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o-------------------o
o-------------------o
| |
| |
| a b |
| a b |
| |
| |
o-------------------o
o-------------------o
| |
| |
| b c |
| b c |
<br>
<br>
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
|+ <math>\text{Table A.}~~\text{Existential Interpretation}</math>
|+ <math>\text{Table A.}~~\text{Existential Interpretation}</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| <math>\text{Interpretation}\!</math>
| <math>\text{Interpretation}\!</math>
|-
|-
|
| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
| |
| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
| <math>\operatorname{true}.</math>
| <math>\operatorname{true}.</math>
|-
|-
|
| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
| |
| <math>\texttt{(~)}</math>
| <math>\texttt{(~)}</math>
| <math>\operatorname{false}.</math>
| <math>\operatorname{false}.</math>
|-
|-
|
| height="100px" | [[Image:Cactus A Big.jpg|20px]]
| |
| <math>a\!</math>
| <math>a\!</math>
| <math>a.\!</math>
| <math>a.\!</math>
|-
|-
|
| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
| |
| <math>\texttt{(} a \texttt{)}</math>
| <math>\texttt{(} a \texttt{)}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\tilde{a}
\tilde{a}
\\[6pt]
\\[2pt]
a^\prime
a^\prime
\\[6pt]
\\[2pt]
\lnot a
\lnot a
\\[6pt]
\\[2pt]
\operatorname{not}~ a.
\operatorname{not}~ a.
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
| |
| <math>a~b~c</math>
| <math>a~b~c</math>
|
|
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
|
|
<math>\begin{matrix}
a \lor b \lor c
\\[6pt]
a ~\operatorname{or}~ b ~\operatorname{or}~ c.
<math>\begin{matrix}
a \lor b \lor c
\\[6pt]
a ~\operatorname{or}~ b ~\operatorname{or}~ c.
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]]
| |
| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a \Rightarrow b
a \Rightarrow b
\\[6pt]
\\[2pt]
a ~\operatorname{implies}~ b.
a ~\operatorname{implies}~ b.
\\[6pt]
\\[2pt]
\operatorname{if}~ a ~\operatorname{then}~ b.
\operatorname{if}~ a ~\operatorname{then}~ b.
\\[6pt]
\\[2pt]
\operatorname{not}~ a ~\operatorname{without}~ b.
\operatorname{not}~ a ~\operatorname{without}~ b.
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
| |
| <math>\texttt{(} a, b \texttt{)}</math>
| <math>\texttt{(} a, b \texttt{)}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a + b
a + b
\\[6pt]
\\[2pt]
a \neq b
a \neq b
\\[6pt]
\\[2pt]
a ~\operatorname{exclusive-or}~ b.
a ~\operatorname{exclusive-or}~ b.
\\[6pt]
\\[2pt]
a ~\operatorname{not~equal~to}~ b.
a ~\operatorname{not~equal~to}~ b.
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
| |
| <math>\texttt{((} a, b \texttt{))}</math>
| <math>\texttt{((} a, b \texttt{))}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a = b
a = b
\\[6pt]
\\[2pt]
a \iff b
a \iff b
\\[6pt]
\\[2pt]
a ~\operatorname{equals}~ b.
a ~\operatorname{equals}~ b.
\\[6pt]
\\[2pt]
a ~\operatorname{if~and~only~if}~ b.
a ~\operatorname{if~and~only~if}~ b.
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
| <math>\texttt{(} a, b, c \texttt{)}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{just~one~of}
\operatorname{just~one~of}
\\
\\
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]]
| |
| <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
| <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
|
|
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="160px" | [[Image:Cactus (A,(B),(C)) Big.jpg|65px]]
| |
| <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
| <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
|
|
<br>
<br>
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
|+ <math>\text{Table B.}~~\text{Entitative Interpretation}</math>
|+ <math>\text{Table B.}~~\text{Entitative Interpretation}</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| <math>\text{Interpretation}\!</math>
| <math>\text{Interpretation}\!</math>
|-
|-
|
| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
| |
| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
| <math>\operatorname{false}.</math>
| <math>\operatorname{false}.</math>
|-
|-
| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
| <math>\texttt{(~)}</math>
| <math>\operatorname{true}.</math>
|-
| height="100px" | [[Image:Cactus A Big.jpg|20px]]
| <math>a\!</math>
| <math>a.\!</math>
|-
| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
| <math>\texttt{(} a \texttt{)}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\tilde{a}
\tilde{a}
\\[6pt]
\\[2pt]
a^\prime
a^\prime
\\[6pt]
\\[2pt]
\lnot a
\lnot a
\\[6pt]
\\[2pt]
\operatorname{not}~ a.
\operatorname{not}~ a.
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
| |
| <math>a~b~c</math>
| <math>a~b~c</math>
|
|
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
| |
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
|
|
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| height="120px" | [[Image:Cactus (A)B Big.jpg|35px]]
| <math>\texttt{(} a \texttt{)} b</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a \Rightarrow b
a \Rightarrow b
\\[6pt]
\\[2pt]
a ~\operatorname{implies}~ b.
a ~\operatorname{implies}~ b.
\\[6pt]
\\[2pt]
\operatorname{if}~ a ~\operatorname{then}~ b.
\operatorname{if}~ a ~\operatorname{then}~ b.
\\[6pt]
\\[2pt]
\operatorname{not}~ a, ~\operatorname{or}~ b.
\operatorname{not}~ a, ~\operatorname{or}~ b.
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
| |
| <math>\texttt{(} a, b \texttt{)}</math>
| <math>\texttt{(} a, b \texttt{)}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a = b
a = b
\\[6pt]
\\[2pt]
a \iff b
a \iff b
\\[6pt]
\\[2pt]
a ~\operatorname{equals}~ b.
a ~\operatorname{equals}~ b.
\\[6pt]
\\[2pt]
a ~\operatorname{if~and~only~if}~ b.
a ~\operatorname{if~and~only~if}~ b.
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
| |
| <math>\texttt{((} a, b \texttt{))}</math>
| <math>\texttt{((} a, b \texttt{))}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a + b
a + b
\\[6pt]
\\[2pt]
a \neq b
a \neq b
\\[6pt]
\\[2pt]
a ~\operatorname{exclusive-or}~ b.
a ~\operatorname{exclusive-or}~ b.
\\[6pt]
\\[2pt]
a ~\operatorname{not~equal~to}~ b.
a ~\operatorname{not~equal~to}~ b.
