Transitive closure, proximity and intransitivities

Assignments of weak orders to complete binary relations are considered. Firstly, it is shown that assigning the transitive closure of a complete binary relation does not always assign the closest weak order according to any reasonable metric on complete binary relations. It is then shown that the assignment of a weak order to a complete binary relation assigns its transitive closure if and only if it assigns the closest weak order according to a particular distance function that is not a metric. This permits more direct comparisons between the Transitive Closure Rule and other rules such as the Slater Rule. Copyright Springer-Verlag Berlin/Heidelberg 2003