Coherent rejection functions for arbitrary things

This paper investigates how to characterize (axiomatize) coherent rejection functions for arbitrary objects. In this very general setting, we assume that there is some sensible notion of preference over these objects, the existence of an “objective” background order treated as a constraint on preferences, and that the preferences constitute a strict partial order; we assume nothing else about the structure of the objects. The first insight is that we can represent binary comparisons of objects by ordered pairs – which are just another kind of thing; it is simple to represent coherent preference orders directly in terms of these higher-order objects. Once we have coherence axioms for sets of desirable things of this kind, we can immediately see what the corresponding coherence axioms are for sets of desirable sets (SDS) of these things. But these are not in one-to-one correspondence with rejection functions for the original things; they express more. The second insight is that rejection functions do correspond (almost) exactly to objects we will call “set preferences”. So I give coherence axioms for set preferences, which equivalently fully characterize coherence for rejection functions. I present two main results: (1) coherence axioms for set preferences, and (2) the connection between coherent rejection functions for the original things and coherence for SDS of the ordered-pair things.