Plausibility = Just Double | DoNotKnow
Recently I gave a talk on “Probability theory as an extension of logic” where I stated the Cox’s theorem.
The first assumption of the theorem is that plausibilities are real numbers.
The following objection to the first assumption was raised:
Why should the plausibility be a real number rather than just “I don’t know” — Olly
Such idea hasn’t been ignored by the people who developed this theory, but let us first remind ourselves of the purpose of the exercise: to process the incomplete information optimally.
Hence, if you suspect that you don’t have enough information for a decision to be made and you can postpone making the decision than spare your brain from unnecessary computations.
Another way to put it: garbage in, garbage out.
But what if your decision can’t wait?
So you have a tough decision to make. The deadline is looming and you
decided to get a robot to help you. And so let us pretend that we managed to
develop the thinking robot based on the principles of Cox’s theorem with the
modified first axiom. We also taught it to process English, tapped into your
brain and uploaded your life experience. And after such expensive exercise, you
finally ask the robot the question: what should I do, Mr. Robot? And the robot
goes 
for 6 hours and then spits out
DoNotKnow. And so you are left with making the decision yourself. Useless.
“I don’t know” is implied
The reason why we resort to the use of probability theory in reasoning is because we don’t know. You give the information to the robot and wait for its judgement. If it comes out with \(p = 0\) or \(p = 1\) it means that you know. There is no uncertainty: it’s just a matter of chaining logical steps together. Once \(0 < p < 1\) then you don’t know1.
Refusal to verbalise probability assignments is just assigning them implicitly
If you say that the universe didn’t give you enough information to do action A so you choose to do B it has given you enough information.
Day after day we are taking the plunge and actually do something. Whether it’s a decision to cycle to work or the refusal to do so, whether it’s a decision to join effective altruism movement or a refusal to do so. Whether it’s a decision to stop eating meat or a refusal to do so. There is no such thing as “unresolved issue”. Once you are aware of the issue but choose to ignore it you resolved it.
So the criticism of the first assumption boils down the criticism of forcing your resolutions to be explicit rather than implicit.
Further readings
Having to pick from “I don’t know” or a real number is somewhat akin to using a two-dimensional theory, where second dimension is just a boolean flag.
The idea of having a two-dimensional theory for plausibilities hasn’t been ignored, but the results of a simpler, one-dimensional theory speak for themselves. The proponents of two-dimensional theories haven’t yet demonstrated practical advantage of such framework.
But for a further discussion on the necessity of two-dimensional theories, look at A guide to Cox’s Theorem pages 4-7.
Title explained
data Plausibility = Just Double | DoNotKnowThis is just a haskell-way of saying that plausibility of a statement is either a real number or “I don’t know”.