Conditional logics play an important role in recent attempts to investigate default reasoning. We show that a straightforward definition of first-order conditional logic using preferential structures, such as Delgrande's conditional logic, encounters several problems. In particular, these logics cannot handle exceptional individuals, a problem that is typified by an infinitary version of the lottery paradox. We take a different approach, using the notion of plausibility spaces. This is a notion that generalizes a number of different semantics for defaults, including preferential structures. As we show here in a first-order setting, plausibility spaces provide the necessary expressive power we need to deal with the lottery paradox and its variants. We claim that our approach provides the most natural extension of the KLM properties to a first-order language. In fact, we provide a sound and complete axiomatization of our logic that contains only the KLM properties and standard axioms of first-order modal logic. Plausibility spaces give us the tools to analyze the obvious first-order extensions of standard propositional approaches, and the approaches of of Delgrande and of Lehmann & Magidor, and to understand the difficulties they encounter.