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Finally, in conformity with the use of fiber notation to indicate sets of models, it is natural to use notations like the following to denote sets of propositions that satisfy the umpires in question.
Finally, in conformity with the use of fiber notation to indicate sets of models, it is natural to use notations like the following to denote sets of propositions that satisfy the umpires in question.
{| 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.