12,080
edits
Changes
MyWikiBiz, Author Your Legacy — Friday April 04, 2025
Jump to navigationJump to searchDirectory:Jon Awbrey/Papers/Functional Logic : Quantification Theory (view source)
Revision as of 03:08, 14 December 2008
, 03:08, 14 December 2008→Measure for Measure: reset formula display
Line 542:
Line 542:
An acquaintance with the functions of the umpire operator can be gained from Tables 4 and 5, where the 2-dimensional case is worked out in full.
: <math>[| \alpha_i |] = (\alpha_i)^{-1}(1),\!</math>+ +: <math>[| \beta_i |] = (\beta_i)^{-1}(1),\!</math>+ +: <math>[| \Upsilon_p |] = (\Upsilon_p)^{-1}(1),\!</math>+
===Measure for Measure===
===Measure for Measure===
−Define two families of measures:
Define two families of measures:
Line 1,287:
Line 1,285:
\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.
Line 1,292:
Line 1,291:
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.
