# Directory talk:Jon Awbrey/Papers/Functional Logic : Quantification Theory

The umpire measure of type $${\Upsilon : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}}\!$$ links the constant proposition $$1 : \mathbb{B}^2 \to \mathbb{B}\!$$ to a value of $$1\!$$ and every other proposition to a value of $$0.\!$$ Expressed in symbolic form:
 $$\Upsilon (f) = 1_\mathbb{B} \quad \Leftrightarrow \quad u = 1_{\mathbb{B}^2 \to \mathbb{B}}.\!$$
The umpire operator of type $${\Upsilon : (\mathbb{B}^2 \to \mathbb{B})^2 \to \mathbb{B}}\!$$ links pairs of propositions in which the first implies the second to a value of $$1\!$$ and every other pair to a value of $$0.\!$$ Expressed in symbolic form:
 $$\Upsilon (e, f) = 1 \quad \Leftrightarrow \quad e \Rightarrow f.\!$$