An Attempt to Answer Bryan's Question

I don't think you really mean that the difference between the known unknowns and the unknown unknowns is whether or not there actually is a futures market on the question. You say this:

"I think that we will not see contingent claims markets emerge in the case of unknown unknowns, because the bets are too hard to define. If you can define a bet, then you have created a known unknown. If you cannot define a bet, you have an unknown unknown."

That suggests that you think the nonexistence of futures markets for the unknowns unknowns isn't what makes them unknown unknowns; it's a symptom of the bets being "too hard to define." But then what's really doing the work in differentiating between known unknowns and unknown unknowns is whether bets are too hard to define, not whether there's actually a market. But I think Brian was asking for a more precise characterization of the unknown unknowns than saying that they're propositions on which bets are too hard to define--he wanted an answer in terms of probability theory.

As an answer to Brian's question, while maybe I should be posting this on his thread rather than here, I'd suggest the following paper by James Joyce (no relation): Joyce, J. 2005. ‘How Probabilities Reflect Evidence’ Philosophical Perspectives 19: 153-178.

The relevant part is section 6, on specificity, but you probably need to read what comes before for it to help.

Update: I posted a summary of the relevant stuff down on the original thread where Brian posted the question.

"Explain his point using standard probability language": why would anyone assume that that language is powerful enough to be used in every discourse? Hell, it is only a mathematical model.

I think "no traded contingent claim" is too narrow. People have Bayesian priors over all sorts of events, but modeling this would be too complex, so economists assume away things that hopefully don't matter. If some group, be they Austrians or whatever, want to give up model simplicity to build in structures for Bayesian updating on some dimensions, and these models do better than competing models with the data, then more power too them!

Is this problem really so hard?

Couldn't we just say that the "unknown unknowns" of a model are always either

1) Events that you didn't anticipate and thus have probability zero in the model
2) Events you didn't anticipate and thus don't fit neatly into the event space, so they make you want to redesign your model.

For an example, let's go back to the famous debate from decades ago among physicists, about the nature of light. Some say it's a particle and some say it's a wave. Let's say each scientist is polled about the relative likelihood of the two possibilities. Scientist X says: I believe there is a 75% chance we'll determine it's a particle, and 25% chance we'll find out it's a wave.

The problem with Scientist X's model is that light is both a particle and a wave. There are plenty of ways to diagnose the problem. One is to say: this event ("light is discovered to be both a particle and a wave") was implicitly given probability zero, since it was not an explicitly possible event in the space.

Bryan is surprised that we don't mentally explode when a "probability zero" event occurs. But the fact is, we don't fully believe our models. We know they are oversimplified. Maybe this bothers Bryan. So you can either:

1) Say that we do fully believe our models: then, there is some "fudge factor" event which we explicitly model, which represents the union of all events that can't be described as any of the other events.
2) Say that we don't fully believe our models: then, modelling is a two-level process. We have a model, and then we have a level of belief in the model. In this rendering, a probability-zero event according to the model is not actually a probability-zero event, because I don't fully believe the model.

The possibility of terror attacks was a "known unknown," but the specific use of planes on 9-11 was certainly an unknown unknown.

What we're talking about is hidden information, of which there is a great amount in a complex system. More, there are known knowns, known unknowns, and unknown unknowns for one individual that may not be the case for another (my example above is a case in point, as what was an unknown unknown to most Americans was in fact a known known to the al Qaida planners). An unknown unknown is an "unforeseen coincidence" or "unforseen situation." We talk about these things using these terms all the time. The fact is the world is full of unknown information -- which is why economic planning is completely impossible.

This doesn't answer Brian's point. The point is related to Austrian economists versus Neoclassicists. To answer it you need an examples where Austrians study something or add some to cases where people, "don't know what they don't know" as opposed to neoclassical economists who allegedly aren't equipped to handle such cases.

Also, lack of a contingent claims market is a really bad measure of whether or not something is an unknown unknown. There are many market failure reasons that a contingent claims market may not exist. Example: I can't by insurance that pays my legal fees in case I'm charged with murder. Though I'd like to be cheaply insured for a low probability but costly event that I know could happen, it turns out such a contract would attract too many murders.

Also, contingent claims markets entail costs. Some low probability events just aren't worth creating a market for. Ie. the probability the capital building is hit by a terrorist attack. And, of course, some of these markets are illegal.