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Directory:Jon Awbrey/Papers/Functional Logic : Inquiry and Analogy (view source)

Revision as of 15:12, 22 June 2013

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'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''

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==Abstract==

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| [[Image:Peirce's Formulation of Analogy (Version 2).jpg|600px]]

|-

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| <math>\operatorname{Figure~8.~Peirce's~Formulation~of~Analogy~(Version~2)}</math>

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| <math>\operatorname{Figure~8.~Peirce's~Formulation~of~Analogy~(Version~2)}\!</math>

|}

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====Higher Order Propositions and Logical Operators (''n'' = 1)====

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A ''higher order proposition'' is, very roughly speaking, a proposition about propositions. If the original order of propositions is a class of indicator functions <math>f : X \to \mathbb{B},</math> then the next higher order of propositions consists of maps of the type <math>m : (X \to \mathbb{B}) \to \mathbb{B}.</math>

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A ''higher order proposition'' is, very roughly speaking, a proposition about propositions. If the original order of propositions is a class of indicator functions <math>{f : X \to \mathbb{B}},\!</math> then the next higher order of propositions consists of maps of the type <math>{m : (X \to \mathbb{B}) \to \mathbb{B}}.\!</math>

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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>

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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>

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Table 10 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion: Columns 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 1 displaying the names of the functions <math>f_i,\!</math> for <math>i\!</math> = 1 to 4, while the entries in Column 2 give the values of each function for the argument values that are listed in the corresponding column head. Column 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>

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Table 11 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion: Columns 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 1 displaying the names of the functions <math>{f_i},\!</math> for <math>{i}\!</math> = 1 to 4, while the entries in Column 2 give the values of each function for the argument values that are listed in the corresponding column head. Column 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:white; color:black; font-weight:bold; text-align:center; width:90%"

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|+ ~~'''~~Table 10. Higher Order Propositions (''n'' = 1)~~'''~~

+

|+ style="height:25px" |

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<math>\text{Table 11. Higher Order Propositions} ~~ (n = 1)\!</math>

|- style="background:ghostwhite"

| align="right" | <math>x:</math>

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I am going to put off explaining Table 11, 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.

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I am going to put off explaining Table 12, 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="font-weight:bold; text-align:center; width:90%"

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|+ ~~'''~~Table 11. Interpretive Categories for Higher Order Propositions (''n'' = 1)~~'''~~

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|+ style="height:25px" |

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<math>\text{Table 12. Interpretive Categories for Higher Order Propositions} ~~ (n = 1)~\!</math>

|- style="background:ghostwhite"

| Measure

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

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By way of reviewing notation and preparing to extend it to higher order universes of discourse, let us first consider the universe of discourse <math>X^\Box = [\mathcal{X}] = [x_1, x_2] = [u, v],</math> based on two logical features or boolean variables <math>u\!</math> and <math>v.\!</math>

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By way of reviewing notation and preparing to extend it to higher order universes of discourse, let us first consider the universe of discourse <math>{X^\Box = [\mathcal{X}] = [x_1, x_2] = [u, v]},\!</math> based on two logical features or boolean variables <math>{u}\!</math> and <math>{v}.\!</math>

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The universe of discourse <math>X^\Box\!</math> consists of two parts, a set of ''points'' and a set of ''propositions''.

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The universe of discourse <math>{X^\Box}\!</math> consists of two parts, a set of ''points'' and a set of ''propositions''.

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The points of <math>X^\Box</math> form the space:

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The points of <math>{X^\Box}\!</math> form the space:

{| align="center" cellpadding="8"

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| <math>X \quad = \quad \langle \mathcal{X} \rangle \quad = \quad \langle u, v \rangle \quad = \quad \{ (u, v) \} \quad \cong \quad \mathbb{B}^2.</math>

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| <math>{X \quad = \quad \langle \mathcal{X} \rangle \quad = \quad \langle u, v \rangle \quad = \quad \{ (u, v) \} \quad \cong \quad \mathbb{B}^2}.~\!</math>

|}

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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:

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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:

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

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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.

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There are <math>2^{16} = 65536\!</math> measures of the type <math>m : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}.</math> Table 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.

