Changes

====Option 1 : Less General====

====Option 1 : Less General====

We now examine measures at the high end of the standard ordering.  Instrumental to this purpose we define a couple of higher order operators, <math>\Upsilon_1 : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> and <math>\Upsilon : (\mathbb{B}^2 \to \mathbb{B}) \times (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B},</math> both symbolized by cursive upsilon characters and referred to as the absolute and relative "umpire operators", respectively.  If either one of these operators is defined in terms of more primitive notions then the remaining operator can be defined in terms of the one first established.

We now examine measures at the high end of the standard ordering.  Instrumental to this purpose we define a couple of higher order operators, <math>\Upsilon_1 : (X \to \mathbb{B}) \to \mathbb{B}</math> and <math>\Upsilon : (X \to \mathbb{B}) \times (X \to \mathbb{B}) \to \mathbb{B},</math> both symbolized by cursive upsilon characters and referred to as the absolute and relative "umpire operators", respectively.  If either one of these operators is defined in terms of more primitive notions then the remaining operator can be defined in terms of the one first established.

The relative operator takes two propositions as arguments and reports the value "true" if the first implies the second, otherwise "false".

The relative operator takes two propositions of type <math>X \to \mathbb{B}</math> as arguments and reports the value 1 if the first implies the second, otherwise 0.

{| align="center" cellpadding="8"

{| align="center" cellpadding="8"

| <math>\Upsilon \langle e, f \rangle = 1 \quad \operatorname{if~and~only~if} \quad e \Rightarrow f.</math>

| <math>\Upsilon (e, f) = 1\!</math>

| <math>\operatorname{if~and~only~if}</math>

| <math>e \Rightarrow f.\!</math>

|}

|}

{| align="center" cellpadding="8"

{| align="center" cellpadding="8"

| <math>\Upsilon \langle e, f \rangle = 1 \quad \Leftrightarrow \quad (e (f)) = 1.</math>

| <math>\Upsilon (e, f) = 1\!</math>

| <math>\Leftrightarrow</math>

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

|}

|}

{| align="center" cellpadding="8"

{| align="center" cellpadding="8"

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

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

| <math>\Leftrightarrow</math>

| <math>\underline{(e (f))} = 1 : X \to \mathbb{B}.</math>

|}

|}