Difference between revisions of "User:Jon Awbrey/TABLE"

MyWikiBiz, Author Your Legacy — Thursday March 28, 2024
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(→‎Ascii Tables: cleanup)
 
(31 intermediate revisions by the same user not shown)
Line 338: Line 338:
 
<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"
Line 345: Line 345:
 
| <math>\text{Interpretation}\!</math>
 
| <math>\text{Interpretation}\!</math>
 
|-
 
|-
|
+
| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
<pre>
 
o-------------------o
 
|                   |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <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]]
<pre>
 
o-------------------o
 
|                   |
 
|        o        |
 
|        |        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{(~)}</math>
 
| <math>\texttt{(~)}</math>
 
| <math>\operatorname{false}.</math>
 
| <math>\operatorname{false}.</math>
 
|-
 
|-
|
+
| height="100px" | [[Image:Cactus A Big.jpg|20px]]
<pre>
 
o-------------------o
 
|                   |
 
|        a        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>a\!</math>
 
| <math>a\!</math>
 
| <math>a.\!</math>
 
| <math>a.\!</math>
 
|-
 
|-
|
+
| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
<pre>
 
o-------------------o
 
|                   |
 
|        a        |
 
|        o        |
 
|        |        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <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]]
<pre>
 
o-------------------o
 
|                   |
 
|      a b c      |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>a~b~c</math>
 
| <math>a~b~c</math>
 
|
 
|
Line 421: Line 379:
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
|
+
| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
<pre>
 
o-------------------o
 
|                   |
 
|      a b c      |
 
|      o o o      |
 
|        \|/        |
 
|        o        |
 
|        |        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
 
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
 
|
 
|
Line 442: Line 388:
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
|
+
| height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]]
<pre>
 
o-------------------o
 
|                   |
 
|        a  b    |
 
|        o---o    |
 
|        |        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <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]]
<pre>
 
o-------------------o
 
|                   |
 
|      a  b      |
 
|      o---o      |
 
|        \ /        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <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]]
<pre>
 
o-------------------o
 
|                   |
 
|      a  b      |
 
|      o---o      |
 
|        \ /        |
 
|        o        |
 
|        |        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <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]]
<pre>
 
o-------------------o
 
|                   |
 
|      a  b  c      |
 
|      o--o--o      |
 
|      \  /      |
 
|        \ /        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{(} a, b, c \texttt{)}</math>
 
| <math>\texttt{(} a, b, c \texttt{)}</math>
 
|
 
|
Line 535: Line 438:
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
|
+
| height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]]
<pre>
 
o-------------------o
 
|                   |
 
|      a  b  c      |
 
|      o  o  o      |
 
|      |  |  |      |
 
|      o--o--o      |
 
|      \  /      |
 
|        \ /        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
 
| <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
 
|
 
|
Line 559: Line 449:
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
|
+
| height="160px" | [[Image:Cactus (A,(B),(C)) Big.jpg|65px]]
<pre>
 
o-------------------o
 
|                   |
 
|        b  c      |
 
|        o  o      |
 
|      a  |  |      |
 
|      o--o--o      |
 
|      \  /      |
 
|        \ /        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
 
| <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
 
|
 
|
Line 586: Line 463:
 
<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"
Line 593: Line 470:
 
| <math>\text{Interpretation}\!</math>
 
| <math>\text{Interpretation}\!</math>
 
|-
 
|-
|
+
| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
<pre>
 
o-------------------o
 
|                   |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <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]]
<pre>
 
o-------------------o
 
|                   |
 
|        o        |
 
|        |        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{(~)}</math>
 
| <math>\texttt{(~)}</math>
 
| <math>\operatorname{true}.</math>
 
| <math>\operatorname{true}.</math>
 
|-
 
|-
|
+
| height="100px" | [[Image:Cactus A Big.jpg|20px]]
<pre>
 
o-------------------o
 
|                   |
 
|        a        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>a\!</math>
 
| <math>a\!</math>
 
| <math>a.\!</math>
 
| <math>a.\!</math>
 
|-
 
|-
|
+
| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
<pre>
 
o-------------------o
 
|                   |
 
|        a        |
 
|        o        |
 
|        |        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <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]]
<pre>
 
o-------------------o
 
|                   |
 
|      a b c      |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>a~b~c</math>
 
| <math>a~b~c</math>
 
|
 
|
Line 669: Line 504:
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
|
+
| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
<pre>
 
o-------------------o
 
|                   |
 
|      a b c      |
 
|      o o o      |
 
|        \|/        |
 
|        o        |
 
|        |        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
 
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
 
|
 
|
Line 690: Line 513:
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
|
+
| height="120px" | [[Image:Cactus (A)B Big.jpg|35px]]
<pre>
 
o-------------------o
 
|                   |
 
|        o a      |
 
|        |        |
 
|        @ b      |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{(} a \texttt{)} b</math>
 
| <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]]
<pre>
 
o-------------------o
 
|                   |
 
|      a  b      |
 
|      o---o      |
 
|        \ /        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <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]]
<pre>
 
o-------------------o
 
|                   |
 
|      a  b      |
 
|      o---o      |
 
|        \ /        |
 
|        o        |
 
|        |        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <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]]
<pre>
 
o-------------------o
 
|                   |
 
|      a  b  c      |
 
|      o--o--o      |
 
|      \  /      |
 
|        \ /        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{(} a, b, c \texttt{)}</math>
 