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There are <math>2^{16} = 65536\!</math> measures of the type <math>m : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}.</math> Table 13 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" style="background:white; color:black; text-align:center; width:90%"

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

+

|+ style="height:25px" |

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<math>\text{Table 13. Higher Order Propositions} ~~ (n = 2)\!</math>

|- style="background:ghostwhite"

| align="right" | <math>\begin{matrix}u\!:\\v\!:\end{matrix}</math>

| <math>\begin{matrix}1100\\1010\end{matrix}</math>

| <math>f\!</math>

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| <math>\underset{0}{m}</math>

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| <math>{\underset{0}{m}}</math>

−

| <math>\underset{1}{m}</math>

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| <math>{\underset{1}{m}}</math>

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| <math>\underset{2}{m}</math>

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| <math>{\underset{2}{m}}</math>

−

| <math>\underset{3}{m}</math>

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| <math>{\underset{3}{m}}</math>

−

| <math>\underset{4}{m}</math>

+

| <math>{\underset{4}{m}}</math>

−

| <math>\underset{5}{m}</math>

+

| <math>{\underset{5}{m}}</math>

−

| <math>\underset{6}{m}</math>

+

| <math>{\underset{6}{m}}</math>

−

| <math>\underset{7}{m}</math>

+

| <math>{\underset{7}{m}}</math>

−

| <math>\underset{8}{m}</math>

+

| <math>{\underset{8}{m}}</math>

−

| <math>\underset{9}{m}</math>

+

| <math>{\underset{9}{m}}</math>

−

| <math>\underset{10}{m}</math>

+

| <math>{\underset{10}{m}}</math>

−

| <math>\underset{11}{m}</math>

+

| <math>{\underset{11}{m}}</math>

−

| <math>\underset{12}{m}</math>

+

| <math>{\underset{12}{m}}</math>

−

| <math>\underset{13}{m}</math>

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| <math>{\underset{13}{m}}</math>

−

| <math>\underset{14}{m}</math>

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| <math>{\underset{14}{m}}</math>

−

| <math>\underset{15}{m}</math>

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| <math>{\underset{15}{m}}</math>

−

| <math>\underset{16}{m}</math>

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| <math>{\underset{16}{m}}</math>

−

| <math>\underset{17}{m}</math>

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| <math>{\underset{17}{m}}</math>

−

| <math>\underset{18}{m}</math>

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| <math>{\underset{18}{m}}</math>

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| <math>\underset{19}{m}</math>

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| <math>{\underset{19}{m}}</math>

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| <math>\underset{20}{m}</math>

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| <math>{\underset{20}{m}}</math>

−

| <math>\underset{21}{m}</math>

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| <math>{\underset{21}{m}}</math>

−

| <math>\underset{22}{m}</math>

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| <math>{\underset{22}{m}}</math>

−

| <math>\underset{23}{m}</math>

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| <math>{\underset{23}{m}}</math>

|-

−

| <math>f_0</math>

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| <math>f_0\!</math>

−

| <math>0000</math>

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| <math>0000\!</math>

−

| <math>\texttt{(~)}</math>

+

| <math>\texttt{(~)}\!</math>

| 0 || style="background:black; color:white" | 1

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| 0 || style="background:black; color:white" | 1

|-

−

| <math>f_1</math>

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| <math>f_1\!</math>

−

| <math>0001</math>

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| <math>0001\!</math>

−

| <math>\texttt{(u)(v)}</math>

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| <math>\texttt{(u)(v)}\!</math>

| 0 || 0

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

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

|-

−

| <math>f_2</math>

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| <math>f_2\!</math>

−

| <math>0010</math>

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| <math>{0010}\!</math>

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| <math>\texttt{(u) v}</math>

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| <math>\texttt{(u) v}\!</math>

| 0 || 0 || 0 || 0

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

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

|-

−

| <math>f_3</math>

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| <math>f_3\!</math>

−

| <math>0011</math>

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| <math>0011\!</math>

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| <math>\texttt{(u)}</math>

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| <math>\texttt{(u)}\!</math>

| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0

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

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| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0

|-

−

| <math>f_4</math>

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| <math>f_4\!</math>

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| <math>0100</math>

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| <math>0100\!</math>

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| <math>\texttt{u (v)}</math>

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| <math>\texttt{u (v)}\!</math>

| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0

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

|-

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| <math>f_5</math>

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| <math>f_5\!</math>

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| <math>0101</math>

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| <math>{0101}\!</math>

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| <math>\texttt{(v)}</math>

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| <math>\texttt{(v)}\!</math>

| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0

|-

−

| <math>f_6</math>

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| <math>f_6\!</math>

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| <math>0110</math>

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| <math>0110\!</math>

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| <math>\texttt{(u, v)}</math>

