Changes

For example, consider the case where <math>X = \mathbb{B}.</math>  Then there are exactly four propositions <math>F : \mathbb{B} \to \mathbb{B},</math> and exactly sixteen higher order propositions that are based on this set, all bearing the type <math>m : (\mathbb{B} \to \mathbb{B}) \to \mathbb{B}.</math>

For example, consider the case where <math>X = \mathbb{B}.</math>  Then there are exactly four propositions <math>F : \mathbb{B} \to \mathbb{B},</math> and exactly sixteen higher order propositions that are based on this set, all bearing the type <math>m : (\mathbb{B} \to \mathbb{B}) \to \mathbb{B}.</math>

Table&nbsp;10 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion:  Columns&nbsp;1 and 2 form a truth table for the four <math>F : \mathbb{B} \to \mathbb{B},</math> turned on its side from the way that one is most likely accustomed to see truth tables, with the row leaders in Column&nbsp;1 displaying the names of the functions <math>F_i,\!</math> for <math>i\!</math> = 1 to 4, while the entries in Column&nbsp;2 give the values of each function for the argument values that are listed in the corresponding column head.  Column&nbsp;3 displays one of the more usual expressions for the proposition in question.  The last sixteen columns are topped by a collection of conventional names for the higher order propositions, also known as the ''measures'' <math>m_j,\!</math> for <math>j\!</math> = 0 to 15, where the entries in the body of the Table record the values that each <math>m_j\!</math> assigns to each <math>F_i.\!</math>

Table&nbsp;1 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion:  Columns&nbsp;1 and 2 form a truth table for the four <math>F : \mathbb{B} \to \mathbb{B},</math> turned on its side from the way that one is most likely accustomed to see truth tables, with the row leaders in Column&nbsp;1 displaying the names of the functions <math>F_i,\!</math> for <math>i\!</math> = 1 to 4, while the entries in Column&nbsp;2 give the values of each function for the argument values that are listed in the corresponding column head.  Column&nbsp;3 displays one of the more usual expressions for the proposition in question.  The last sixteen columns are topped by a collection of conventional names for the higher order propositions, also known as the ''measures'' <math>m_j,\!</math> for <math>j\!</math> = 0 to 15, where the entries in the body of the Table record the values that each <math>m_j\!</math> assigns to each <math>F_i.\!</math>

{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"

{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"

|+ '''Table 7.  Higher Order Propositions (n = 1)'''

|+ '''Table 1.  Higher Order Propositions (''n'' = 1)'''

|- style="background:paleturquoise"

|- style="background:paleturquoise"

| \ ''x'' || 1 0 || ''F''

| \ ''x'' || 1 0 || ''F''

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I am going to put off explaining Table 8, that presents a sample of what I call "Interpretive Categories for Higher Order Propositions", until after we get beyond the 1-dimensional case, since these lower dimensional cases tend to be a bit "condensed" or "degenerate" in their structures, and a lot of what is going on here will almost automatically become clearer as soon as we get even two logical variables into the mix.

I am going to put off explaining Table&mdash;2, that presents a sample of what I call ''interpretive categories'' for higher order propositions, until after we get beyond the 1-dimensional case, since these lower dimensional cases tend to be a bit ''condensed'' or ''degenerate'' in their structures, and a lot of what is going on here will almost automatically become clearer as soon as we get even two logical variables into the mix.

{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"

{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"

|+ '''Table 8.  Interpretive Categories for Higher Order Propositions (n = 1)'''

|+ '''Table 2.  Interpretive Categories for Higher Order Propositions (''n'' = 1)'''

|- style="background:paleturquoise"

|- style="background:paleturquoise"

|Measure||Happening||Exactness||Existence||Linearity||Uniformity||Information

|Measure||Happening||Exactness||Existence||Linearity||Uniformity||Information

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====Higher Order Propositions and Logical Operators (n <nowiki>=</nowiki> 2)====

====Higher Order Propositions and Logical Operators (''n'' = 2)====

By way of reviewing notation and preparing to extend it to higher order universes of discourse, let us first consider the universe of discourse ''X''° = [''X''] = [''x''₁, ''x''₂] = [''x'', ''y''], based on two logical features or boolean variables ''x'' and ''y''.

By way of reviewing notation and preparing to extend it to higher order universes of discourse, let us first consider the universe of discourse ''X''° = [''X''] = [''x''₁, ''x''₂] = [''x'', ''y''], based on two logical features or boolean variables ''x'' and ''y''.

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 [''x'', ''y''] may be regarded as a gallery of sixteen venn diagrams, while the measures ''m'' : (''X'' &rarr; '''B''') &rarr; '''B''' 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 m_j partitions the gallery of pictures into two aesthetic portions, the pictures (''m''_''j'')^–1(1) that ''m''_''j'' likes and the pictures (''m''_''j'')^–1(0) that ''m''_''j'' 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 [''x'', ''y''] may be regarded as a gallery of sixteen venn diagrams, while the measures ''m'' : (''X'' &rarr; '''B''') &rarr; '''B''' 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 m_j partitions the gallery of pictures into two aesthetic portions, the pictures (''m''_''j'')^–1(1) that ''m''_''j'' likes and the pictures (''m''_''j'')^–1(0) that ''m''_''j'' dislikes.

There are 2¹⁶ = 65536 measures of the type ''m'' : ('''B'''² &rarr; '''B''') &rarr; '''B'''.  Table 9 introduces the first 24 of these measures in the fashion of the higher order truth table that I used before.  The column headed "''m''_''j''" shows the values of the measure ''m''_''j'' on each of the propositions ''f''_''i'' : '''B'''² &rarr; '''B''', for ''i'' = 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 ''j'' of the measure ''m''_''j'' is the decimal equivalent of the bit string that is associated with ''m''_''j'''s functional values, which can be obtained in turn by reading the ''j''^th 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 2¹⁶ = 65536 measures of the type ''m'' : ('''B'''² &rarr; '''B''') &rarr; '''B'''.  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 "''m''_''j''" shows the values of the measure ''m''_''j'' on each of the propositions ''f''_''i'' : '''B'''² &rarr; '''B''', for ''i'' = 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 ''j'' of the measure ''m''_''j'' is the decimal equivalent of the bit string that is associated with ''m''_''j'''s functional values, which can be obtained in turn by reading the ''j''^th 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:lightcyan; font-weight:bold; text-align:center; width:90%"

{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"

|+ '''Table 9.  Higher Order Propositions (n = 2)'''

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

|- style="background:paleturquoise"

|- style="background:paleturquoise"

| align=right | ''x'' : || 1100 || ''f''

| align=right | ''x'' : || 1100 || ''f''