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
| |
| <math>\texttt{(} a, b, c \texttt{)}</math>
| <math>\texttt{(} a, b, c \texttt{)}</math>
|
|
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| height="160px" | [[Image:Cactus ((A,B,C)) Big.jpg|65px]]
| <math>\texttt{((} a, b, c \texttt{))}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{just~one~of}
\operatorname{just~one~of}
\\
\\
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|65px]]
| |
| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
|
|
|}
|}
<br>
===Ascii Tables===
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
|+ <math>\text{Table C.}~~\text{Dualing Interpretations}</math>
<pre>
|- style="background:#f0f0ff"
Table A1. Propositional Forms On Two Variables
| <math>\text{Graph}\!</math>
| <math>\text{String}\!</math>
| L_1 | L_2 | L_3 | L_4 | L_5 | L_6 |
| <math>\text{Existential}\!</math>
| | | | | | |
| <math>\text{Entitative}\!</math>
| Decimal | Binary | Vector | Cactus | English | Ordinary |
|-
| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
| | x : 1 1 0 0 | | | |
| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
| | y : 1 0 1 0 | | | |
| <math>\operatorname{true}.</math>
| <math>\operatorname{false}.</math>
| | | | | | |
|-
| f_0 | f_0000 | 0 0 0 0 | () | false | 0 |
| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
| | | | | | |
| <math>\texttt{(~)}</math>
| f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y |
| <math>\operatorname{false}.</math>
| | | | | | |
| <math>\operatorname{true}.</math>
| f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y |
|-
| | | | | | |
| height="100px" | [[Image:Cactus A Big.jpg|20px]]
| f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x |
| <math>a\!</math>
| | | | | | |
| <math>a.\!</math>
| <math>a.\!</math>
| | | | | | |
|-
| f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y |
| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
| | | | | | |
| <math>\texttt{(} a \texttt{)}</math>
| f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y |
| <math>\lnot a</math>
| | | | | | |
| <math>\lnot a</math>
| f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y |
|-
| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
| f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y |
| <math>a~b~c</math>
| | | | | | |
| <math>a \land b \land c</math>
| f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| <math>a \lor b \lor c</math>
|-
| f_10 | f_1010 | 1 0 1 0 | y | y | y |
| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
| | | | | | |
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
| f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y |
| <math>a \lor b \lor c</math>
| <math>a \land b \land c</math>
| f_12 | f_1100 | 1 1 0 0 | x | x | x |
|-
| | | | | | |
| height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]]
| f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y |
| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
| | | | | | |
| <math>a \Rightarrow b</math>
| f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y |
|
| | | | | | |
|-
| f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 |
| height="120px" | [[Image:Cactus (A)B Big.jpg|35px]]
| | | | | | |
| <math>\texttt{(} a \texttt{)} b</math>
|
</pre>
| <math>a \Rightarrow b</math>
|-
<pre>
| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
Table A2. Propositional Forms On Two Variables
| <math>\texttt{(} a, b \texttt{)}</math>
o---------o---------o---------o----------o------------------o----------o
| <math>a \neq b</math>
| L_1 | L_2 | L_3 | L_4 | L_5 | L_6 |
| <math>a = b\!</math>
|-
| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
| <math>\texttt{((} a, b \texttt{))}</math>
| <math>a = b\!</math>
| <math>a \neq b\!</math>
|-
| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
| <math>\texttt{(} a, b, c \texttt{)}</math>
|
<math>\begin{matrix}
\operatorname{just~one}
\\
\operatorname{of}~ a, b, c
\\
\operatorname{is~false}.
\end{matrix}</math>
|
<math>\begin{matrix}
\operatorname{not~just~one}
\\
\operatorname{of}~ a, b, c
\\
\operatorname{is~true}.
\end{matrix}</math>
|-
| height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]]
| <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
|
<math>\begin{matrix}
\operatorname{just~one}
\\
\operatorname{of}~ a, b, c
\\
\operatorname{is~true}.
\end{matrix}</math>
|
<math>\begin{matrix}
\operatorname{not~just~one}
\\
\operatorname{of}~ a, b, c
\\
\operatorname{is~false}.
\end{matrix}</math>
|-
| height="160px" | [[Image:Cactus ((A,B,C)) Big.jpg|65px]]
| <math>\texttt{((} a, b, c \texttt{))}</math>
|
<math>\begin{matrix}
\operatorname{not~just~one}
\\
\operatorname{of}~ a, b, c
\\
\operatorname{is~false}.
\end{matrix}</math>
|
<math>\begin{matrix}
\operatorname{just~one}
\\
\operatorname{of}~ a, b, c
\\
\operatorname{is~true}.
\end{matrix}</math>
|-
| height="200px" | [[Image:Cactus (((A),(B),(C))) Big.jpg|65px]]
| <math>\texttt{(((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{)))}</math>
|
<math>\begin{matrix}
\operatorname{not~just~one}
\\
\operatorname{of}~ a, b, c
\\
\operatorname{is~true}.
\end{matrix}</math>
|
<math>\begin{matrix}
\operatorname{just~one}
\\
\operatorname{of}~ a, b, c
\\
\operatorname{is~false}.
\end{matrix}</math>
|-
| height="160px" | [[Image:Cactus (A,(B),(C)) Big.jpg|65px]]
| <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
|
<math>\begin{matrix}
\operatorname{partition}~ a
\\
\operatorname{into}~ b, c.
\end{matrix}</math>
|
|-
| height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|65px]]
| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
|
|
<math>\begin{matrix}
\operatorname{partition}~ a
\\
\operatorname{into}~ b, c.
\end{matrix}</math>
|}
<br>
==Differential Logic==
===Ascii Tables===
<pre>
Table A1. Propositional Forms On Two Variables
o---------o---------o---------o----------o------------------o----------o
| L_1 | L_2 | L_3 | L_4 | L_5 | L_6 |
| | | | | | |
| | | | | | |
| f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x |
| Decimal | Binary | Vector | Cactus | English | Ordinary |
o---------o---------o---------o----------o------------------o----------o
| | x : 1 1 0 0 | | | |
| | y : 1 0 1 0 | | | |
o---------o---------o---------o----------o------------------o----------o
| | | | | | |
| | | | | | |
| f_12 | f_1100 | 1 1 0 0 | x | x | x |
| f_0 | f_0000 | 0 0 0 0 | () | false | 0 |
| | | | | | |
| | | | | | |
| f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y |
| | | | | | |
| | | | | | |
| f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y |
| f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y |
| | | | | | |
| | | | | | |
| f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x |
| | | | | | |
| | | | | | |
| f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y |
| | | | | | |
| | | | | | |
| f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y |
| f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y |
| | | | | | |
| | | | | | |
| f_10 | f_1010 | 1 0 1 0 | y | y | y |
| f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y |
| | | | | | |
| | | | | | |
| f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y |
| f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y |
| | | | | | |
| f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y |
| | | | | | |
| f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| | | | | | |
| f_10 | f_1010 | 1 0 1 0 | y | y | y |
| | | | | | |
| | | | | | |
| f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y |
| f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y |
| | | | | | |
| f_12 | f_1100 | 1 1 0 0 | x | x | x |
| | | | | | |
| | | | | | |
| f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y |
| f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y |
| | | | | | |
| | | | | | |
| f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y |
| f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y |
| | | | | | |
| | | | | | |
| f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 |
| f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 |
<pre>
<pre>
Table A3. Ef Expanded Over Differential Features {dx, dy}
Table A2. Propositional Forms On Two Variables