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| <math>\texttt{(u, v)}\!</math>

| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0

|-

−

| <math>f_7</math>

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| <math>f_7\!</math>

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| <math>0111</math>

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| <math>0111\!</math>

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| <math>\texttt{(u v)}</math>

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| <math>\texttt{(u v)}\!</math>

| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0

|-

−

| <math>f_8</math>

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| <math>f_8\!</math>

−

| <math>1000</math>

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| <math>1000\!</math>

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| <math>\texttt{u v}</math>

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| <math>\texttt{u v}\!</math>

| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0

|-

−

| <math>f_9</math>

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| <math>f_9\!</math>

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| <math>1001</math>

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| <math>1001\!</math>

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| <math>\texttt{((u, v))}</math>

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| <math>\texttt{((u, v))}\!</math>

| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0

|-

−

| <math>f_{10}</math>

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| <math>f_{10}\!</math>

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| <math>1010</math>

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| <math>1010\!</math>

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| <math>\texttt{v}</math>

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| <math>\texttt{v}\!</math>

| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0

|-

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| <math>f_{11}</math>

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| <math>f_{11}\!</math>

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| <math>1011</math>

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| <math>1011\!</math>

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| <math>\texttt{(u (v))}</math>

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| <math>\texttt{(u (v))}\!</math>

| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0

|-

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| <math>f_{12}</math>

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| <math>f_{12}\!</math>

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| <math>1100</math>

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| <math>1100\!</math>

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| <math>\texttt{u}</math>

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| <math>\texttt{u}\!</math>

| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0

|-

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| <math>f_{13}</math>

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| <math>f_{13}\!</math>

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| <math>1101</math>

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| <math>1101\!</math>

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| <math>\texttt{((u) v)}</math>

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| <math>\texttt{((u) v)}\!</math>

| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0

|-

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| <math>f_{14}</math>

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| <math>f_{14}\!</math>

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| <math>1110</math>

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| <math>1110\!</math>

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| <math>\texttt{((u)(v))}</math>

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| <math>\texttt{((u)(v))}\!</math>

| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0

|-

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| <math>f_{15}</math>

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| <math>f_{15}\!</math>

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| <math>1111</math>

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| <math>1111\!</math>

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| <math>\texttt{((~))}</math>

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| <math>\texttt{((~))}\!</math>

| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0

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|}

−

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 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.

+

+

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 14. 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%"

−

|+ ~~'''~~Table 13. Qualifiers of Implication Ordering:~~  <math>~~\alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f)</math>~~'''~~

+

|+ style="height:25px" |

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<math>\text{Table 14. Qualifiers of Implication Ordering:} ~~ \alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f)\!</math>

|- style="background:ghostwhite"

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

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| <math>\alpha_{15}</math>

|-

−

| <math>f_0</math>

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| <math>f_0\!</math>

| 0000

| <math>\texttt{(~)}</math>

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| style="background:white; color:black" | 0

|-

−

| <math>f_1</math>

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| <math>f_1\!</math>

| 0001

| <math>\texttt{(} u \texttt{)(} v \texttt{)}</math>

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| style="background:white; color:black" | 0

|-

−

| <math>f_2</math>

+

| <math>f_2\!</math>

| 0010

| <math>\texttt{(} u \texttt{)} ~ v</math>

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| style="background:white; color:black" | 0

|-

−

| <math>f_3</math>

+

| <math>f_3\!</math>

| 0011

| <math>\texttt{(} u \texttt{)}</math>

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| style="background:white; color:black" | 0

|-

−

| <math>f_4</math>

+

| <math>f_4\!</math>

| 0100

| <math>u ~ \texttt{(} v \texttt{)}</math>

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| style="background:white; color:black" | 0

|-

−

| <math>f_5</math>

+

| <math>f_5\!</math>

| 0101

| <math>\texttt{(} v \texttt{)}</math>

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| style="background:white; color:black" | 0

|-

−

| <math>f_6</math>

+

| <math>f_6\!</math>

| 0110

| <math>\texttt{(} u ~ \texttt{,} ~ v \texttt{)}</math>

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| style="background:white; color:black" | 0

|-

−

| <math>f_7</math>

+

| <math>f_7\!</math>

| 0111

| <math>\texttt{(} u ~ v \texttt{)}</math>

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| style="background:white; color:black" | 0

|-

−

| <math>f_8</math>

+

| <math>f_8\!</math>

| 1000

| <math>u ~ v</math>

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| style="background:white; color:black" | 0

