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As a special application of this operator, we next define the absolute umpire operator, also called the ''umpire measure''.  This is a higher order proposition <math>\Upsilon_1 : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> which is given by the relation <math>\Upsilon_1 (f) = \Upsilon (1, f).\!</math>  Here, the subscript 1 on the left and the argument 1 on the right both refer to the constant proposition <math>1 : \mathbb{B}^2 \to \mathbb{B}.</math>  In most contexts where <math>\Upsilon_1\!</math> is actually applied the subscript 1 is safely omitted, since the number of arguments indicates which type of operator is intended.  Thus, we have the following identities and equivalents:

As a special application of this operator, we next define the absolute umpire operator, also called the ''umpire measure''.  This is a higher order proposition <math>\Upsilon_1 : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> which is given by the relation <math>\Upsilon_1 (f) = \Upsilon (1, f).\!</math>  Here, the subscript 1 on the left and the argument 1 on the right both refer to the constant proposition <math>1 : \mathbb{B}^2 \to \mathbb{B}.</math>  In most contexts where <math>\Upsilon_1\!</math> is actually applied the subscript 1 is safely omitted, since the number of arguments indicates which type of operator is intended.  Thus, we have the following identities and equivalents:

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| <math>\Upsilon f = \Upsilon_1 (f) = 1 \in \mathbb{B} \quad \Leftrightarrow \quad (1 (f)) = 1 \quad \Leftrightarrow \quad f = 1 : \mathbb{B}^2 \to \mathbb{B}.</math>

| <math>\Upsilon f = \Upsilon_1 (f) = 1 \in \mathbb{B}</math>

| <math>\Leftrightarrow</math>

| <math>\underline{(1~(f))} = \underline{1}</math>

| <math>\Leftrightarrow</math>

| <math>f = 1 : \mathbb{B}^2 \to \mathbb{B}.</math>

|}

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