Changes

==Functional Quantifiers==

==Functional Quantifiers==

The '''umpire measure''' of type <math>\Upsilon : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> is a higher order proposition that holds for the constant proposition <math>1 : \mathbb{B}^2 \to \mathbb{B}</math> and fails for the rest.

The ''umpire measure'' of type <math>\Upsilon : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> assigns the constant proposition <math>1 : \mathbb{B}^2 \to \mathbb{B}</math> a value of 1 and everything else of that type a value of 0. Expressed in symbolic form:

: <math>\Upsilon p = 1 \quad \Leftrightarrow \quad p = 1.</math>

| <math>\Upsilon p = 1 \quad \Leftrightarrow \quad p = 1.</math>

The '''umpire operator''' of type <math>\Upsilon : (\mathbb{B}^2 \to \mathbb{B})^2 \to \mathbb{B}</math> is a higher order proposition that holds for ordered pairs of propositions in which the first implies the second and fails for the rest.

The ''umpire operator'' of type <math>\Upsilon : (\mathbb{B}^2 \to \mathbb{B})^2 \to \mathbb{B}</math> assigns ordered pairs of propositions in which the first implies the second a value of 1 and everything else of that type a value of 0. Expressed in symbolic form:

: <math>\Upsilon \langle p, q \rangle = 1 \quad \Leftrightarrow \quad p \Rightarrow q.</math>

| <math>\Upsilon \langle p, q \rangle = 1 \quad \Leftrightarrow \quad p \Rightarrow q.</math>

===Tables===

===Tables===