o------o------------o------------o------------o------------o------------o
o---------o---------o---------o----------o------------------o----------o
| | | | | | |
| L_1 | L_2 | L_3 | L_4 | L_5 | L_6 |
| | f | T_11 f | T_10 f | T_01 f | T_00 f |
| | | | | | |
| | | | | | |
| Decimal | Binary | Vector | Cactus | English | Ordinary |
| | | Ef| dx dy | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
o---------o---------o---------o----------o------------------o----------o
| | | | | | |
| | x : 1 1 0 0 | | | |
o------o------------o------------o------------o------------o------------o
| | y : 1 0 1 0 | | | |
| | | | | | |
o---------o---------o---------o----------o------------------o----------o
| f_0 | () | () | () | () | () |
| | | | | | |
| | | | | | |
| f_0 | f_0000 | 0 0 0 0 | () | false | 0 |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | | | | | |
o---------o---------o---------o----------o------------------o----------o
| f_1 | (x)(y) | x y | x (y) | (x) y | (x)(y) |
| | | | | | |
| | | | | | |
| f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y |
| f_2 | (x) y | x (y) | x y | (x)(y) | (x) y |
| | | | | | |
| | | | | | |
| f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y |
| f_4 | x (y) | (x) y | (x)(y) | x y | x (y) |
| | | | | | |
| | | | | | |
| f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y |
| f_8 | x y | (x)(y) | (x) y | x (y) | x y |
| | | | | | |
| | | | | | |
| f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | | | | | |
o---------o---------o---------o----------o------------------o----------o
| f_3 | (x) | x | x | (x) | (x) |
| | | | | | |
| f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x |
| | | | | | |
| f_12 | f_1100 | 1 1 0 0 | x | x | x |
| | | | | | |
o---------o---------o---------o----------o------------------o----------o
| | | | | | |
| f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y |
| | | | | | |
| f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| | | | | | |
o---------o---------o---------o----------o------------------o----------o
| | | | | | |
| f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y |
| | | | | | |
| f_10 | f_1010 | 1 0 1 0 | y | y | y |
| | | | | | |
o---------o---------o---------o----------o------------------o----------o
| | | | | | |
| f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y |
| | | | | | |
| f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y |
| | | | | | |
| f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y |
| | | | | | |
| f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y |
| | | | | | |
o---------o---------o---------o----------o------------------o----------o
| | | | | | |
| f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 |
| | | | | | |
o---------o---------o---------o----------o------------------o----------o
</pre>
<pre>
Table A3. Ef Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | T_11 f | T_10 f | T_01 f | T_00 f |
| | | | | | |
| | | Ef| dx dy | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | x y | x (y) | (x) y | (x)(y) |
| | | | | | |
| f_2 | (x) y | x (y) | x y | (x)(y) | (x) y |
| | | | | | |
| f_4 | x (y) | (x) y | (x)(y) | x y | x (y) |
| | | | | | |
| f_8 | x y | (x)(y) | (x) y | x (y) | x y |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | x | x | (x) | (x) |
| | | | | | |
| | | | | | |
| f_12 | x | (x) | (x) | x | x |
| f_12 | x | (x) | (x) | x | x |
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math>
|+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math>
|+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math>
|+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ <math>\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ p, q \}</math>
|+ <math>\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ p, q \}</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ p, q \}</math>
|+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ p, q \}</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math>
|+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math>
|+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math>
|+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ <math>\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
|+ <math>\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
|+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
|- style="height:50px"
|- style="height:50px"
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math>
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
| width="22%" style="border-bottom:1px solid black" |
| width="22%" style="border-bottom:1px solid black" |
<math>\operatorname{T}_{00}</math>
<math>\operatorname{T}_{00}</math>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
|- style="height:50px"
|- style="height:50px"
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math>
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
| width="22%" style="border-bottom:1px solid black" |
| width="22%" style="border-bottom:1px solid black" |
<math>\operatorname{e}</math>
<math>\operatorname{e}</math>
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}</math>
|+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ <math>\text{Matrix Representations of Permutations in}~ \operatorname{Sym}(3)</math>
|+ <math>\text{Matrix Representations of Permutations in}~ \operatorname{Sym}(3)</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
\end{tabular}\end{quote}
\end{tabular}\end{quote}
</pre>
</pre>
==Group Operation Tables==
<br>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%"
|+ <math>\text{Table 32.1}~~\text{Scheme of a Group Operation Table}</math>
|- style="height:50px"
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>*\!</math>
| style="border-bottom:1px solid black" | <math>x_0\!</math>
| style="border-bottom:1px solid black" | <math>\cdots\!</math>
| style="border-bottom:1px solid black" | <math>x_j\!</math>
| style="border-bottom:1px solid black" | <math>\cdots\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>x_0\!</math>
| <math>x_0 * x_0\!</math>
| <math>\cdots\!</math>
| <math>x_0 * x_j\!</math>
| <math>\cdots\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\cdots\!</math>
| <math>\cdots\!</math>
| <math>\cdots\!</math>
| <math>\cdots\!</math>
| <math>\cdots\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>x_i\!</math>
| <math>x_i * x_0\!</math>
| <math>\cdots\!</math>
| <math>x_i * x_j\!</math>
| <math>\cdots\!</math>
|- style="height:50px"
| width="12%" style="border-right:1px solid black" | <math>\cdots\!</math>
| width="22%" | <math>\cdots\!</math>
| width="22%" | <math>\cdots\!</math>
| width="22%" | <math>\cdots\!</math>
| width="22%" | <math>\cdots\!</math>
|}
<br>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%"
|+ <math>\text{Table 32.2}~~\text{Scheme of the Regular Ante-Representation}</math>
|- style="height:50px"
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>x_0\!</math>
| <math>\{\!</math>
| <math>(x_0 ~,~ x_0 * x_0),\!</math>
| <math>\cdots\!</math>
| <math>(x_j ~,~ x_0 * x_j),\!</math>
| <math>\cdots\!</math>
| <math>\}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\cdots\!</math>
| <math>\{\!</math>
| <math>\cdots\!</math>
| <math>\cdots\!</math>
| <math>\cdots\!</math>
| <math>\cdots\!</math>
| <math>\}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>x_i\!</math>
| <math>\{\!</math>
| <math>(x_0 ~,~ x_i * x_0),\!</math>
| <math>\cdots\!</math>
| <math>(x_j ~,~ x_i * x_j),\!</math>
| <math>\cdots\!</math>
| <math>\}\!</math>
|- style="height:50px"
| width="12%" style="border-right:1px solid black" | <math>\cdots\!</math>
| width="4%" | <math>\{\!</math>
| width="18%" | <math>\cdots\!</math>
| width="22%" | <math>\cdots\!</math>
| width="22%" | <math>\cdots\!</math>
| width="18%" | <math>\cdots\!</math>
| width="4%" | <math>\}\!</math>
|}
<br>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%"
|+ <math>\text{Table 32.3}~~\text{Scheme of the Regular Post-Representation}</math>
|- style="height:50px"
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>x_0\!</math>
| <math>\{\!</math>
| <math>(x_0 ~,~ x_0 * x_0),\!</math>
| <math>\cdots\!</math>
| <math>(x_j ~,~ x_j * x_0),\!</math>
| <math>\cdots\!</math>
