Partitions and Graphs

A partition of a set is a way of dividing that set into several non-empty, non-overlapping subsets. For example, the partitions of the set {A,B,C} are as follows:

  • A|B|C
  • AB|C
  • AC|B
  • BC|A
  • ABC

The Bell numbers (OEIS/A000110) describe the number of unique partitions for a set of n elements. The sequence grows pretty fast:

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322, 1382958545, ...

Set partitions are interesting for a number of reasons. One reason is that set partitions also describe the number of equivalence relations on a set. So, for an arbitrary set there are B_n ways to treat the n elements of that set as equivalent. This puts an upper bound on measurement, since we can understand these equivalence relations as modes of indistinguishability.

Another reason is that Bell numbers also describe the number of unique component graphs for n nodes. E.g. for a simple graph with 3 labeled nodes, there are 5 unique graphs:

  • A B C
  • A-B C
  • A-C B
  • B-C A
  • A-B-C

Set partitions also describe the possible ways of dividing a set of events into equivalent outcomes for Bayesian learning. If I want to know how to think about the possible outcomes of some experiment, for example, I partition the set of possible observations. However, your choice of partition matters a lot to how you should set your credences. This suggests that you should follow a rule or be otherwise internally consistent in some way in how you partition the set of possible observations. What I'd like to know is whether we can give well-defined conditions under which this requirement is met.

A Causal Model for the Principle of Alternative Possibilities

The Principle of Alternative Possibilities says that S is morally responsible for P iff it is possible that S did not perform P.

In his 1969 book Alternate Possibilities and Moral Responsibility Harry Frankfurt introduced a much discusssed counterexample to the Principle. His counterexample runs as follows:

Suppose that Smith really wants Jones to perform some action. He sets out to ensure that Jones will perform this action. Smith resolves to wait until Jones makes up his mind about what to do before he acts. Once Jones resolves what to do, Smith will step in to ensure that Jones performs the desired action. As Smith watches, Jones deliberates and then elects to do the action Smith desired all along!

Frankfurt concludes from this example that Jones is certainly morally responsible for his action, even though it's true that Jones could not have done otherwise.

In this post I'd like to offer an explanation for the force of the counterexample based on its causal model.

The Model

An oversimplified first pass at the causal model of the situation Jones is in looks like this:

deliberation --> action

In Frankfurt's original case, Smith knows the value that deliberation will take. If it takes an certain value (say 0), then Jones will act in the way undesired by Smith. Smith observes the value that Jones takes in deliberation and then Smith will step in to ensure that action takes the desired value, in case deliberation takes the value 0. If deliberation takes the value 1, then Smith does nothing. This is effectively a plan to intervene on the variable action.

While it's true that Jones could not have done otherwise, this fact isn't due to a feature of the causal model itself (I'm just going to stipulate this, although you might think if a model like this was veridical and deterministic there would be bigger threats to Jones's freedom than Smith) but due to the way the intervention is being applied to the model. The performance of the intervention is conditional on the values for deliberation. It seems to me that the force of Frankfurt's example is that Jones is morally responsible for his action (despite being unable to do otherwise) because of the value that deliberation takes.

However, suppose that Smith intervened on deliberation instead. My reaction to this intervention is that Jones isn't responsible at all. Part of this is the way intervention gets formally modeled. There's no way for Smith to observe that deliberation has taken value 0 and then intervene "before" that value is transmitted to action. The intervention and subsequent downstream effects are simultaneous. The only way to get Frankfurt's case is to suppose another exogeneous variable exists:

brain activity --> deliberation --> action

Now Smith can observe the brain activity variable for its value and intervene on deliberation to prevent the value from getting passed downstream. I think the main problem at stake with this approach is that it begins to look less and less plausible that Frankfurt's counterexample is going to do the work of undermining the Principle. If the causal model for Jones' behavior looks like the one here, then it's already false that Jones could have done otherwise.

A few lingering thoughts:

  1. It would be interesting to see how the model would differ in a stochastic case.
  2. Is the best way to understand Smith's action as an intervention?
  3. Could Smith be intervening not on a variable in the model but on the model's structure?
Adam Edwards
Notion Template for Drafts & Notes

I started using Notion this year and my Very Short Review is that it's incredible and everyone should be using it. My only complaint is that I want to use Notion so much that it has tended to be cluttered faster than I'd hoped when I first started using it this summer.

The point of this post isn't to get you to use Notion (although you can try it for free and you use this link for $10 in credit towards a paid account), it's to document an incredibly simple and effective writing template I discovered recently that's really changed how I write.

The template is based off work by Edward Tufte and the tufte LaTeX package and in particular the tufte-handout document class in LaTeX (and Rmarkdown!). I've used the Tufte-styled handouts for a few years and I think they're fantastic. In particular, the side-margin for notes, references, graphs, and other marginalia is something I actually adopted for hand-written notes much longer ago. I adopted the Cornell method between undergrad and grad school, and anecdotally it seems to have made a big difference.

The original purpose of Tufte's design, as far as I can tell, isn't the same as the purpose of the margin in teh Cornell method. However, the basic principle of dividing your writing space into two unequal columns -- one for primary content and the other for a secondary stream -- is the same.

The Template

When I write in Notion, the template I use is based on a synthesis of these ideas. First, I create a new blank page and set the page to fullscreen width. While Notion does have built-in notes and blog post templates, these don't match how I typically think about writing notes.

Next, I add two headers: Draft and Notes. The titles of these may change depending on the purpose, but usually it's these.

Then I use the drag-and-drop functionality in Notion to create two columns from these headers and I adjust the width of the columns to reflect the Tufte/Cornell margins. From what I can tell, Notion currently divides the vertical space in a page into 16 equal parts, and I use a ratio of 13/3 on the page. The video below shows how this works in practice.


Once the draft is written and ready to be sent out or converted into another document, Notion has a built-in Markdown export for any page that will easily convert the document into human-readable Markdown. This is where the template really shines, because the "Notes" column is converted to a separate section at the bottom of the page.

Since one of the main uses of the "Notes" column is things like references and footnotes, the links from these get listed at the bottom of the Markdown page in single spot, which is really nice. Even better is using the MMD syntax for footnotes to make sure that when the document is used for other things those notes (now endnotes rather than marginalia) are associated with the correct sentence or paragraph.

Finally, I use John MacFarlane's pandoc software to convert the exported Markdown document into a *.doc, *.html, or even a *.pdf with the tufte-handout class! The footnote syntax makes sure that the notes get connected to the right part of the document, and it lets me export from Notion into any format I want.

An example chapter from my dissertation that uses this template.

Because Notion has a template-building feature with the "Button" blocks, I don't have to go through this process every time. I have a template button that creates a new page with these dimensions and then I just drag the page wherever I want it to live in Notion.

If you have your own Notion account you can see an example of the template page here.