|-

−

| <math>f_9</math>

+

| <math>f_9\!</math>

| 1001

| <math>\texttt{((} u ~ \texttt{,} ~ v \texttt{))}</math>

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| style="background:white; color:black" | 0

|-

−

| <math>f_{10}</math>

+

| <math>f_{10}\!</math>

| 1010

| <math>v\!</math>

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| style="background:white; color:black" | 0

|-

−

| <math>f_{11}</math>

+

| <math>f_{11}\!</math>

| 1011

| <math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math>

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| style="background:white; color:black" | 0

|-

−

| <math>f_{12}</math>

+

| <math>f_{12}\!</math>

| 1100

| <math>u\!</math>

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| style="background:white; color:black" | 0

|-

−

| <math>f_{13}</math>

+

| <math>f_{13}\!</math>

| 1101

| <math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math>

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| style="background:white; color:black" | 0

|-

−

| <math>f_{14}</math>

+

| <math>f_{14}\!</math>

| 1110

| <math>\texttt{((} u \texttt{)(} v \texttt{))}</math>

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| style="background:white; color:black" | 0

|-

−

| <math>f_{15}</math>

+

| <math>f_{15}\!</math>

| 1111

| <math>\texttt{((~))}</math>

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

−

|}~~ ~~

+

|}

−

The values of the sixteen <math>\beta_i\!</math> on each of the sixteen boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}</math> are shown in Table 14. Expressed in terms of the implication ordering on the sixteen functions, <math>\beta_i f = 1\!</math> says that <math>f\!</math> is ''below or identical to'' <math>f_i\!</math> in the implication lattice, that is, <math>\le f_i\!</math> in the implication ordering.

+

+

The values of the sixteen <math>{\beta_i}\!</math> on each of the sixteen boolean functions <math>{f : \mathbb{B}^2 \to \mathbb{B}}\!</math> are shown in Table 15. Expressed in terms of the implication ordering on the sixteen functions, <math>{\beta_i f = 1}\!</math> says that <math>{f}\!</math> is ''below or identical to'' <math>{f_i}\!</math> in the implication lattice, that is, <math>{\le f_i}\!</math> in the implication ordering.

+

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

−

|+ ~~'''~~Table 14. Qualifiers of Implication Ordering:~~  <math>~~\beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)</math>~~'''~~

+

|+ style="height:25px" |

+

<math>\text{Table 15. Qualifiers of Implication Ordering:} ~~ \beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)\!</math>

|- style="background:ghostwhite"

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

Line 1,336: Line 1,351:

| <math>\beta_{15}</math>

|-

−

| <math>f_0</math>

+

| <math>f_0\!</math>

| 0000

| <math>\texttt{(~)}</math>

Line 1,356: Line 1,371:

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

|-

−

| <math>f_1</math>

+

| <math>f_1\!</math>

| 0001

| <math>\texttt{(} u \texttt{)(} v \texttt{)}</math>

Line 1,376: Line 1,391:

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

|-

−

| <math>f_2</math>

+

| <math>f_2\!</math>

| 0010

| <math>\texttt{(} u \texttt{)} ~ v</math>

Line 1,396: Line 1,411:

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

|-

−

| <math>f_3</math>

+

| <math>f_3\!</math>

| 0011

| <math>\texttt{(} u \texttt{)}</math>

Line 1,416: Line 1,431:

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

|-

−

| <math>f_4</math>

+

| <math>f_4\!</math>

| 0100

| <math>u ~ \texttt{(} v \texttt{)}</math>

Line 1,436: Line 1,451:

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

|-

−

| <math>f_5</math>

+

| <math>f_5\!</math>

| 0101

| <math>\texttt{(} v \texttt{)}</math>

Line 1,456: Line 1,471:

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

|-

−

| <math>f_6</math>

+

| <math>f_6\!</math>

| 0110

| <math>\texttt{(} u ~ \texttt{,} ~ v \texttt{)}</math>

Line 1,476: Line 1,491:

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

|-

−

| <math>f_7</math>

+

| <math>f_7\!</math>

| 0111

| <math>\texttt{(} u ~ v \texttt{)}</math>

Line 1,496: Line 1,511:

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

|-

−

| <math>f_8</math>

+

| <math>f_8\!</math>

| 1000

| <math>u ~ v</math>

Line 1,516: Line 1,531:

