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.

Nonsummativism about group belief

Here is a quick argument that a group can believe something no individual does. You hire a team of three consultants to tell you whether a potential employee, Alice, is smart and honest. The team takes on the task. The team leader first leads a discussion as to which of the other two team members is best qualified to investigate which attribute, and unanimous agreement is reached on that question. Both of these then investigate and come to a decision. The team leader writes “Alice is” on a piece of paper, and then passes the piece of paper around to the second team member, who writes down the attribute she investigated or its negation, depending on what she found, followed by “and”. The leader then passes the piece of paper to the third team member, who writes down the attribute they investigated or its negation, followed by a period, without reading (and hence being biased by) what was written already. Job done, the leader without reading folds the paper in half and hands it to you, saying: “Here’s what we think.”

You open the paper and read the verdict of the consulting team: “Alice is smart and not honest.” The team agrees unanimously that the division of labor was the right way to produce an epistemically responsible group verdict, but nobody on the consulting team believes or even knows the verdict. The team leader has no opinions on Alice: she delegated the opinions to the intelligence and integrity experts. The intelligence expert has no view on Alice’s integrity and vice versa.

One could say that the team doesn’t believe its verdict. But to issue a verdict that one does not believe is to fail in sincerity. But there need be no failure in the above procedures.

(My own view is that when we say the team “believes” something, we are using “believes” in an analogical sense. But the points stand.)

Group "belief"

Even though nobody thinks Strong AI has been achieved, we attribute beliefs to computer systems and software:

• Microsoft Word thinks that I mistyped that word.

• Google knows where I’ve been shopping.

The attribution is communicatively useful and natural, but is not literal.

It seems to me, however, that the difference in kind between the beliefs of computers and the beliefs of persons is no greater than the difference in kind between the beliefs of groups and the beliefs of persons.

Given this, the attribution of beliefs to groups should also not be taken to be literal.

Groups and roles

I’ve had a grad student, Nathan Mueller, do an independent study in social epistemology in the hope of learning from him about the area (and indeed, I have learned much from him), so I’ve been thinking about group stuff once a week (at least). Here’s something that hit me today during our meeting. There is an interesting disanalogy between individuals and groups. Each group is partly but centrally defined by a role, with different groups often having different defining roles. The American Philosophical Association has a role defined by joint philosophical engagement, while the Huaco Bowmen have a role defined by joint archery. But this is not the case for individuals. While individuals have roles, the only roles that it is very plausible to say that they are partly and centrally defined by are general roles that all human beings have, roles like human being or child of God.

This means that if we try to draw analogies between group and individual concepts such as belief or intention, we should be careful to draw the analogy between the group concept and the concept as it applies not just to an individual but to an individual-in-a-role. Thus, the analogy is not between, say, the APA believing some proposition and my believing some proposition, but between the APA believing some proposition and my believing that proposition qua father (or qua philosopher or qua mathematician).

If this is right, then it suggests an interesting research program: Study the attribution of mental properties to individuals-in-roles as a way of making progress on the attribution of analogous properties to groups. For instance, there are well-founded worries in the social epistemology literature about simple ways of moving from the belief of the members of the group to the belief of the group (e.g., attributing to the group any belief held by the majority of the members). These might be seen to parallel the obvious fact that one cannot move from my believing p to my believing p qua father (or qua mathematician). And perhaps if we better understand what one needs to add to my believing p to get that I believe p qua father, this addition will help us understand the group case.

(I should say, for completeness, that my claim that the only roles that human beings are partly and centrally defined by are general roles like human being is controversial. Our recent graduate Mengyao Yan in her very interesting dissertation argues that we are centrally defined by token roles like child of x. She may even be right about the specific case of descent-based roles like child of x, given essentiality of origins, but I do not think it is helpful to analyze the attribution of mental properties to us in general in terms of us having these roles.)

Educational institutions and football

In the light of the brain damage resulting from football, it is a serious question whether it's morally permissible to participate in or support the sport at all. Still, one can make a case that there are human excellences that this sport provides a particularly good opportunity for (I am grateful to Dan Johnson for this point), and the brain damage is an unintended side-effect, so there might be a defense of the sport in general on the basis of the Principle of Double Effect.

