Functions of type n are characteristic functions on n-ary relations. In Beyond the Frege Boundary , Keenan established their importance for natural language semantics, by showing that natural language has many examples of irreducible type n functions, where he called a function of type n reducible if it can be represented as a composition of functions of type 1 . We will give a normal form theorem for functions of type n , and use this to show that natural language has many examples of irreducible type n functions in a much stronger sense, where we take a function to be reducible if it can be represented as a composition of functions of lower types.
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