| <math>\}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\cdots\!</math>
| <math>\{\!</math>
| <math>\cdots\!</math>
| <math>\cdots\!</math>
| <math>\cdots\!</math>
| <math>\cdots\!</math>
| <math>\}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>x_i\!</math>
| <math>\{\!</math>
| <math>(x_0 ~,~ x_0 * x_i),\!</math>
| <math>\cdots\!</math>
| <math>(x_j ~,~ x_j * x_i),\!</math>
| <math>\cdots\!</math>
| <math>\}\!</math>
|- style="height:50px"
| width="12%" style="border-right:1px solid black" | <math>\cdots\!</math>
| width="4%" | <math>\{\!</math>
| width="18%" | <math>\cdots\!</math>
| width="22%" | <math>\cdots\!</math>
| width="22%" | <math>\cdots\!</math>
| width="18%" | <math>\cdots\!</math>
| width="4%" | <math>\}\!</math>
|}
<br>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
|+ <math>\text{Table 33.1}~~\text{Multiplication Operation of the Group}~V_4</math>
|- style="height:50px"
| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{e}</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{f}</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{g}</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{h}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{e}</math>
| <math>\operatorname{e}</math>
| <math>\operatorname{f}</math>
| <math>\operatorname{g}</math>
| <math>\operatorname{h}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{f}</math>
| <math>\operatorname{f}</math>
| <math>\operatorname{e}</math>
| <math>\operatorname{h}</math>
| <math>\operatorname{g}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{g}</math>
| <math>\operatorname{g}</math>
| <math>\operatorname{h}</math>
| <math>\operatorname{e}</math>
| <math>\operatorname{f}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{h}</math>
| <math>\operatorname{h}</math>
| <math>\operatorname{g}</math>
| <math>\operatorname{f}</math>
| <math>\operatorname{e}</math>
|}
<br>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
|+ <math>\text{Table 33.2}~~\text{Regular Representation of the Group}~V_4</math>
|- style="height:50px"
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
|- style="height:50px"
| width="20%" style="border-right:1px solid black" | <math>\operatorname{e}</math>
| width="4%" | <math>\{\!</math>
| width="16%" | <math>(\operatorname{e}, \operatorname{e}),</math>
| width="20%" | <math>(\operatorname{f}, \operatorname{f}),</math>
| width="20%" | <math>(\operatorname{g}, \operatorname{g}),</math>
| width="16%" | <math>(\operatorname{h}, \operatorname{h})</math>
| width="4%" | <math>\}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{f}</math>
| <math>\{\!</math>
| <math>(\operatorname{e}, \operatorname{f}),</math>
| <math>(\operatorname{f}, \operatorname{e}),</math>
| <math>(\operatorname{g}, \operatorname{h}),</math>
| <math>(\operatorname{h}, \operatorname{g})</math>
| <math>\}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{g}</math>
| <math>\{\!</math>
| <math>(\operatorname{e}, \operatorname{g}),</math>
| <math>(\operatorname{f}, \operatorname{h}),</math>
| <math>(\operatorname{g}, \operatorname{e}),</math>
| <math>(\operatorname{h}, \operatorname{f})</math>
| <math>\}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{h}</math>
| <math>\{\!</math>
| <math>(\operatorname{e}, \operatorname{h}),</math>
| <math>(\operatorname{f}, \operatorname{g}),</math>
| <math>(\operatorname{g}, \operatorname{f}),</math>
| <math>(\operatorname{h}, \operatorname{e})</math>
| <math>\}\!</math>
|}
<br>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
|+ <math>\text{Table 33.3}~~\text{Regular Representation of the Group}~V_4</math>
|- style="height:50px"
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Symbols}\!</math>
|- style="height:50px"
| width="20%" style="border-right:1px solid black" | <math>\operatorname{e}</math>
| width="4%" | <math>\{\!</math>
| width="16%" | <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>
| width="20%" | <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>
| width="20%" | <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>
| width="16%" | <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime})</math>
| width="4%" | <math>\}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{f}</math>
| <math>\{\!</math>
| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>
| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>
| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>
| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime})</math>
| <math>\}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{g}</math>
| <math>\{\!</math>
| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>
| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>
| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>
| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime})</math>
| <math>\}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{h}</math>
| <math>\{\!</math>
| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>
| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>
| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>
| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime})</math>
| <math>\}\!</math>
|}
<br>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
|+ <math>\text{Table 34.1}~~\text{Multiplicative Presentation of the Group}~Z_4(\cdot)</math>
|- style="height:50px"
| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{1}</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{a}</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{b}</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{c}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{1}</math>
| <math>\operatorname{1}</math>
| <math>\operatorname{a}</math>
| <math>\operatorname{b}</math>
| <math>\operatorname{c}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{a}</math>
| <math>\operatorname{a}</math>
| <math>\operatorname{b}</math>
| <math>\operatorname{c}</math>
| <math>\operatorname{1}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{b}</math>
| <math>\operatorname{b}</math>
| <math>\operatorname{c}</math>
| <math>\operatorname{1}</math>
| <math>\operatorname{a}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{c}</math>
| <math>\operatorname{c}</math>
| <math>\operatorname{1}</math>
| <math>\operatorname{a}</math>
| <math>\operatorname{b}</math>
|}
<br>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
|+ <math>\text{Table 34.2}~~\text{Regular Representation of the Group}~Z_4(\cdot)</math>
|- style="height:50px"
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
|- style="height:50px"
| width="20%" style="border-right:1px solid black" | <math>\operatorname{1}</math>
| width="4%" | <math>\{\!</math>
| width="16%" | <math>(\operatorname{1}, \operatorname{1}),</math>
| width="20%" | <math>(\operatorname{a}, \operatorname{a}),</math>
| width="20%" | <math>(\operatorname{b}, \operatorname{b}),</math>
| width="16%" | <math>(\operatorname{c}, \operatorname{c})</math>
| width="4%" | <math>\}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{a}</math>
| <math>\{\!</math>
| <math>(\operatorname{1}, \operatorname{a}),</math>
| <math>(\operatorname{a}, \operatorname{b}),</math>
| <math>(\operatorname{b}, \operatorname{c}),</math>
| <math>(\operatorname{c}, \operatorname{1})</math>
| <math>\}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{b}</math>
| <math>\{\!</math>
| <math>(\operatorname{1}, \operatorname{b}),</math>
| <math>(\operatorname{a}, \operatorname{c}),</math>
| <math>(\operatorname{b}, \operatorname{1}),</math>
| <math>(\operatorname{c}, \operatorname{a})</math>
| <math>\}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{c}</math>
| <math>\{\!</math>
| <math>(\operatorname{1}, \operatorname{c}),</math>
| <math>(\operatorname{a}, \operatorname{1}),</math>
| <math>(\operatorname{b}, \operatorname{a}),</math>
| <math>(\operatorname{c}, \operatorname{b})</math>
| <math>\}\!</math>
|}
<br>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
|+ <math>\text{Table 35.1}~~\text{Additive Presentation of the Group}~Z_4(+)</math>
|- style="height:50px"
| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>+\!</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{0}</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{1}</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{2}</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{3}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{0}</math>
| <math>\operatorname{0}</math>
| <math>\operatorname{1}</math>
| <math>\operatorname{2}</math>
| <math>\operatorname{3}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{1}</math>
| <math>\operatorname{1}</math>
| <math>\operatorname{2}</math>
| <math>\operatorname{3}</math>
| <math>\operatorname{0}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{2}</math>
| <math>\operatorname{2}</math>
| <math>\operatorname{3}</math>
| <math>\operatorname{0}</math>
| <math>\operatorname{1}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{3}</math>
| <math>\operatorname{3}</math>
| <math>\operatorname{0}</math>
| <math>\operatorname{1}</math>
| <math>\operatorname{2}</math>
|}
<br>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
|+ <math>\text{Table 35.2}~~\text{Regular Representation of the Group}~Z_4(+)</math>
|- style="height:50px"
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
|- style="height:50px"
| width="20%" style="border-right:1px solid black" | <math>\operatorname{0}</math>
| width="4%" | <math>\{\!</math>
| width="16%" | <math>(\operatorname{0}, \operatorname{0}),</math>
| width="20%" | <math>(\operatorname{1}, \operatorname{1}),</math>
| width="20%" | <math>(\operatorname{2}, \operatorname{2}),</math>
| width="16%" | <math>(\operatorname{3}, \operatorname{3})</math>
| width="4%" | <math>\}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{1}</math>
| <math>\{\!</math>
| <math>(\operatorname{0}, \operatorname{1}),</math>
| <math>(\operatorname{1}, \operatorname{2}),</math>
| <math>(\operatorname{2}, \operatorname{3}),</math>
| <math>(\operatorname{3}, \operatorname{0})</math>
| <math>\}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{2}</math>
| <math>\{\!</math>
| <math>(\operatorname{0}, \operatorname{2}),</math>
| <math>(\operatorname{1}, \operatorname{3}),</math>
| <math>(\operatorname{2}, \operatorname{0}),</math>
| <math>(\operatorname{3}, \operatorname{1})</math>
| <math>\}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{3}</math>
| <math>\{\!</math>
| <math>(\operatorname{0}, \operatorname{3}),</math>
| <math>(\operatorname{1}, \operatorname{0}),</math>
| <math>(\operatorname{2}, \operatorname{1}),</math>
| <math>(\operatorname{3}, \operatorname{2})</math>
| <math>\}\!</math>
|}
<br>
==Higher Order Propositions==
<br>
<table align="center" cellpadding="4" cellspacing="0" style="text-align:center; width:90%">
<caption><font size="+2"><math>\text{Table 1.} ~~ \text{Higher Order Propositions} ~ (n = 1)</math></font></caption>
<tr>
<td style="border-bottom:2px solid black" align="right"><math>x:</math></td>
<td style="border-bottom:2px solid black"><math>1 ~ 0</math></td>
<td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td>
<td style="border-bottom:2px solid black"><math>m_{0}</math></td>
<td style="border-bottom:2px solid black"><math>m_{1}</math></td>
<td style="border-bottom:2px solid black"><math>m_{2}</math></td>
<td style="border-bottom:2px solid black"><math>m_{3}</math></td>
<td style="border-bottom:2px solid black"><math>m_{4}</math></td>
<td style="border-bottom:2px solid black"><math>m_{5}</math></td>
<td style="border-bottom:2px solid black"><math>m_{6}</math></td>
<td style="border-bottom:2px solid black"><math>m_{7}</math></td>
<td style="border-bottom:2px solid black"><math>m_{8}</math></td>
<td style="border-bottom:2px solid black"><math>m_{9}</math></td>
<td style="border-bottom:2px solid black"><math>m_{10}</math></td>
<td style="border-bottom:2px solid black"><math>m_{11}</math></td>
<td style="border-bottom:2px solid black"><math>m_{12}</math></td>
<td style="border-bottom:2px solid black"><math>m_{13}</math></td>
<td style="border-bottom:2px solid black"><math>m_{14}</math></td>
<td style="border-bottom:2px solid black"><math>m_{15}</math></td></tr>
<tr>
<td><math>f_{0}</math></td>
<td><math>0 ~ 0</math></td>
<td style="border-right:2px solid black"><math>\texttt{(~)}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{1}</math></td>
<td><math>0 ~ 1</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} x \texttt{)}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{2}</math></td>
<td><math>1 ~ 0</math></td>
<td style="border-right:2px solid black"><math>x</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{3}</math></td>
<td><math>1 ~ 1</math></td>
<td style="border-right:2px solid black"><math>\texttt{((~))}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
</table>
<br>
<table align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:center; width:90%">
<caption><font size="+2"><math>\text{Table 2.} ~~ \text{Interpretive Categories for Higher Order Propositions} ~ (n = 1)</math></font></caption>
<tr>
<td style="border-bottom:2px solid black; border-right:2px solid black">Measure</td>
<td style="border-bottom:2px solid black">Happening</td>
<td style="border-bottom:2px solid black">Exactness</td>
<td style="border-bottom:2px solid black">Existence</td>
<td style="border-bottom:2px solid black">Linearity</td>
<td style="border-bottom:2px solid black">Uniformity</td>
<td style="border-bottom:2px solid black">Information</td></tr>
<tr>
<td style="border-right:2px solid black"><math>m_{0}</math></td>
<td>Nothing happens</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td></tr>
<tr>
<td style="border-right:2px solid black"><math>m_{1}</math></td>
<td> </td>
<td>Just false</td>
<td>Nothing exists</td>
<td> </td>
<td> </td>
<td> </td></tr>
<tr>
<td style="border-right:2px solid black"><math>m_{2}</math></td>
<td> </td>
<td>Just not <math>x</math></td>
<td> </td>
<td> </td>
<td> </td>
<td> </td></tr>
<tr>
<td style="border-right:2px solid black"><math>m_{3}</math></td>
<td> </td>
<td> </td>
<td>Nothing is <math>x</math></td>
<td> </td>
<td> </td>
<td> </td></tr>
<tr>
<td style="border-right:2px solid black"><math>m_{4}</math></td>
<td> </td>
<td>Just <math>x</math></td>
<td> </td>
<td> </td>
<td> </td>
<td> </td></tr>
<tr>
<td style="border-right:2px solid black"><math>m_{5}</math></td>
<td> </td>
<td> </td>
<td>Everything is <math>x</math></td>
<td><math>f</math> is linear</td>
<td> </td>
<td> </td></tr>
<tr>
<td style="border-right:2px solid black"><math>m_{6}</math></td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td><math>f</math> is not uniform</td>
<td><math>f</math> is informed</td></tr>
<tr>
<td style="border-right:2px solid black"><math>m_{7}</math></td>
<td> </td>
<td>Not just true</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td></tr>
<tr>
<td style="border-right:2px solid black"><math>m_{8}</math></td>
<td> </td>
<td>Just true</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td></tr>
<tr>
<td style="border-right:2px solid black"><math>m_{9}</math></td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td><math>f</math> is uniform</td>
<td><math>f</math> is not informed</td></tr>
<tr>
<td style="border-right:2px solid black"><math>m_{10}</math></td>
<td> </td>
<td> </td>
<td>Something is not <math>x</math></td>
<td><math>f</math> is not linear</td>
<td> </td>
<td> </td></tr>
<tr>
<td style="border-right:2px solid black"><math>m_{11}</math></td>
<td> </td>
<td>Not just <math>x</math></td>
<td> </td>
<td> </td>
<td> </td>
<td> </td></tr>
<tr>
<td style="border-right:2px solid black"><math>m_{12}</math></td>
<td> </td>
<td> </td>
<td>Something is <math>x</math></td>
<td> </td>
<td> </td>
<td> </td></tr>
<tr>
<td style="border-right:2px solid black"><math>m_{13}</math></td>
<td> </td>
<td>Not just not <math>x</math></td>
<td> </td>
<td> </td>
<td> </td>
<td> </td></tr>
<tr>
<td style="border-right:2px solid black"><math>m_{14}</math></td>
<td> </td>
<td>Not just false</td>
<td>Something exists</td>
<td> </td>
<td> </td>
<td> </td></tr>
<tr>
<td style="border-right:2px solid black"><math>m_{15}</math></td>
<td>Anything happens</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td></tr>
</table>
<br>
<table align="center" cellpadding="1" cellspacing="0" style="background:white; color:black; text-align:center; width:90%">
