  We introduce a new form of logical relation which, in the spirit of metric
relations, allows us to assign each pair of programs a quantity measuring their
distance, rather than a boolean value standing for their being equivalent. The
novelty of differential logical relations consists in measuring the distance
between terms not (necessarily) by a numerical value, but by a mathematical
object which somehow reflects the interactive complexity, i.e. the type, of the
compared terms. We exemplify this concept in the simply-typed lambda-calculus,
and show a form of soundness theorem. We also see how ordinary logical
relations and metric relations can be seen as instances of differential logical
relations. Finally, we show that differential logical relations can be
organised in a cartesian closed category, contrarily to metric relations, which
are well-known not to have such a structure, but only that of a monoidal closed
category.
