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Directory:Jon Awbrey/Papers/Functional Logic : Quantification Theory (view source)
Revision as of 01:04, 19 November 2008
, 01:04, 19 November 2008→Measure for Measure
Applied to a given proposition <math>f,\!</math> the qualifiers <math>\alpha_i\!</math> and <math>\beta_i\!</math> tell whether <math>f\!</math> rests <math>\operatorname{above}\ f_i</math> or <math>\operatorname{below}\ f_i,</math> respectively, in the implication ordering. By way of example, let us trace the effects of several such measures, namely, those that occupy the limiting positions of the Tables.
Applied to a given proposition <math>f,\!</math> the qualifiers <math>\alpha_i\!</math> and <math>\beta_i\!</math> tell whether <math>f\!</math> rests <math>\operatorname{above}\ f_i</math> or <math>\operatorname{below}\ f_i,</math> respectively, in the implication ordering. By way of example, let us trace the effects of several such measures, namely, those that occupy the limiting positions of the Tables.
<math>\begin{matrix}
<center><math>\begin{matrix}
\alpha_0 f = 1 &
\alpha_0 f = 1 &
\mathit{iff} &
\mathrm{iff} &
f_0 \Rightarrow f &
f_0 \Rightarrow f &
\mathit{iff} &
\mathrm{iff} &
0 \Rightarrow f. &
0 \Rightarrow f. &
\mathrm{Therefore} &
\therefore &
\alpha_0 f = 1 &
\alpha_0 f = 1 &
\operatorname{for~all}\ f. \\
\operatorname{for~all}\ f. \\
\alpha_{15} f = 1 &
\alpha_{15} f = 1 &
\mathit{iff} &
\mathrm{iff} &
f_{15} \Rightarrow f &
f_{15} \Rightarrow f &
\mathit{iff} &
\mathrm{iff} &
1 \Rightarrow f. &
1 \Rightarrow f. &
\mathrm{Therefore} &
\therefore &
\alpha_{15} f = 1 &
\alpha_{15} f = 1 &
\mathit{iff} f = 1. \\
\mathrm{iff} f = 1. \\
\beta_0 f = 1 &
\beta_0 f = 1 &
\mathit{iff} &
\mathrm{iff} &
f \Rightarrow f_0 &
f \Rightarrow f_0 &
\mathit{iff} &
\mathrm{iff} &
f \Rightarrow 0. &
f \Rightarrow 0. &
\mathrm{Therefore} &
\therefore &
\beta_0 f = 1 &
\beta_0 f = 1 &
\mathit{iff} f = 0. \\
\mathrm{iff} f = 0. \\
\beta_{15} f = 1 &
\beta_{15} f = 1 &
\mathit{iff} &
\mathrm{iff} &
f \Rightarrow f_{15} &
f \Rightarrow f_{15} &
\mathit{iff} &
\mathrm{iff} &
f \Rightarrow 1. &
f \Rightarrow 1. &
\mathrm{Therefore} &
\therefore &
\beta_{15} f = 1 &
\beta_{15} f = 1 &
\operatorname{for~all}\ f. \\
\operatorname{for~all}\ f. \\
\end{matrix}</math>
\end{matrix}</math></center>
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.