<caption><font size="+2"><math>\text{Table 3.} ~~ \text{Higher Order Propositions} ~ (n = 2)</math></font></caption>
<tr>
<td style="border-bottom:2px solid black" align="right"><math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td>
<td style="border-bottom:2px solid black">
<math>\begin{matrix}1100\\1010\end{matrix}</math></td>
<td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td>
<td style="border-bottom:2px solid black"><math>\underset{0}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{1}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{2}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{3}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{4}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{5}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{6}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{7}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{8}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{9}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{10}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{11}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{12}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{13}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{14}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{15}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{16}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{17}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{18}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{19}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{20}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{21}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{22}{m}</math></td>
<td style="border-bottom:2px solid black"><math>\underset{23}{m}</math></td>
</tr>
<tr>
<td><math>f_{0}</math></td>
<td><math>0000</math></td>
<td style="border-right:2px solid black"><math>\texttt{(~)}</math></td>
<td>0</td>
<td style="background:black; color:white">1</td>
<td>0</td>
<td style="background:black; color:white">1</td>
<td>0</td>
<td style="background:black; color:white">1</td>
<td>0</td>
<td style="background:black; color:white">1</td>
<td>0</td>
<td style="background:black; color:white">1</td>
<td>0</td>
<td style="background:black; color:white">1</td>
<td>0</td>
<td style="background:black; color:white">1</td>
<td>0</td>
<td style="background:black; color:white">1</td>
<td>0</td>
<td style="background:black; color:white">1</td>
<td>0</td>
<td style="background:black; color:white">1</td>
<td>0</td>
<td style="background:black; color:white">1</td>
<td>0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{1}</math></td>
<td><math>0001</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
<td>0</td><td>0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td>0</td><td>0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td>0</td><td>0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td>0</td><td>0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td>0</td><td>0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td>0</td><td>0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{2}</math></td>
<td><math>0010</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{3}</math></td>
<td><math>0011</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)}</math></td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
<tr>
<td><math>f_{4}</math></td>
<td><math>0100</math></td>
<td style="border-right:2px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{5}</math></td>
<td><math>0101</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} v \texttt{)}</math></td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
<tr>
<td><math>f_{6}</math></td>
<td><math>0110</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
<tr>
<td><math>f_{7}</math></td>
<td><math>0111</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
<tr>
<td><math>f_{8}</math></td>
<td><math>1000</math></td>
<td style="border-right:2px solid black"><math>u ~ v</math></td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
<tr>
<td><math>f_{9}</math></td>
<td><math>1001</math></td>
<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
<tr>
<td><math>f_{10}</math></td>
<td><math>1010</math></td>
<td style="border-right:2px solid black"><math>v</math></td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
<tr>
<td><math>f_{11}</math></td>
<td><math>1011</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
<tr>
<td><math>f_{12}</math></td>
<td><math>1100</math></td>
<td style="border-right:2px solid black"><math>u</math></td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
<tr>
<td><math>f_{13}</math></td>
<td><math>1101</math></td>
<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
<tr>
<td><math>f_{14}</math></td>
<td><math>1110</math></td>
<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
<tr>
<td><math>f_{15}</math></td>
<td><math>1111</math></td>
<td style="border-right:2px solid black"><math>\texttt{((~))}</math></td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td>
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
</table>
<br>
<table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%">
<caption><font size="+2"><math>\text{Table 4.} ~~ \text{Qualifiers of the Implication Ordering:} ~ \alpha_{i} f = \Upsilon (f_{i}, f) = \Upsilon (f_{i} \Rightarrow f)</math></font></caption>
<tr>
<td style="border-bottom:2px solid black" align="right">
<math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td>
<td style="border-bottom:2px solid black">
<math>\begin{matrix}1100\\1010\end{matrix}</math></td>
<td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td>
<td style="border-bottom:2px solid black"><math>\alpha_{15}</math></td>
<td style="border-bottom:2px solid black"><math>\alpha_{14}</math></td>
<td style="border-bottom:2px solid black"><math>\alpha_{13}</math></td>
<td style="border-bottom:2px solid black"><math>\alpha_{12}</math></td>
<td style="border-bottom:2px solid black"><math>\alpha_{11}</math></td>
<td style="border-bottom:2px solid black"><math>\alpha_{10}</math></td>
<td style="border-bottom:2px solid black"><math>\alpha_{9}</math></td>
<td style="border-bottom:2px solid black"><math>\alpha_{8}</math></td>
<td style="border-bottom:2px solid black"><math>\alpha_{7}</math></td>
<td style="border-bottom:2px solid black"><math>\alpha_{6}</math></td>
<td style="border-bottom:2px solid black"><math>\alpha_{5}</math></td>
<td style="border-bottom:2px solid black"><math>\alpha_{4}</math></td>
<td style="border-bottom:2px solid black"><math>\alpha_{3}</math></td>
<td style="border-bottom:2px solid black"><math>\alpha_{2}</math></td>
<td style="border-bottom:2px solid black"><math>\alpha_{1}</math></td>
<td style="border-bottom:2px solid black"><math>\alpha_{0}</math></td></tr>
<tr>
<td><math>f_{0}</math></td>
<td><math>0000</math></td>
<td style="border-right:2px solid black"><math>\texttt{(~)}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{1}</math></td>
<td><math>0001</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{2}</math></td>
<td><math>0010</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{3}</math></td>
<td><math>0011</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{4}</math></td>
<td><math>0100</math></td>
<td style="border-right:2px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{5}</math></td>
<td><math>0101</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} v \texttt{)}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{6}</math></td>
<td><math>0110</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{7}</math></td>
<td><math>0111</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{8}</math></td>
<td><math>1000</math></td>
<td style="border-right:2px solid black"><math>u ~ v</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{9}</math></td>
<td><math>1001</math></td>
<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{10}</math></td>
<td><math>1010</math></td>
<td style="border-right:2px solid black"><math>v</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{11}</math></td>
<td><math>1011</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{12}</math></td>
<td><math>1100</math></td>
