Changes

Finally, it is often convenient to write the first argument as a subscript, hence <math>\Upsilon_e \langle f \rangle = \Upsilon \langle e, f \rangle.</math>

Finally, it is often convenient to write the first argument as a subscript, hence <math>\Upsilon_e \langle f \rangle = \Upsilon \langle e, f \rangle.</math>

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 \langle f \rangle = \Upsilon \langle 1, f \rangle.</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>

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 \langle f \rangle = \Upsilon \langle 1, f \rangle.</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 reference to "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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====Option 2 : More General====

====Option 2 : More General====