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===Measure for Measure===
===Measure for Measure===
Define two families of measures:
Define two families of measures:
\operatorname{for~all}\ f. \\
\operatorname{for~all}\ f. \\
\end{matrix}</math></center>
\end{matrix}</math></center>
<br>
Thus, <math>\alpha_0 = \beta_{15}\!</math> is a totally indiscriminate measure, one that accepts all propositions <math>f : \mathbb{B}^2 \to \mathbb{B},</math> whereas <math>\alpha_{15}\!</math> and <math>\beta_0\!</math> are measures that value the constant propositions <math>1 : \mathbb{B}^2 \to \mathbb{B}</math> and <math>0 : \mathbb{B}^2 \to \mathbb{B},</math> respectively, above all others.
Thus, <math>\alpha_0 = \beta_{15}\!</math> is a totally indiscriminate measure, one that accepts all propositions <math>f : \mathbb{B}^2 \to \mathbb{B},</math> whereas <math>\alpha_{15}\!</math> and <math>\beta_0\!</math> are measures that value the constant propositions <math>1 : \mathbb{B}^2 \to \mathbb{B}</math> and <math>0 : \mathbb{B}^2 \to \mathbb{B},</math> respectively, above all others.
Finally, in conformity with the use of the fiber notation to indicate sets of models, it is natural to use notations like:
Finally, in conformity with the use of the fiber notation to indicate sets of models, it is natural to use notations like:
{| align="center" cellpadding="8"
| <math>[| \alpha_i |]\!</math>
| <math>=\!</math>
| <math>(\alpha_i)^{-1}(1),\!</math>
|-
| <math>[| \beta_i |]\!</math>
| <math>=\!</math>
| <math>(\beta_i)^{-1}(1),\!</math>
|-
| <math>[| \Upsilon_p |]\!</math>
| <math>=\!</math>
| <math>(\Upsilon_p)^{-1}(1),\!</math>
|}
to denote sets of propositions that satisfy the umpires in question.
to denote sets of propositions that satisfy the umpires in question.