<td style="border-right:2px solid black"><math>u</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{13}</math></td>
<td><math>1101</math></td>
<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{14}</math></td>
<td><math>1110</math></td>
<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{15}</math></td>
<td><math>1111</math></td>
<td style="border-right:2px solid black"><math>\texttt{((~))}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
</table>
<br>
<table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%">
<caption><font size="+2"><math>\text{Table 5.} ~~ \text{Qualifiers of the Implication Ordering:} ~ \beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)</math></font></caption>
<tr>
<td style="border-bottom:2px solid black" align="right">
<math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td>
<td style="border-bottom:2px solid black">
<math>\begin{matrix}1100\\1010\end{matrix}</math></td>
<td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td>
<td style="border-bottom:2px solid black"><math>\beta_{0}</math></td>
<td style="border-bottom:2px solid black"><math>\beta_{1}</math></td>
<td style="border-bottom:2px solid black"><math>\beta_{2}</math></td>
<td style="border-bottom:2px solid black"><math>\beta_{3}</math></td>
<td style="border-bottom:2px solid black"><math>\beta_{4}</math></td>
<td style="border-bottom:2px solid black"><math>\beta_{5}</math></td>
<td style="border-bottom:2px solid black"><math>\beta_{6}</math></td>
<td style="border-bottom:2px solid black"><math>\beta_{7}</math></td>
<td style="border-bottom:2px solid black"><math>\beta_{8}</math></td>
<td style="border-bottom:2px solid black"><math>\beta_{9}</math></td>
<td style="border-bottom:2px solid black"><math>\beta_{10}</math></td>
<td style="border-bottom:2px solid black"><math>\beta_{11}</math></td>
<td style="border-bottom:2px solid black"><math>\beta_{12}</math></td>
<td style="border-bottom:2px solid black"><math>\beta_{13}</math></td>
<td style="border-bottom:2px solid black"><math>\beta_{14}</math></td>
<td style="border-bottom:2px solid black"><math>\beta_{15}</math></td></tr>
<tr>
<td><math>f_{0}</math></td>
<td><math>0000</math></td>
<td style="border-right:2px solid black"><math>\texttt{(~)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{1}</math></td>
<td><math>0001</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{2}</math></td>
<td><math>0010</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{3}</math></td>
<td><math>0011</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{4}</math></td>
<td><math>0100</math></td>
<td style="border-right:2px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{5}</math></td>
<td><math>0101</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} v \texttt{)}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{6}</math></td>
<td><math>0110</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{7}</math></td>
<td><math>0111</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{8}</math></td>
<td><math>1000</math></td>
<td style="border-right:2px solid black"><math>u ~ v</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{9}</math></td>
<td><math>1001</math></td>
<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{10}</math></td>
<td><math>1010</math></td>
<td style="border-right:2px solid black"><math>v</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{11}</math></td>
<td><math>1011</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{12}</math></td>
<td><math>1100</math></td>
<td style="border-right:2px solid black"><math>u</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{13}</math></td>
<td><math>1101</math></td>
<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{14}</math></td>
<td><math>1110</math></td>
<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{15}</math></td>
<td><math>1111</math></td>
<td style="border-right:2px solid black"><math>\texttt{((~))}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
</table>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ <math>\text{Table 7.} ~~ \text{Syllogistic Premisses as Higher Order Indicator Functions}</math>
|
<math>\begin{array}{clcl}
\mathrm{A}
& \mathrm{Universal~Affirmative}
& \mathrm{All} ~ u ~ \mathrm{is} ~ v
& \mathrm{Indicator~of} ~ u \texttt{(} v \texttt{)} = 0
\\
\mathrm{E}
& \mathrm{Universal~Negative}
& \mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}
& \mathrm{Indicator~of} ~ u \cdot v = 0
\\
\mathrm{I}
& \mathrm{Particular~Affirmative}
& \mathrm{Some} ~ u ~ \mathrm{is} ~ v
& \mathrm{Indicator~of} ~ u \cdot v = 1
\\
\mathrm{O}
& \mathrm{Particular~Negative}
& \mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}
& \mathrm{Indicator~of} ~ u \texttt{(} v \texttt{)} = 1
\end{array}</math>
|}
<br>
<table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%">
<caption><font size="+2"><math>\text{Table 8.} ~~ \text{Simple Qualifiers of Propositions (Version 1)}</math></font></caption>
<tr>
<td width="4%" style="border-bottom:1px solid black" align="right">
<math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td>
<td width="6%" style="border-bottom:1px solid black">
<math>\begin{matrix}1100\\1010\end{matrix}</math></td>
<td width="10%" style="border-bottom:1px solid black; border-right:1px solid black">
<math>f</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\texttt{(} \ell_{11} \texttt{)}
\\
\mathrm{No} ~ u
\\
\mathrm{is} ~ v
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\texttt{(} \ell_{10} \texttt{)}
\\
\mathrm{No} ~ u
\\
\mathrm{is} ~ \texttt{(} v \texttt{)}
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\texttt{(} \ell_{01} \texttt{)}
\\
\mathrm{No} ~ \texttt{(} u \texttt{)}
\\
\mathrm{is} ~ v
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\texttt{(} \ell_{00} \texttt{)}
\\
\mathrm{No} ~ \texttt{(} u \texttt{)}
\\
\mathrm{is} ~ \texttt{(} v \texttt{)}
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\ell_{00}
\\
\mathrm{Some} ~ \texttt{(} u \texttt{)}
\\
\mathrm{is} ~ \texttt{(} v \texttt{)}
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\ell_{01}
\\
\mathrm{Some} ~ \texttt{(} u \texttt{)}
\\
\mathrm{is} ~ v
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\ell_{10}
\\
\mathrm{Some} ~ u
\\
\mathrm{is} ~ \texttt{(} v \texttt{)}
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\ell_{11}
\\
\mathrm{Some} ~ u
\\
\mathrm{is} ~ v
\end{smallmatrix}</math></td></tr>
<tr>
<td><math>f_{0}</math></td>
<td><math>0000</math></td>
<td style="border-right:1px solid black"><math>\texttt{(~)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td><math>f_{1}</math></td>
<td><math>0001</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td><math>f_{2}</math></td>
<td><math>0010</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td><math>f_{3}</math></td>
<td><math>0011</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td><math>f_{4}</math></td>
<td><math>0100</math></td>
<td style="border-right:1px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td><math>f_{5}</math></td>
<td><math>0101</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} v \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td><math>f_{6}</math></td>
<td><math>0110</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td><math>f_{7}</math></td>
<td><math>0111</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td><math>f_{8}</math></td>
<td><math>1000</math></td>
<td style="border-right:1px solid black"><math>u ~ v</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{9}</math></td>
<td><math>1001</math></td>
<td style="border-right:1px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{10}</math></td>
<td><math>1010</math></td>
<td style="border-right:1px solid black"><math>v</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{11}</math></td>
<td><math>1011</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{12}</math></td>
<td><math>1100</math></td>
<td style="border-right:1px solid black"><math>u</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{13}</math></td>
<td><math>1101</math></td>
<td style="border-right:1px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{14}</math></td>
<td><math>1110</math></td>
