  Regular logic can be regarded as the internal language of regular categories,
but the logic itself is generally not given a categorical treatment. In this
paper, we understand the syntax and proof rules of regular logic in terms of
the free regular category FRg(T) on a set T. From this point of view, regular
theories are certain monoidal 2-functors from a suitable 2-category of contexts
-- the 2-category of relations in FRg(T) -- to that of posets. Such functors
assign to each context the set of formulas in that context, ordered by
entailment. We refer to such a 2-functor as a regular calculus because it
naturally gives rise to a graphical string diagram calculus in the spirit of
Joyal and Street. We shall show that every natural category has an associated
regular calculus, and conversely from every regular calculus one can construct
a regular category.
