Jul 10, 2018
First, I should explain what I mean by space-like separated from you. Imagine a world that looks like a Bayesian network, and imagine that you are a node in that Bayesian network. If there is a path from you to another node following edges in the network, I will say that node is time-like separated from you, and in your future. If there is a path from another node to you, I will say that node is time-like separated from you, and in your past. Otherwise, I will say that the node is space-like separated from you.
Nodes in your past can be thought of as things that you observe. When you think about physics, it sure does seem like there are a lot of things in your past that you do not observe, but I am not thinking about physics-time, I am thinking about logical-time. If something is in your past, but has no effect on what algorithm you are running on what observations you get, then it might as well be considered as space-like separated from you. If you compute how everything in the universe evaluates, the space-like separated things are the things that can be evaluated either before or after you, since their output does not change yours or vice-versa. If you partially observe a fact, then I want to say you can decompose that fact into the part that you observed and the part that you didn't, and say that the part you observed is in your past, while the part you didn't observe is space-like separated from you. (Whether or not you actually can decompose things like this is complicated, and related to whether or not you can use the tickle defense is the smoking lesion problem.)
Nodes in your future can be thought of as things that you control. These are not always things that you want to control. For example, you control the output of "You assign probability less than 1/2 to this sentence," but perhaps you wish you didn't. Again, if you partially control a fact, I want to say that (maybe) you can break that fact into multiple nodes, some of which you control, and some of which you don't.
So, you know the things in your past, so there is no need for probability there. You don't know the things in your future, or things that are space-like separated from you. (Maybe. I'm not sure that talking about knowing things you control is not just a type error.) You may have cached that you should use Bayesian probability to deal with things you are uncertain about. You may have this justified by the fact that if you don't use Bayesian probability, there is a Pareto improvement that will cause you to predict better in all worlds. The problem is that the standard justifications of Bayesian probability are in a framework where the facts that you are uncertain about are not in any way affected by whether or not you believe them! Therefore, our reasons for liking Bayesian probability do not apply to our uncertainty about the things that are in our future! Note that many things in our future (like our future observations) are also in the future of things that are space-like separated from us, so we want to use Bayes to reason about those things in order to have better beliefs about our observations.
I claim that logical inductors do not feel entirely Bayesian, and this might be why. They can't if they are able to think about sentences like "You assign probability less than 1/2 to this sentence."