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

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→‎Higher Order Propositions and Logical Operators (''n'' = 2): reformat table

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

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{| align="center" style="background:white; color:black; text-align:center; width:90%"

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

|- style="background:ghostwhite"

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

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

−

~~| 1100<br>1010~~

+

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

| <math>f\!</math>

−

| <math>~~m_0~~</math>

+

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

−

| <math>~~m_1~~</math>

+

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

−

| <math>~~m_2~~</math>

+

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

−

| <math>~~m_3~~</math>

+

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

−

| <math>~~m_4~~</math>

+

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

−

| <math>~~m_5~~</math>

+

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

−

| <math>~~m_6~~</math>

+

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

−

| <math>~~m_7~~</math>

+

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

−

| <math>~~m_8~~</math>

+

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

−

| <math>~~m_9~~</math>

+

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

−

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

+

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

−

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

+

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

−

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

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

−

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

+

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

−

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

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

−

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

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

−

| <math>m_{16}</math>

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

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| <math>m_{17}</math>

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

−

| <math>m_{18}</math>

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

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| <math>m_{19}</math>

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

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| <math>m_{20}</math>

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

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| <math>m_{21}</math>

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

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| <math>m_{22}</math>

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

−

| <math>m_{23}</math>

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

|-

| <math>f_0</math>

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| 0000

+

| <math>0000</math>

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

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

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|-

| <math>f_1</math>

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| 0001

+

| <math>0001</math>

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

| 0 || 0

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|-

| <math>f_2</math>

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| 0010

+

| <math>0010</math>

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

| 0 || 0 || 0 || 0

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|-

| <math>f_3</math>

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| 0011

+

| <math>0011</math>

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

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

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|-

| <math>f_4</math>

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| 0100

+

| <math>0100</math>

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

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

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|-

| <math>f_5</math>

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| 0101

+

| <math>0101</math>

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

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

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|-

| <math>f_6</math>

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| 0110

+

| <math>0110</math>

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

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

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|-

| <math>f_7</math>

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| 0111

+

| <math>0111</math>

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

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

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|-

| <math>f_8</math>

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| 1000

+

| <math>1000</math>

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

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

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|-

| <math>f_9</math>

−

| 1001

+

| <math>1001</math>

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

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

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|-

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

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| 1010

+

| <math>1010</math>

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

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

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|-

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

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| 1011

+

| <math>1011</math>

| <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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| 1100

+

| <math>1100</math>

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

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

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|-

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

−

| 1101

+

| <math>1101</math>

| <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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| 1110

+

| <math>1110</math>

| <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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| 1111

+

| <math>1111</math>

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

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

−

|}<br>

+

|}

+

<br>

===Umpire Operators===

Jon Awbrey

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edits