Normal forms for characteristic functions on n-ary relations

Functions of type n are characteristic functions on n-ary relations. Keenan [5] established their importance for natural language semantics, by showing that natural language has many examples of irreducible type n functions, i.e., functions of type n that cannot be represented as compositions of unary functions. Keenan proposed some tests for reducibility, and Dekker [3] improved on these by proposing an invariance condition that characterizes the functions with a reducible counterpart with the same behaviour on product relations. The present paper generalizes the notion of reducibility (a quantifier is reducible if it can be represented as a composition of quantifiers of lesser, but not necessarily unary, types), proposes a direct criterion for reducibility, and establishes a diamond theorem and a normal form theorem for reduction. These results are then used to show that every positive n function has a unique representation as a composition of positive irreducible functions, and to give an algorithm for finding this representation. With these formal tools it can be established that natural language has examples of n-ary quantificational expressions that cannot be reduced to any composition of quantifiers of lesser degree.