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==Functional Quantifiers==
==Functional Quantifiers==
The '''relative umpire operator''' <math>\Upsilon : (\mathbb{B}^2 \to \mathbb{B})^2 \to \mathbb{B}</math> takes two propositions as arguments and gives the value <math>1\!</math> if and only if the first implies the second. In symbols:
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.
: <math>\Upsilon \langle e, f \rangle = 1 \quad \operatorname{iff} \quad e \Rightarrow f.</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.
: <math>\Upsilon \langle p, q \rangle = 1 \quad \Leftrightarrow \quad p \Rightarrow q.</math>
===Tables===
===Tables===
The auxiliary notations:
The auxiliary notations:
: <math>\alpha_i f = \Upsilon (f_i, f),\!</math>
: <math>\alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f)</math>
: <math>\beta_i f = \Upsilon (f, f_i),\!</math>
: <math>\beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)</math>
define two series of measures:
define two series of measures:
