Changes

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 as arguments and reports the value "true" if the first implies the second, otherwise "false".

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

<math>\Upsilon \langle e, f \rangle = 1 \quad \operatorname{iff} \quad e \Rightarrow f.</math>

Expressing it another way, we may also write:

Expressing it another way, we may also write:

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

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

In writing this, however, it is important to notice that the 1's appearing on the left and right have different meanings.  Filling in the details, we have:

In writing this, however, it is important to notice that the 1's appearing on the left and right have different meanings.  Filling in the details, we have:

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

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>  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:

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

The umpire measure is defined at the level of truth functions, but can also be understood in terms of its implied judgments at the syntactic level.  Interpreted this way, <math>\Upsilon_1\!</math> recognizes theorems of the propositional calculus over <math>[u, v],\!</math> giving a score of "1" to tautologies and a score of "0" to everything else, regarding all contingent statements as no better than falsehoods.

The umpire measure is defined at the level of truth functions, but can also be understood in terms of its implied judgments at the syntactic level.  Interpreted this way, <math>\Upsilon_1\!</math> recognizes theorems of the propositional calculus over <math>[u, v],\!</math> giving a score of "1" to tautologies and a score of "0" to everything else, regarding all contingent statements as no better than falsehoods.