  Inspired by the theory of apartness relations of Scott, we establish a
positive theory of dissimilarity valued in an involutive quantale $\mathsf{Q}$
without the aid of negation. It is demonstrated that a set equipped with a
$\mathsf{Q}$-valued dissimilarity is precisely a symmetric category enriched in
a subquantaloid of the quantaloid of back diagonals of $\mathsf{Q}$.
Interactions between $\mathsf{Q}$-valued dissimilarities and
$\mathsf{Q}$-valued similarities (which are equivalent to $\mathsf{Q}$-valued
equalities in the sense of H{\"o}hle--Kubiak) are investigated with the help of
lax functors. In particular, it is shown that similarities and dissimilarities
are interdefinable if $\mathsf{Q}$ is a Girard quantale with a hermitian and
cyclic dualizing element.
