In order to get a handle on the space of higher order propositions and eventually to carry out a functional approach to quantification theory, it serves to construct some specialized tools. Specifically, I define a higher order operator Υ, called the "umpire operator", which takes up to three propositions as arguments and returns a single truth value as the result. Formally, this so-called "multi-grade" property of Υ can be expressed as a union of function types, in the following manner: | In order to get a handle on the space of higher order propositions and eventually to carry out a functional approach to quantification theory, it serves to construct some specialized tools. Specifically, I define a higher order operator Υ, called the "umpire operator", which takes up to three propositions as arguments and returns a single truth value as the result. Formally, this so-called "multi-grade" property of Υ can be expressed as a union of function types, in the following manner: |