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Directory:Jon Awbrey/Papers/Functional Logic : Inquiry and Analogy (view source)
Revision as of 18:44, 17 November 2009
, 18:44, 17 November 2009→Measure for Measure: refprmat display
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.
{| align="center" cellpadding="8"
\alpha_0 f = 1 &
|
\mathrm{iff} &
<math>\begin{matrix}
f_0 \Rightarrow f &
\alpha_0 f = 1
\mathrm{iff} &
& \text{iff} &
0 \Rightarrow f. &
f_0 \Rightarrow f
\therefore &
& \text{iff} &
\alpha_0 f = 1 &
0 \Rightarrow f,
\operatorname{for~all}\ f. \\
& \text{hence} &
\alpha_{15} f = 1 &
\alpha_0 f = 1
\mathrm{iff} &
& \text{for all}~ f.
f_{15} \Rightarrow f &
\\
\mathrm{iff} &
\alpha_{15} f = 1
1 \Rightarrow f. &
& \text{iff} &
\therefore &
f_{15} \Rightarrow f
\alpha_{15} f = 1 &
& \text{iff} &
\mathrm{iff} f = 1. \\
1 \Rightarrow f,
\beta_0 f = 1 &
& \text{hence} &
\mathrm{iff} &
\alpha_{15} f = 1
f \Rightarrow f_0 &
& \text{iff}~ f = 1.
\mathrm{iff} &
\\
f \Rightarrow 0. &
\beta_0 f = 1
\therefore &
& \text{iff} &
\beta_0 f = 1 &
f \Rightarrow f_0
\mathrm{iff} f = 0. \\
& \text{iff} &
\beta_{15} f = 1 &
f \Rightarrow 0,
\mathrm{iff} &
& \text{hence} &
f \Rightarrow f_{15} &
\beta_0 f = 1
\mathrm{iff} &
& \text{iff}~ f = 0.
f \Rightarrow 1. &
\\
\therefore &
\beta_{15} f = 1
\beta_{15} f = 1 &
& \text{iff} &
\operatorname{for~all}\ f. \\
f \Rightarrow f_{15}
\end{matrix}</math></center>
& \text{iff} &
f \Rightarrow 1,
& \text{hence} &
\beta_{15} f = 1
& \text{for all}~ f.
\end{matrix}</math>
|}
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.