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

|-

−

| <math>f_9</math>

+

| <math>f_9\!</math>

| 1001

| <math>\texttt{((} u ~ \texttt{,} ~ v \texttt{))}</math>

Line 1,536: Line 1,551:

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

|-

−

| <math>f_{10}</math>

+

| <math>f_{10}\!</math>

| 1010

| <math>v\!</math>

Line 1,556: Line 1,571:

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

|-

−

| <math>f_{11}</math>

+

| <math>f_{11}\!</math>

| 1011

| <math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math>

Line 1,576: Line 1,591:

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

|-

−

| <math>f_{12}</math>

+

| <math>f_{12}\!</math>

| 1100

| <math>u\!</math>

Line 1,596: Line 1,611:

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

|-

−

| <math>f_{13}</math>

+

| <math>f_{13}\!</math>

| 1101

| <math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math>

Line 1,616: Line 1,631:

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

|-

−

| <math>f_{14}</math>

+

| <math>f_{14}\!</math>

| 1110

| <math>\texttt{((} u \texttt{)(} v \texttt{))}</math>

Line 1,636: Line 1,651:

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

|-

−

| <math>f_{15}</math>

+

| <math>f_{15}\!</math>

| 1111

| <math>\texttt{((~))}</math>

Line 1,655: Line 1,670:

| style="background:white; color:black" | 0

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

−

|}

+

|}

+

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.

Line 1,753: Line 1,770:

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

−

Intuitively, the <math>\ell_{ij}\!</math> operators may be thought of as qualifying propositions according to the elements of the universe of discourse that each proposition positively values. Taken together, these measures provide us with the means to express many useful observations about the propositions in <math>X^\Box = [u, v],</math> and so they mediate a subtext <math>[\ell_{00}, \ell_{01}, \ell_{10}, \ell_{11}]\!</math> that takes place within the higher order universe of discourse <math>X^{\Box\,2} = [X^\Box] = [[u, v]].\!</math> Figure 15 summarizes the action of the <math>\ell_{ij}\!</math> operators on the <math>f_i\!</math> within <math>X^{\Box\,2}.\!</math>

+

Intuitively, the <math>\ell_{ij}\!</math> operators may be thought of as qualifying propositions according to the elements of the universe of discourse that each proposition positively values. Taken together, these measures provide us with the means to express many useful observations about the propositions in <math>X^\Box = [u, v],</math> and so they mediate a subtext <math>[\ell_{00}, \ell_{01}, \ell_{10}, \ell_{11}]\!</math> that takes place within the higher order universe of discourse <math>X^{\Box\,2} = [X^\Box] = [[u, v]].\!</math> Figure 16 summarizes the action of the <math>\ell_{ij}\!</math> operators on the <math>f_i\!</math> within <math>X^{\Box\,2}.\!</math>

{| align="center" cellpadding="10" style="text-align:center"

| [[Image:Venn Diagram 4 Dimensions UV Cacti 8 Inch.jpg]]

|-

−

| <math>\text{Figure 6.} ~~ \text{Higher Order Universe of Discourse} ~ \left[ \ell_{00}, \ell_{01}, \ell_{10}, \ell_{11} \right] \subseteq \left[\left[ u, v \right]\right]</math>

+

| <math>\text{Figure 16.} ~~ \text{Higher Order Universe of Discourse} ~ \left[ \ell_{00}, \ell_{01}, \ell_{10}, \ell_{11} \right] \subseteq \left[\left[ u, v \right]\right]</math>

|}

Line 1,768: Line 1,785:

With this interpretation in mind we note the following correspondences between classical quantifications and higher order indicator functions:

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

−

|+ ~~'''~~Table 16. Syllogistic Premisses as Higher Order Indicator Functions~~'''~~

+

|+ style="height:25px" |

+

<math>\text{Table 17. Syllogistic Premisses as Higher Order Indicator Functions}\!</math>

|

<math>\begin{array}{clcl}

Line 1,790: Line 1,810:

\mathrm{Indicator~of}\ u (v) = 1 \\

\end{array}</math>

−

|}

+

|}

+

The following Tables develop these ideas in more detail.

