I am studying along your Research Agenda, and I am very excited about your whole plan.
As for this one, I am puzzled that how to formalize preferences like this together, “I prefer apples to both bananas and peaches”, since the function l is one-to-one here. In contrast, the model you proposed in the “Toy model piece #1: Partial preferences revisited” deals with this quite easily.
Does this imply I prefer X apples to Y bananas and Z pears, where Y+Z=X?
If it’s just for a single fruit, I’d decompose that preference into two separate ones? Apple vs Banana, Apple vs Pear.
Sorry for my vague expressions here. What I try to say is that “I prefer apples to bananas and I prefer apples to peaches”. My original thought is that: If this statement is formalized in a single world, since it is not clear that whether I prefer bananas to peaches, it seems that the function l has to map apples to bananas and peaches at the same time, which violates its one-to-one property.
But maybe I also asked a bad question: I mistook the definition of partial preferences for any simple statement about preferences, and tried to apply the model proposed in this post to the “composite” preferences, which actually expressed two preferences.