<td style="border-right:1px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{15}</math></td>
<td><math>1111</math></td>
<td style="border-right:1px solid black"><math>\texttt{((~))}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
</table>
<br>
<table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%">
<caption><font size="+2"><math>\text{Table 9.} ~~ \text{Simple Qualifiers of Propositions (Version 2)}</math></font></caption>
<tr>
<td width="4%" style="border-bottom:1px solid black" align="right">
<math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td>
<td width="6%" style="border-bottom:1px solid black">
<math>\begin{matrix}1100\\1010\end{matrix}</math></td>
<td width="10%" style="border-bottom:1px solid black; border-right:1px solid black">
<math>f</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\texttt{(} \ell_{11} \texttt{)}
\\
\mathrm{No} ~ u
\\
\mathrm{is} ~ v
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\texttt{(} \ell_{10} \texttt{)}
\\
\mathrm{No} ~ u
\\
\mathrm{is} ~ \texttt{(} v \texttt{)}
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\texttt{(} \ell_{01} \texttt{)}
\\
\mathrm{No} ~ \texttt{(} u \texttt{)}
\\
\mathrm{is} ~ v
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\texttt{(} \ell_{00} \texttt{)}
\\
\mathrm{No} ~ \texttt{(} u \texttt{)}
\\
\mathrm{is} ~ \texttt{(} v \texttt{)}
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\ell_{00}
\\
\mathrm{Some} ~ \texttt{(} u \texttt{)}
\\
\mathrm{is} ~ \texttt{(} v \texttt{)}
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\ell_{01}
\\
\mathrm{Some} ~ \texttt{(} u \texttt{)}
\\
\mathrm{is} ~ v
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\ell_{10}
\\
\mathrm{Some} ~ u
\\
\mathrm{is} ~ \texttt{(} v \texttt{)}
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\ell_{11}
\\
\mathrm{Some} ~ u
\\
\mathrm{is} ~ v
\end{smallmatrix}</math></td></tr>
<tr>
<td style="border-bottom:1px solid black"><math>f_{0}</math></td>
<td style="border-bottom:1px solid black"><math>0000</math></td>
<td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{(~)}</math></td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td></tr>
<tr>
<td><math>f_{1}</math></td>
<td><math>0001</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td><math>f_{2}</math></td>
<td><math>0010</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td><math>f_{4}</math></td>
<td><math>0100</math></td>
<td style="border-right:1px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td style="border-bottom:1px solid black"><math>f_{8}</math></td>
<td style="border-bottom:1px solid black"><math>1000</math></td>
<td style="border-bottom:1px solid black; border-right:1px solid black"><math>u ~ v</math></td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
<tr>
<td><math>f_{3}</math></td>
<td><math>0011</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td style="border-bottom:1px solid black"><math>f_{12}</math></td>
<td style="border-bottom:1px solid black"><math>1100</math></td>
<td style="border-bottom:1px solid black; border-right:1px solid black"><math>u</math></td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
<tr>
<td><math>f_{6}</math></td>
<td><math>0110</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td style="border-bottom:1px solid black"><math>f_{9}</math></td>
<td style="border-bottom:1px solid black"><math>1001</math></td>
<td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
<tr>
<td><math>f_{5}</math></td>
<td><math>0101</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} v \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td style="border-bottom:1px solid black"><math>f_{10}</math></td>
<td style="border-bottom:1px solid black"><math>1010</math></td>
<td style="border-bottom:1px solid black; border-right:1px solid black"><math>v</math></td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
<tr>
<td><math>f_{7}</math></td>
<td><math>0111</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td><math>f_{11}</math></td>
<td><math>1011</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{13}</math></td>
<td><math>1101</math></td>
<td style="border-right:1px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td style="border-bottom:1px solid black"><math>f_{14}</math></td>
<td style="border-bottom:1px solid black"><math>1110</math></td>
<td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
<tr>
<td><math>f_{15}</math></td>
<td><math>1111</math></td>
<td style="border-right:1px solid black"><math>\texttt{((~))}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
</table>
<br>
<table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%">
<caption><font size="+2"><math>\text{Table 10.} ~~ \text{Relation of Quantifiers to Higher Order Propositions}</math></font></caption>
<tr>
<td style="border-bottom:1px solid black"><math>\mathrm{Mnemonic}</math></td>
<td style="border-bottom:1px solid black"><math>\mathrm{Category}</math></td>
<td style="border-bottom:1px solid black"><math>\mathrm{Classical~Form}</math></td>
<td style="border-bottom:1px solid black"><math>\mathrm{Alternate~Form}</math></td>
<td style="border-bottom:1px solid black"><math>\mathrm{Symmetric~Form}</math></td>
<td style="border-bottom:1px solid black"><math>\mathrm{Operator}</math></td></tr>
<tr>
<td><math>\begin{matrix}
\mathrm{E}
\\
\mathrm{Exclusive}
\end{matrix}</math></td>
<td><math>\begin{matrix}
\mathrm{Universal}
\\
\mathrm{Negative}
\end{matrix}</math></td>
<td><math>\mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td> </td>
<td><math>\mathrm{No} ~ u ~ \mathrm{is} ~ v</math></td>
<td><math>\texttt{(} \ell_{11} \texttt{)}</math></td></tr>
<tr>
<td style="border-bottom:1px solid black">
<math>\begin{matrix}
\mathrm{A}
\\
\mathrm{Absolute}
\end{matrix}</math></td>
<td style="border-bottom:1px solid black">
<math>\begin{matrix}
\mathrm{Universal}
\\
\mathrm{Affirmative}
\end{matrix}</math></td>
<td style="border-bottom:1px solid black"><math>\mathrm{All} ~ u ~ \mathrm{is} ~ v</math></td>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td style="border-bottom:1px solid black"><math>\texttt{(} \ell_{10} \texttt{)}</math></td></tr>
<tr>
<td> </td>
<td> </td>
<td><math>\mathrm{All} ~ v ~ \mathrm{is} ~ u</math></td>
<td><math>\mathrm{No} ~ v ~ \mathrm{is} ~ \texttt{(} u \texttt{)}</math></td>
<td><math>\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
<td><math>\texttt{(} \ell_{01} \texttt{)}</math></td></tr>
<tr>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"><math>\mathrm{All} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ u</math></td>
<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ \texttt{(} u \texttt{)}</math></td>
<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td style="border-bottom:1px solid black"><math>\texttt{(} \ell_{00} \texttt{)}</math></td></tr>
<tr>
<td> </td>
<td> </td>
<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td> </td>
<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td><math>\ell_{00}</math></td></tr>
<tr>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
<td style="border-bottom:1px solid black"><math>\ell_{01}</math></td></tr>
<tr>
<td><math>\begin{matrix}
\mathrm{O}
\\
\mathrm{Obtrusive}
\end{matrix}</math></td>
<td><math>\begin{matrix}
\mathrm{Particular}
\\
\mathrm{Negative}
\end{matrix}</math></td>
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td> </td>
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td><math>\ell_{10}</math></td></tr>
<tr>
<td><math>\begin{matrix}
\mathrm{I}
\\
\mathrm{Indefinite}
\end{matrix}</math></td>
<td><math>\begin{matrix}
\mathrm{Particular}
\\
\mathrm{Affirmative}
\end{matrix}</math></td>
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td>
<td> </td>
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td>
<td><math>\ell_{11}</math></td></tr>
</table>
<br>
==Inquiry Driven Systems==
==Inquiry Driven Systems==
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:100%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:100%"
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Object}\!</math>
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:100%"
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Object}\!</math>