+

{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

−

|+ ~~'''~~Table 17. Simple Qualifiers of Propositions (Version 1)~~'''~~

+

|+ style="height:25px" |

+

<math>\text{Table 18. Simple Qualifiers of Propositions (Version 1)}\!</math>

|- style="background:ghostwhite"

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

Line 1,809: Line 1,834:

| <math> \ell_{11} </math> <math>\text{Some } u </math> <math>\text{is } v </math>

|-

−

| <math>f_0</math>

+

| <math>f_0\!</math>

| 0000

−

| <math>(~)</math>

+

| <math>(~)\!</math>

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

Line 1,821: Line 1,846:

| style="background:white; color:black" | 0

|-

−

| <math>f_1</math>

+

| <math>f_1\!</math>

| 0001

| <math>(u)(v)\!</math>

Line 1,833: Line 1,858:

| style="background:white; color:black" | 0

|-

−

| <math>f_2</math>

+

| <math>f_2\!</math>

| 0010

| <math>(u) v\!</math>

Line 1,845: Line 1,870:

| style="background:white; color:black" | 0

|-

−

| <math>f_3</math>

+

| <math>f_3\!</math>

| 0011

−

| <math>(u)\!</math>

+

| <math>(u)~\!</math>

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

Line 1,857: Line 1,882:

| style="background:white; color:black" | 0

|-

−

| <math>f_4</math>

+

| <math>f_4\!</math>

| 0100

| <math>u (v)\!</math>

Line 1,869: Line 1,894:

| style="background:white; color:black" | 0

|-

−

| <math>f_5</math>

+

| <math>f_5\!</math>

| 0101

| <math>(v)\!</math>

Line 1,881: Line 1,906:

| style="background:white; color:black" | 0

|-

−

| <math>f_6</math>

+

| <math>f_6\!</math>

| 0110

| <math>(u, v)\!</math>

Line 1,893: Line 1,918:

| style="background:white; color:black" | 0

|-

−

| <math>f_7</math>

+

| <math>f_7\!</math>

| 0111

| <math>(u v)\!</math>

Line 1,905: Line 1,930:

| style="background:white; color:black" | 0

|-

−

| <math>f_8</math>

+

| <math>f_8\!</math>

| 1000

| <math>u v\!</math>

Line 1,917: Line 1,942:

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

|-

−

| <math>f_9</math>

+

| <math>f_9\!</math>

| 1001

| <math>((u, v))\!</math>

Line 1,929: Line 1,954:

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

|-

−

| <math>f_{10}</math>

+

| <math>f_{10}\!</math>

| 1010

| <math>v\!</math>

Line 1,941: Line 1,966:

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

|-

−

| <math>f_{11}</math>

+

| <math>f_{11}\!</math>

| 1011

| <math>(u (v))\!</math>

Line 1,953: Line 1,978:

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

|-

−

| <math>f_{12}</math>

+

| <math>f_{12}\!</math>

| 1100

| <math>u\!</math>

Line 1,965: Line 1,990:

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

|-

−

| <math>f_{13}</math>

+

| <math>f_{13}\!</math>

| 1101

| <math>((u) v)\!</math>

Line 1,977: Line 2,002:

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

|-

−

| <math>f_{14}</math>

+

| <math>f_{14}\!</math>

| 1110

−

| <math>((u)(v))\!</math>

+

| <math>((u)(v))~\!</math>

| style="background:white; color:black" | 0

Line 1,989: Line 2,014:

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

|-

−

| <math>f_{15}</math>

+

| <math>f_{15}\!</math>

| 1111

−

| <math>((~))</math>

+

| <math>((~))\!</math>

| style="background:white; color:black" | 0

Line 2,000: Line 2,025:

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

−

|}

+

|}

+

{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

−

|+ ~~'''~~Table 18. Simple Qualifiers of Propositions (Version 2)~~'''~~

+

|+ style="height:25px" |

+

<math>\text{Table 19. Simple Qualifiers of Propositions (Version 2)}\!</math>

|- style="background:ghostwhite"

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

Line 2,017: Line 2,045:

| <math> \ell_{11} </math> <math>\text{Some } u </math> <math>\text{is } v </math>

|-

−

| <math>f_0</math>

+

| <math>f_0\!</math>

| 0000

−

| <math>(~)</math>

+

| <math>(~)\!</math>

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

Line 2,029: Line 2,057:

| style="background:white; color:black" | 0

|-

−

| <math>f_1</math>

+

| <math>f_1\!</math>

| 0001

| <math>(u)(v)\!</math>

Line 2,041: Line 2,069:

| style="background:white; color:black" | 0

|-

−

| <math>f_2</math>

+

| <math>f_2\!</math>

| 0010

| <math>(u) v\!</math>

Line 2,053: Line 2,081:

| style="background:white; color:black" | 0

|-

−

| <math>f_4</math>

+

| <math>f_4\!</math>

| 0100

| <math>u (v)\!</math>

Line 2,065: Line 2,093:

| style="background:white; color:black" | 0

|-

−

| <math>f_8</math>

+

| <math>f_8\!</math>

| 1000

| <math>u v\!</math>

Line 2,077: Line 2,105:

