she shoots Fred the innocent non-combatant. Can we say that Alice didn’t do anything wrong, but merely accidentally did something that it would have been wrong to do non-accidentally? Perhaps. Perhaps we can describe Alice’s action in two ways:

5. Trying to permissibly shoot the innocent non-combatant Fred to save lives

6. Shooting the innocent non-combatant Fred to save lives.

Action (5) is permissible, but doomed to failure. And it is impermissible to non-accidentally do (6). But now it seems that we cannot make the move of saying that Alice only accidentally did (6). For she was trying to do (6), though she was trying to do more than just what is included in (6): she was trying to do (6) permissibly.

But perhaps there is a similar move possible to the one we made before. Perhaps what is impermissible is to do (6) as such, where the “as such” includes both non-accidentality and the assumption that no further relevant factors are involved. And Alice wasn’t trying to do (6) as such: she was trying to do (6) permissibly.

If so, this would give us a nice account of what happens in cases of honestly mistaken conscience. We are intending to do something permissibly, and we fail at this. Instead we accidentally end up performing only a part of our intention. That part would be something that it would be impermissible to attempt as such, but we didn’t attempt it as such, but we intended it only qua permissible.

For this account to work, it has to be the case that if we are to act well, we should positively include permissibility among our intentions. Virtue may help here.

Physicalism and vice

1. If physicalism is true, then vice is an instance of medium-to-long-term poor bodily function.

2. Instances of medium-to-long-term poor bodily function are illnesses.

3. Vice is not an illness.

4. So, physicalism is not true.

The Non-Identity Theodicy (Scott Hill) SCP session

The Analytic Collective and SCP are having an inaugural online session on August 21 at 4-5:30 pm Eastern Time to discuss Scott Hill's fascinating paper "The Non-Identity Theodicy". I will be commenting on the paper.

Abstract: This paper defends a theodicy based on ideas discussed in the literature on the non-identity problem and the literature on origin essentialism. I then address a series of objections about the ethics of God's acts in my theodicy and about the metaphysics of origins on which my theodicy depends.

Join the Analytic Collective facebook group for a Zoom link.

This is a pre-read session. The paper is here. And my comments are here.