For the disjoint chain part, first imagine all the worlds in the from (n,v) and split them into equivalence classes based on equality of v. The preference P can only compare two worlds that are in the same equivalence class, so each equivalence class will be totally ordered, but no class will be comparable to any other (hence decomposition into disjoint chains).
I thought the (n, v) thing was just an example. If all the worlds are meant to be represented via (n, v), then it seems like you are only ever allowed to give a partial preference based on whatever feature n represents, and you can never talk about any of the other features in v. This seems bad.
(I do agree that if you only give partial preferences on (n, v) worlds in the way you describe, then you get a decomposition into disjoint chains.)
I do believe the OP is talking about partial pref on (n,v) form worlds. Yeah, this seems bad in the “How do I act when looking at different v?” sense, but I get the sense that it’s not supposed to answer that question. Or at least Stuart plans to build a lot from here before it will answer that sort of question.
For the disjoint chain part, first imagine all the worlds in the from (n,v) and split them into equivalence classes based on equality of v. The preference P can only compare two worlds that are in the same equivalence class, so each equivalence class will be totally ordered, but no class will be comparable to any other (hence decomposition into disjoint chains).
I thought the (n, v) thing was just an example. If all the worlds are meant to be represented via (n, v), then it seems like you are only ever allowed to give a partial preference based on whatever feature n represents, and you can never talk about any of the other features in v. This seems bad.
(I do agree that if you only give partial preferences on (n, v) worlds in the way you describe, then you get a decomposition into disjoint chains.)
I do believe the OP is talking about partial pref on (n,v) form worlds. Yeah, this seems bad in the “How do I act when looking at different v?” sense, but I get the sense that it’s not supposed to answer that question. Or at least Stuart plans to build a lot from here before it will answer that sort of question.
That makes the math make sense, but I really object to calling this the “sensible” case.