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

|-

−

| <math>f_3</math>

+

| <math>f_3\!</math>

| 0011

−

| <math>(u)\!</math>

+

| <math>(u)~\!</math>

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

Line 2,089: Line 2,117:

| style="background:white; color:black" | 0

|-

−

| <math>f_{12}</math>

+

| <math>f_{12}\!</math>

| 1100

| <math>u\!</math>

Line 2,101: Line 2,129:

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

|-

−

| <math>f_6</math>

+

| <math>f_6\!</math>

| 0110

| <math>(u, v)\!</math>

Line 2,113: Line 2,141:

| style="background:white; color:black" | 0

|-

−

| <math>f_9</math>

+

| <math>f_9\!</math>

| 1001

| <math>((u, v))\!</math>

Line 2,125: Line 2,153:

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

|-

−

| <math>f_5</math>

+

| <math>f_5\!</math>

| 0101

| <math>(v)\!</math>

Line 2,137: Line 2,165:

| style="background:white; color:black" | 0

|-

−

| <math>f_{10}</math>

+

| <math>f_{10}\!</math>

| 1010

| <math>v\!</math>

Line 2,149: Line 2,177:

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

|-

−

| <math>f_7</math>

+

| <math>f_7\!</math>

| 0111

| <math>(u v)\!</math>

Line 2,161: Line 2,189:

| style="background:white; color:black" | 0

|-

−

| <math>f_{11}</math>

+

| <math>f_{11}\!</math>

| 1011

| <math>(u (v))\!</math>

Line 2,173: Line 2,201:

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

|-

−

| <math>f_{13}</math>

+

| <math>f_{13}\!</math>

| 1101

| <math>((u) v)\!</math>

Line 2,185: Line 2,213:

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

|-

−

| <math>f_{14}</math>

+

| <math>f_{14}\!</math>

| 1110

−

| <math>((u)(v))\!</math>

+

| <math>((u)(v))~\!</math>

| style="background:white; color:black" | 0

Line 2,197: Line 2,225:

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

|-

−

| <math>f_{15}</math>

+

| <math>f_{15}\!</math>

| 1111

−

| <math>((~))</math>

+

| <math>((~))\!</math>

| style="background:white; color:black" | 0

Line 2,208: Line 2,236:

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

−

|}

+

|}

+

{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

−

|+ ~~'''~~Table 19. Relation of Quantifiers to Higher Order Propositions~~'''~~

+

|+ style="height:25px" |

+

<math>\text{Table 20. Relation of Quantifiers to Higher Order Propositions}\!</math>

|- style="background:ghostwhite"

