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Each point in <math>X\!</math> may be described by means of a ''singular proposition'', that is, a proposition that describes it uniquely.  This form of representation leads to the following enumeration of points:

Each point in <math>X\!</math> may be indicated by means of a ''singular proposition'', that is, a proposition that describes it uniquely.  This form of representation leads to the following enumeration of points, presented in two different styles of typography that are used in this text:

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{| align="center" cellpadding="8" style="text-align:center"

| '''<math>X \quad = \quad \{\!</math>&nbsp;<code>(u)(v)</code>&nbsp;,&nbsp;<code>(u)v</code>&nbsp;,&nbsp; <code>u(v)</code>&nbsp;,&nbsp;<code>uv</code>&nbsp;<math>\} \quad \cong \quad \mathbb{B}^2.</math>'''

| <math>X\!</math>

| <math>=\!</math>

| <math>\{\!</math>'''<font size="+1">&nbsp;<code>(u)(v)</code>,&nbsp;&nbsp;<code>(u)v</code>,&nbsp;&nbsp;<code>u(v)</code>,&nbsp;&nbsp;<code>uv</code>&nbsp;</font>'''<math>\}\!</math>

| <math>\cong\!</math>

| <math>\mathbb{B}^2.</math>

|}

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Each point in <math>X\!</math> may also be described by means of its ''coordinates'', that is, by the ordered pair of values in <math>\mathbb{B}</math> that the coordinate propositions <math>u\!</math> and <math>v\!</math> take on that point.  This form of representation leads to the following enumeration of points:

Each point in <math>X\!</math> may also be described by means of its ''coordinates'', that is, by the ordered pair of values in <math>\mathbb{B}</math> that the coordinate propositions <math>u\!</math> and <math>v\!</math> take on that point.  This form of representation leads to the following enumeration of points:

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{| align="center" cellpadding="8" style="text-align:center"

| <math>X \quad = \quad \{\ (0, 0),\ (0, 1),\ (1, 0),\ (1, 1)\ \} \quad \cong \quad \mathbb{B}^2.</math>

| <math>X\!</math>

| <math>=\!</math>

| <math>\{\ (0, 0),\ (0, 1),\ (1, 0),\ (1, 1)\ \}</math>

| <math>\cong\!</math>

| <math>\mathbb{B}^2.</math>

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To save a few words in the remainder of this discussion, I will use the terms ''measure'' and ''qualifier'' to refer to all types of higher order propositions and operators.  To describe the present setting in picturesque terms, the propositions of <math>[u, v]\!</math> may be regarded as a gallery of sixteen venn diagrams, while the measures <math>m : (X \to \mathbb{B}) \to \mathbb{B}</math> are analogous to a body of judges or a panel of critical viewers, each of whom evaluates each of the pictures as a whole and reports the ones that find favor or not.  In this way, each judge <math>m_j\!</math> partitions the gallery of pictures into two aesthetic portions, the pictures <math>m_j^{-1}(1)\!</math> that <math>m_j\!</math> likes and the pictures <math>m_j^{-1}(0)\!</math> that <math>m_j\!</math> dislikes.

To save a few words in the remainder of this discussion, I will use the terms ''measure'' and ''qualifier'' to refer to all types of higher order propositions and operators.  To describe the present setting in picturesque terms, the propositions of <math>[u, v]\!</math> may be regarded as a gallery of sixteen venn diagrams, while the measures <math>m : (X \to \mathbb{B}) \to \mathbb{B}</math> are analogous to a body of judges or a panel of critical viewers, each of whom evaluates each of the pictures as a whole and reports the ones that find favor or not.  In this way, each judge <math>m_j\!</math> partitions the gallery of pictures into two aesthetic portions, the pictures <math>m_j^{-1}(1)\!</math> that <math>m_j\!</math> likes and the pictures <math>m_j^{-1}(0)\!</math> that <math>m_j\!</math> dislikes.

There are <math>2^{16} = 65536\!</math> measures of the type <math>m : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}.</math>  Table&nbsp;12 introduces the first 24 of these measures in the fashion of the higher order truth table that I used before.  The column headed <math>m_j\!</math> shows the values of the measure <math>m_j\!</math> on each of the propositions <math>f_i : \mathbb{B}^2 \to \mathbb{B},</math> for <math>i\!</math> = 0 to 23, with blank entries in the Table being optional for values of zero.  The arrangement of measures that continues according to the plan indicated here is referred to as the ''standard ordering'' of these measures.  In this scheme of things, the index <math>j\!</math> of the measure <math>m_j\!</math> is the decimal equivalent of the bit string that is associated with <math>m_j\!</math>'s functional values, which can be obtained in turn by reading the <math>j^\mathrm{th}\!</math> column of binary digits in the Table as the corresponding range of boolean values, taking them up in the order from bottom to top.

There are <math>2^{16} = 65536\!</math> measures of the type <math>m : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}.</math>  Table&nbsp;3 introduces the first 24 of these measures in the fashion of the higher order truth table that I used before.  The column headed <math>m_j\!</math> shows the values of the measure <math>m_j\!</math> on each of the propositions <math>f_i : \mathbb{B}^2 \to \mathbb{B},</math> for <math>i\!</math> = 0 to 23, with blank entries in the Table being optional for values of zero.  The arrangement of measures that continues according to the plan indicated here is referred to as the ''standard ordering'' of these measures.  In this scheme of things, the index <math>j\!</math> of the measure <math>m_j\!</math> is the decimal equivalent of the bit string that is associated with <math>m_j\!</math>'s functional values, which can be obtained in turn by reading the <math>j^\mathrm{th}\!</math> column of binary digits in the Table as the corresponding range of boolean values, taking them up in the order from bottom to top.

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

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

|+ '''Table 12.  Higher Order Propositions (''n'' = 2)'''

|+ '''Table 3.  Higher Order Propositions (''n'' = 2)'''

|- style="background:ghostwhite"

|- style="background:ghostwhite"

| align="right" | <math>u:</math><br><math>v:</math>

| align="right" | <math>u:</math><br><math>v:</math>