On the Reducibility of Relations: Variations on a Theme of Peirce

The paper presents some mathematical aspects of the question of reducibility of relations. After giving a formal definition of reducibility we present the basic result (due to Herzberger) to the effect that relations of valency at least 3 are always reducible if the cardinality of the relation is at most equal to the cardinality of the underlying set (which is automatically the case if this set is infinite). In contrast to this, if the term "reduction" is given a practicable form, relations on finite sets are "generically" irreducible, as is shown by a simple counting argument. Next we discuss the question of an "intrinsic" criterion for reducibility. Finally we propose a scheme for the graphic representation of reductions.