−

| <math>\text{Mnemonic}</math>

+

| <math>\text{Mnemonic}\!</math>

−

| <math>\text{Category}</math>

+

| <math>\text{Category}\!</math>

−

| <math>\text{Classical Form}</math>

+

| <math>\text{Classical Form}\!</math>

−

| <math>\text{Alternate Form}</math>

+

| <math>\text{Alternate Form}\!</math>

−

| <math>\text{Symmetric Form}</math>

+

| <math>\text{Symmetric Form}\!</math>

−

| <math>\text{Operator}</math>

+

| <math>\text{Operator}\!</math>

|-

−

| <math>\text{E}\!</math> <math>\text{Exclusive}</math>

+

| <math>\text{E}\!</math> <math>\text{Exclusive}\!</math>

−

| <math>\text{Universal}</math> <math>\text{Negative}</math>

+

| <math>\text{Universal}\!</math> <math>\text{Negative}\!</math>

−

| <math>\text{All}\ u\ \text{is}\ (v)</math>

+

| <math>\text{All}~ u ~\text{is}~ (v)</math>

|  

−

| <math>\text{No}\ u\ \text{is}\ v </math>

+

| <math>\text{No}~ u ~\text{is}~ v </math>

−

| <math>(\ell_{11})</math>

+

| <math>{(\ell_{11})}\!</math>

|-

−

| <math>\text{A}\!</math> <math>\text{Absolute}</math>

+

| <math>\text{A}\!</math> <math>\text{Absolute}~\!</math>

−

| <math>\text{Universal}</math> <math>\text{Affirmative}</math>

+

| <math>\text{Universal}\!</math> <math>\text{Affirmative}\!</math>

−

| <math>\text{All}\ u\ \text{is}\ v </math>

+

| <math>\text{All}~ u ~\text{is}~ v </math>

|  

−

| <math>\text{No}\ u\ \text{is}\ (v)</math>

+

| <math>\text{No}~ u ~\text{is}~ (v)</math>

−

| <math>(\ell_{10})</math>

+

| <math>{(\ell_{10})}\!</math>

|-

|  

−

| <math>\text{All}\ v\ \text{is}\ u </math>

+

| <math>\text{All}~ v ~\text{is}~ u </math>

−

| <math>\text{No}\ v\ \text{is}\ (u)</math>

+

| <math>\text{No}~ v ~\text{is}~ (u)</math>

−

| <math>\text{No}\ (u)\ \text{is}\ v </math>

+

| <math>\text{No}~ (u) ~\text{is}~ v </math>

−

| <math>(\ell_{01})</math>

+

| <math>{(\ell_{01})}\!</math>

|-

|  

−

| <math>\text{All}\ (v)\ \text{is}\ u </math>

+

| <math>\text{All}~ (v) ~\text{is}~ u </math>

−

| <math>\text{No}\ (v)\ \text{is}\ (u)</math>

+

| <math>\text{No}~ (v) ~\text{is}~ (u)</math>

−

| <math>\text{No}\ (u)\ \text{is}\ (v)</math>

+

| <math>\text{No}~ (u) ~\text{is}~ (v)</math>

−

| <math>(\ell_{00})</math>

+

| <math>{(\ell_{00})}\!</math>

|-

|  

−

| <math>\text{Some}\ (u)\ \text{is}\ (v)</math>

+

| <math>\text{Some}~ (u) ~\text{is}~ (v)</math>

|  

−

| <math>\text{Some}\ (u)\ \text{is}\ (v)</math>

+

| <math>\text{Some}~ (u) ~\text{is}~ (v)</math>

−

| <math>\ell_{00}\!</math>

+

| <math>{\ell_{00}}\!</math>

|-

|  

−

| <math>\text{Some}\ (u)\ \text{is}\ v</math>

+

| <math>\text{Some}~ (u) ~\text{is}~ v</math>

|  

−

| <math>\text{Some}\ (u)\ \text{is}\ v</math>

+

| <math>\text{Some}~ (u) ~\text{is}~ v</math>

−

| <math>\ell_{01}\!</math>

+

| <math>{\ell_{01}}\!</math>

|-

−

| <math>\text{O}\!</math> <math>\text{Obtrusive}</math>

+

| <math>\text{O}\!</math> <math>\text{Obtrusive}\!</math>

−

| <math>\text{Particular}</math> <math>\text{Negative}</math>

+

| <math>\text{Particular}\!</math> <math>\text{Negative}\!</math>

−

| <math>\text{Some}\ u\ \text{is}\ (v)</math>

+

| <math>\text{Some}~ u ~\text{is}~ (v)</math>

|  

−

| <math>\text{Some}\ u\ \text{is}\ (v)</math>

+

| <math>\text{Some}~ u ~\text{is}~ (v)</math>

−

| <math>\ell_{10}\!</math>

+

| <math>{\ell_{10}}\!</math>

|-

−

| <math>\text{I}\!</math> <math>\text{Indefinite}</math>

+

| <math>\text{I}\!</math> <math>\text{Indefinite}\!</math>

−

| <math>\text{Particular}</math> <math>\text{Affirmative}</math>

+

| <math>\text{Particular}\!</math> <math>\text{Affirmative}\!</math>

−

| <math>\text{Some}\ u\ \text{is}\ v</math>

+

| <math>\text{Some}~ u ~\text{is}~ v</math>

|  

−

| <math>\text{Some}\ u\ \text{is}\ v</math>

+

| <math>\text{Some}~ u ~\text{is}~ v</math>

−

| <math>\ell_{11}\!</math>

+

| <math>{\ell_{11}}\!</math>

−

|}

+

|}

+

==Appendix : Generalized Umpire Operators==

Line 2,386: Line 2,419:

# http://forum.wolframscience.com/showthread.php?postid=1966#post1966

# http://forum.wolframscience.com/showthread.php?postid=1968#post1968

−

~~ <sharethis />~~

[[Category:Adaptive Systems]]

Jon Awbrey

12,080

edits