I think me adding more details will clear things up.
The setup presupposes a certain amount of realism. Start with Possible Worlds Semantics, where logical propositions are attached to / refer to the set of possible worlds in which they are true. A hypothesis is some proposition. We think of data as getting some proposition (in practice this is shaped by the methods/tools you have to look at and measure the world), which narrows down the allowable possible worlds consistent with the data.
Now is the part that I think addresses what you were getting at. I don’t think there’s a direct analog in my setup to your (a). You could consider the hypothesis/proposition, “the set of all worlds compatible with the data I have right now”, but that’s not quite the same. I have more thoughts, but first, do you still want feel like you idea is relevant to the setup I’ve described?
That does seem to change things… Although I’m confused about what simplicity is supposed to refer to, now.
In a pure bayesian version of this setup, I think you’d want some simplicity prior over the worlds, and then discard inconsistent worlds and renormalize every time you encounter new data. But you’re not speaking about simplicity of worlds, you’re speaking about simplicity of propositions, right?
Since a propositions is just a set of worlds, I guess you’re speaking about the combined simplicity of all the worlds. And it makes sense that that would increase if the proposition is consistent with more worlds, since any of the worlds would indeed lead to the proposition being true.
So now I’m at “The simplicity of a proposition is proportional to the prior-weighted number of worlds that it’s consistent with”. That’s starting to sound closer, but you seem to be saying that “The simplicity of a proposition is proportional to the number of other propositions that it’s consistent with”? I don’t understand that yet.
(Also, in my formulation we need some other kind of simplicity for the simplicity prior.)
I’m currently turning my notes from this class into some posts, and I’ll wait to continue this until I’m able to get those up. Then, hopefully, it will be easier to see if this notion of simplicity is lacking. I’ll let you know when that’s done.
I think me adding more details will clear things up.
The setup presupposes a certain amount of realism. Start with Possible Worlds Semantics, where logical propositions are attached to / refer to the set of possible worlds in which they are true. A hypothesis is some proposition. We think of data as getting some proposition (in practice this is shaped by the methods/tools you have to look at and measure the world), which narrows down the allowable possible worlds consistent with the data.
Now is the part that I think addresses what you were getting at. I don’t think there’s a direct analog in my setup to your (a). You could consider the hypothesis/proposition, “the set of all worlds compatible with the data I have right now”, but that’s not quite the same. I have more thoughts, but first, do you still want feel like you idea is relevant to the setup I’ve described?
That does seem to change things… Although I’m confused about what simplicity is supposed to refer to, now.
In a pure bayesian version of this setup, I think you’d want some simplicity prior over the worlds, and then discard inconsistent worlds and renormalize every time you encounter new data. But you’re not speaking about simplicity of worlds, you’re speaking about simplicity of propositions, right?
Since a propositions is just a set of worlds, I guess you’re speaking about the combined simplicity of all the worlds. And it makes sense that that would increase if the proposition is consistent with more worlds, since any of the worlds would indeed lead to the proposition being true.
So now I’m at “The simplicity of a proposition is proportional to the prior-weighted number of worlds that it’s consistent with”. That’s starting to sound closer, but you seem to be saying that “The simplicity of a proposition is proportional to the number of other propositions that it’s consistent with”? I don’t understand that yet.
(Also, in my formulation we need some other kind of simplicity for the simplicity prior.)
I’m currently turning my notes from this class into some posts, and I’ll wait to continue this until I’m able to get those up. Then, hopefully, it will be easier to see if this notion of simplicity is lacking. I’ll let you know when that’s done.