But I think it is particularly difficult to defend educational institutions supporting this sport among students. For the defining task of an educational institution is to develop the minds of the students. But brain damage harms the individual precisely in respect of mental functioning. And it is much harder for an organization to justify an activity that has among its side-effects serious harm to the goods pursuit of which defines the organization.

Gutting on Church authority

Gary Gutting has an interesting opinion piece where he argues that the Bishops don't have the right to define the teachings of the Catholic Church for the purposes of American political discussion, because most American Catholics disagree with them on matters like contraception.

Imagine the Tall Persons' Club, where by well-established and generations-old tradition, the executive council is made up of the three tallest members, and the president is the tallest member. I voluntarily join the Tall Persons' Club, because I love many of its traditionally established activities, such as the annual cleaning of the giraffe enclosure in the local zoo, the discounted tickets to basketball games and the spectacular fireworks on Robert Pershing Wadlow's birthday.

However, I believe that the governing structure is an unfortunate one, because I think (a) height does not correlate with intelligence, (b) a focus on absolute rather than group-relative height is unfair to some ethnic groups, and (c) we should also do more for ostrich conservation than the present leadership does. Moreover, many members are with me on this. But nonetheless, by voluntarily joining the club, I have given its three tallest members a certain right to speak on my behalf on club-related matters. This is particularly true if there are other clubs that engage in similar activities but have a governing structure closer to what I like.

There are a number of important disanalogies, of course. For instance, one might believe that membership in the Catholic Church is necessary for eternal salvation. If one believes that, then one will have a very serious reason to be a member of the Church no matter how much one disagrees with the Magisterium, and the voluntariness that was essential to my story about the Tall Persons' Club is decreased. However, I don't know of any Catholics who disagree with the Magisterium on contraception who think that membership in the Catholic Church is necessary for salvation.

Another disanalogy is that many people become members of the Catholic Church not by their own choice, but by infant baptism (which, as I think Augustine notes, emphasizes that salvation is not by works). However, given a pluralistic society like ours, they are at least typically remaining in the Church voluntarily.

What counts as "the opinion of a group" is a really tough question. But it certainly isn't determined by looking at what the majority believe. For instance, it is false to say that it is the opinion of the Music Department that the earth goes around the sun, though no doubt that is the opinion of the majority of the members of the Music Department. It is not the opinion of the Music Department because the Music Department has not come to this view by the established methods for forming a corporate view of a matter proper to the Music Department. So majority opinion is not a sufficient condition for group opinion. Nor is it a necessary condition for something to be the opinion of a group that the majority believe it, even in the case of an institution whose traditional governance is by simple majority vote. A group can come up with a joint compromise proposition, approved by a majority vote, where in fact no one individual in the group endorses the proposition in its entirety (whether it is ever morally licit to vote in favor of a group resolution to endorse a proposition one takes to be false is a different question).

(Also, the following rather interesting thing can happen in a group. There may be two groups with the same or almost the same membership but with different governance structures, and opinions, preferences and decisions will then be differently attributable to the two groups. For instance, there may be the Music Department as an academic department and the Music Department as a social group. Perhaps the Music Department as a social group likes a particular brand of beer, but that preference is not of the Music Department as an academic group unless they vote for it in a Department meeting. It could be that there is the Tall Persons' Club as such and the Tall Persons' Club as a majority-governed group of individuals. We should then say that ostrich protection is a goal of the second group but not of the first.)

Furthermore, those of us who at least in principle like the idea of constitutional democracies (or monarchies, for that matter--I am Canadian, after all) should not say that the authority of a group derives from synchronic endorsement by the members. For it is a crucial feature (and very important for protecting minorities) of a constitutional system that it persists in authority even when at a particular time the majority fail to respect that authority (in this way, it is like marriage; one also thinks of Ulysses tied to the mast). The military oath in the United States is, importantly, an oath to protect the Constitution, not the present preferences and choices of the American people.

But I am out of my depth in the social/political philosophy stuff.