Changes

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.

<center><math>\begin{matrix}

{| 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.