Changes

Throwing in the lower default value permits the following abbreviations:

Throwing in the lower default value permits the following abbreviations:

{| celpadding="4"

{| cellpadding="4"

| align="right" width="36" | 3.

| align="right" width="36" | 3.

| <math>\Upsilon  q  = \Upsilon (q) = \Upsilon_1 q = \Upsilon (1, q, \textstyle\prod).</math>

| <math>\Upsilon  q  = \Upsilon (q) = \Upsilon_1 q = \Upsilon (1, q, \textstyle\prod).</math>

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}

\alpha_0 f = 1             &

| <math>\alpha_{00} f = 1\!</math>

\mathit{iff}               &

| iff   || <math>f_{00} \Rightarrow f,</math>

f_0 \Rightarrow f         &

| iff   || <math>0 \Rightarrow f,</math>

\mathit{iff}              &

| hence || <math>\alpha_{00} f = 1\ \operatorname{for~all}\ f.</math>

0 \Rightarrow f.          &

\mathrm{Therefore}         &

\alpha_0 f = 1             &

| <math>\alpha_{15} f = 1\!</math>

\operatorname{for~all}\ f. \\

| iff  || <math>f_{15} \Rightarrow f,</math>

\alpha_{15} f = 1         &

| iff   || <math>1 \Rightarrow f,</math>

\mathit{iff}              &

| hence || <math>\alpha_{15} f = 1 \Rightarrow f = 1.</math>

f_{15} \Rightarrow f       &

\mathit{iff}              &

| &nbsp;

1 \Rightarrow f.          &

| <math>\beta_{00} f = 1\!</math>

| iff  || <math>f \Rightarrow f_{00},</math>

\alpha_{15} f = 1         &

| iff  || <math>f \Rightarrow 0,</math>

\mathit{iff} f = 1.       \\

| hence || <math>\beta_{00} f = 1 \Rightarrow f = 0.</math>

\beta_0 f = 1              &

\mathit{iff}               &

f \Rightarrow f_0          &

| <math>\beta_{15} f = 1\!</math>

\mathit{iff}               &

| iff  || <math>f \Rightarrow f_{15},</math>

f \Rightarrow 0.          &

| iff   || <math>f \Rightarrow 1,</math>

\mathrm{Therefore}         &

| hence || <math>\beta_{15} f = 1\ \operatorname{for~all}\ f.</math>

\beta_0 f = 1             &

|}<br>

\mathit{iff} f = 0.       \\

\beta_{15} f = 1           &

\mathit{iff}              &

f \Rightarrow f_{15}       &

\mathit{iff}              &

f \Rightarrow 1.          &

\beta_{15} f = 1           &

\operatorname{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.