Changes

===Measure for Measure===

===Measure for Measure===

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.

Define two families of measures:

The auxiliary notations:

| <math>\alpha_i, \beta_i : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}, i = 0 \ldots 15,</math>

: <math>\alpha_i f = \Upsilon (f_i, f),\!</math>

by means of the following formulas:

: <math>\beta_i f = \Upsilon (f, f_i),\!</math>

 

| <math>\alpha_i f = \Upsilon \langle f_i, f \rangle = \Upsilon \langle f_i \Rightarrow f \rangle,</math>

 

| <math>\beta_i f = \Upsilon \langle f, f_i \rangle = \Upsilon \langle f \Rightarrow f_i \rangle.</math>

: <math>\alpha_i, \beta_i : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B},</math>

incidentally providing compact names for the column headings of the next two Tables.

The values of the sixteen <math>\alpha_i\!</math> on each of the sixteen boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}</math> are shown in Table&nbsp;13.  Expressed in terms of the implication ordering on the sixteen functions, <math>\alpha_i f = 1\!</math> says that <math>f\!</math> is ''above or identical to'' <math>f_i\!</math> in the implication lattice, that is, <math>\ge f_i\!</math> in the implication ordering.

{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

| style="background:black; color:white" | 1

| style="background:black; color:white" | 1

|}<br>

|}<br>

{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

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

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

\end{matrix}</math></center>

\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.

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:

: <math>[| \alpha_i |] = (\alpha_i)^{-1}(1),\!</math>

 

| <math>[| \alpha_i |]\!</math>

: <math>[| \beta_i |] = (\beta_i)^{-1}(1),\!</math>

| <math>=\!</math>

 

| <math>(\alpha_i)^{-1}(1),\!</math>

: <math>[| \Upsilon_p |] = (\Upsilon_p)^{-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.