A First-order conditional logic with qualitative statistical semantics

We define a first-order conditional logic in which conditionals, such as α → β, are interpreted as saying that normal/common/typical objects which satisfy α satisfy β as well. This qualitative 'statistical' interpretation is achieved by imposing additional structure on the domain of a single first-order model in the form of an ordering over domain elements and tuples. α → β then holds if all objects with property α whose ranking is minimal satisfy β as well. These minimally ranked objects represent the typical or common objects having the property α. This semantics differs from that of the more common subjective interpretation of conditionals, in which conditionals are interpreted over sets of standard first-order structures. Our semantics provides a more natural way of modelling qualitative statistical statements, such as 'typical birds fly', or 'normal birds fly'. We provide a sound and complete axiomatization of this logic, and we show that it can be given probabilistic semantics.