  In the category of sets and partial functions, $\mathsf{PAR}$, while the
disjoint union $\sqcup$ is the usual categorical coproduct, the Cartesian
product $\times$ becomes a restriction categorical analogue of the categorical
product: a restriction product. Nevertheless, $\mathsf{PAR}$ does have a usual
categorical product as well in the form $A \& B := A \sqcup B \sqcup (A \times
B)$. Surprisingly, asking that a distributive restriction category (a
restriction category with restriction products $\times$ and coproducts
$\oplus$) has $A \& B$ a categorical product is enough to imply that the
category is a classical restriction category. This is a restriction category
which has joins and relative complements and, thus, supports classical Boolean
reasoning. The first and main observation of the paper is that a distributive
restriction category is classical if and only if $A \& B := A \oplus B \oplus
(A \times B)$ is a categorical product in which case we call $\&$ the
''classical'' product.
  In fact, a distributive restriction category has a categorical product if and
only if it is a classified restriction category. This is in the sense that
every map $A \to B$ factors uniquely through a total map $A \to B \oplus
\mathsf{1}$, where $\mathsf{1}$ is the restriction terminal object. This
implies the second significant observation of the paper, namely, that a
distributive restriction category has a classical product if and only if it is
the Kleisli category of the exception monad $\_ \oplus \mathsf{1}$ for an
ordinary distributive category.
  Thus having a classical product has a significant structural effect on a
distributive restriction category. In particular, the classical product not
only provides an alternative axiomatization for being classical but also for
being the Kleisli category of the exception monad on an ordinary distributive
category.
