  First-order logic with dependent sorts, such as Makkai's first-order logic
with dependent sorts (FOLDS), or Aczel's and Belo's dependently typed
(intuitionistic) first-order logic (DFOL), may be regarded as logic enriched
dependent type theories. Categories with families (cwfs) is an established
semantical structure for dependent type theories, such as Martin-L\"of type
theory. We introduce in this article a notion of hyperdoctrine over a cwf, and
show how FOLDS and DFOL fit in this semantical framework. A soundness and
completeness theorem is proved for DFOL. The semantics is functorial in the
sense of Lawvere, and uses a dependent version of the Lindenbaum-Tarski algebra
for a DFOL theory. Agreement with standard first-order semantics is
established. Applications of DFOL to constructive mathematics and categorical
foundations are given. A key feature is a local propositions-as-types
principle.
