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Revision as of 23:00, 14 December 2008
, 23:00, 14 December 2008→Umpire Operators: style="text-align:center"
Given an ordered pair of propositions <math>e, f : \langle u, v \rangle \to \mathbb{B}</math> as arguments, the relative operator reports the value 1 if the first implies the second, otherwise 0.
Given an ordered pair of propositions <math>e, f : \langle u, v \rangle \to \mathbb{B}</math> as arguments, the relative operator reports the value 1 if the first implies the second, otherwise 0.
{| align="center" cellpadding="8"
{| align="center" cellpadding="8" style="text-align:center"
| <math>\Upsilon (e, f) = 1\!</math>
| <math>\Upsilon (e, f) = 1\!</math>
| <math>\operatorname{if~and~only~if}</math>
| <math>\operatorname{if~and~only~if}</math>
Expressing it another way, we may also write:
Expressing it another way, we may also write:
{| align="center" cellpadding="8"
{| align="center" cellpadding="8" style="text-align:center"
| <math>\Upsilon (e, f) = 1\!</math>
| <math>\Upsilon (e, f) = 1\!</math>
| <math>\Leftrightarrow</math>
| <math>\Leftrightarrow</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:
{| align="center" cellpadding="8"
{| align="center" cellpadding="8" style="text-align:center"
| <math>\Upsilon (e, f) = 1 \in \mathbb{B}</math>
| <math>\Upsilon (e, f) = 1 \in \mathbb{B}</math>
| <math>\Leftrightarrow</math>
| <math>\Leftrightarrow</math>
Writing types as subscripts and using the fact that <math>X = \langle u, v \rangle,</math> it is possible to express this a little more succinctly as follows:
Writing types as subscripts and using the fact that <math>X = \langle u, v \rangle,</math> it is possible to express this a little more succinctly as follows:
{| align="center" cellpadding="8"
{| align="center" cellpadding="8" style="text-align:center"
| <math>\Upsilon (e, f) = 1_\mathbb{B}</math>
| <math>\Upsilon (e, f) = 1_\mathbb{B}</math>
| <math>\Leftrightarrow</math>
| <math>\Leftrightarrow</math>
