Conditional logics play an important role in recent attempts to
investigate default reasoning. This paper investigates
first-order conditional logic. We show that, as for first-order
probabilistic logic, it is important not to confound statistical
conditionals over the domain (such as `most birds fly'), and
subjective conditionals over possible worlds (such as `I believe that
Tweety is unlikely to fly'). We then address the issue of ascribing
semantics to first-order conditional logic. As in the propositional case,
there are many possible semantics. To study the problem in a coherent way,
we use plausibility structures. These provide us with a general
framework in which many of the standard approaches can be embedded. We show
that while these standard approaches are all the same at the propositional
level, they are significantly different in the context of a first-order
language. We show that plausibilities provide the most natural extension of
conditional logic to the first-order case: We provide a sound and complete
axiomatization that contains only the KLM properties and standard axioms of
first-order modal logic. We show that most of the other approaches have
additional properties, which result in an inappropriate treatment of an
infinitary version of the lottery paradox.