| <math>\texttt{(} a, b, c \texttt{)}</math>
 
|
 
|
Line 782: Line 563:
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
|
+
| height="160px" | [[Image:Cactus ((A,B,C)) Big.jpg|65px]]
<pre>
 
o-------------------o
 
|                   |
 
|      a  b  c      |
 
|      o--o--o      |
 
|      \  /      |
 
|        \ /        |
 
|        o        |
 
|        |        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{((} a, b, c \texttt{))}</math>
 
| <math>\texttt{((} a, b, c \texttt{))}</math>
 
|
 
|
Line 806: Line 574:
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
|
+
| height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|65px]]
<pre>
 
o-------------------o
 
|                   |
 
|      a            |
 
|      o            |
 
|      |  b  c      |
 
|      o--o--o      |
 
|      \  /      |
 
|        \ /        |
 
|        o        |
 
|        |        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
 
| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
 
|
 
|
Line 835: Line 588:
 
<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 C.}~~\text{Dualing Interpretations}</math>
 
|+ <math>\text{Table C.}~~\text{Dualing Interpretations}</math>
 
|- style="background:#f0f0ff"
 
|- style="background:#f0f0ff"
Line 843: Line 596:
 
| <math>\text{Entitative}\!</math>
 
| <math>\text{Entitative}\!</math>
 
|-
 
|-
|
+
| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
<pre>
 
o-------------------o
 
|                   |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
 
| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
 
| <math>\operatorname{true}.</math>
 
| <math>\operatorname{true}.</math>
 
| <math>\operatorname{false}.</math>
 
| <math>\operatorname{false}.</math>
 
|-
 
|-
|
+
| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
<pre>
 
o-------------------o
 
|                   |
 
|        o        |
 
|        |        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{(~)}</math>
 
| <math>\texttt{(~)}</math>
 
| <math>\operatorname{false}.</math>
 
| <math>\operatorname{false}.</math>
 
| <math>\operatorname{true}.</math>
 
| <math>\operatorname{true}.</math>
 
|-
 
|-
|
+
| height="100px" | [[Image:Cactus A Big.jpg|20px]]
<pre>
 
o-------------------o
 
|                   |
 
|        a        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>a\!</math>
 
| <math>a\!</math>
 
| <math>a.\!</math>
 
| <math>a.\!</math>
 
| <math>a.\!</math>
 
| <math>a.\!</math>
 
|-
 
|-
|
+
| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
<pre>
 
o-------------------o
 
|                   |
 
|        a        |
 
|        o        |
 
|        |        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{(} a \texttt{)}</math>
 
| <math>\texttt{(} a \texttt{)}</math>
 
| <math>\lnot a</math>
 
| <math>\lnot a</math>
 
| <math>\lnot a</math>
 
| <math>\lnot a</math>
 
|-
 
|-
|
+
| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
<pre>
 
o-------------------o
 
|                   |
 
|      a b c      |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>a~b~c</math>
 
| <math>a~b~c</math>
 
| <math>a \land b \land c</math>
 
| <math>a \land b \land c</math>
 
| <math>a \lor  b \lor  c</math>
 
| <math>a \lor  b \lor  c</math>
 
|-
 
|-
|
+
| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
<pre>
 
o-------------------o
 
|                   |
 
|      a b c      |
 
|      o o o      |
 
|        \|/        |
 
|        o        |
 
|        |        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
 
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
 
| <math>a \lor  b \lor  c</math>
 
| <math>a \lor  b \lor  c</math>
 
| <math>a \land b \land c</math>
 
| <math>a \land b \land c</math>
 
|-
 
|-
|
+
| height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]]
<pre>
 
o-------------------o
 
|                   |
 
|        a  b    |
 
|        o---o    |
 
|        |        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
 
| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
 
| <math>a \Rightarrow b</math>
 
| <math>a \Rightarrow b</math>
 
| &nbsp;
 
| &nbsp;
 
|-
 
|-
|
+
| height="120px" | [[Image:Cactus (A)B Big.jpg|35px]]
<pre>
 
o-------------------o
 
|                   |
 
|        o a      |
 
|        |        |
 
|        @ b      |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{(} a \texttt{)} b</math>
 
| <math>\texttt{(} a \texttt{)} b</math>
 
| &nbsp;
 
| &nbsp;
 
| <math>a \Rightarrow b</math>
 
| <math>a \Rightarrow b</math>
 
|-
 
|-
|
+
| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
<pre>
 
o-------------------o
 
|                   |
 
|      a  b      |
 
|      o---o      |
 
|        \ /        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{(} a, b \texttt{)}</math>
 
| <math>\texttt{(} a, b \texttt{)}</math>
 
| <math>a \neq b</math>
 
| <math>a \neq b</math>
 
| <math>a  =  b\!</math>
 
| <math>a  =  b\!</math>
 
|-
 
|-
|
+
| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
<pre>
 
o-------------------o
 
|                   |
 
|      a  b      |
 
|      o---o      |
 
|        \ /        |
 
|        o        |
 
|        |        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{((} a, b \texttt{))}</math>
 
| <math>\texttt{((} a, b \texttt{))}</math>
 
| <math>a  =  b\!</math>
 
| <math>a  =  b\!</math>
 
| <math>a \neq b\!</math>
 
| <math>a \neq b\!</math>
 
|-
 
|-
|
+
| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
<pre>
 
o-------------------o
 
|                   |
 
|      a  b  c      |
 
|      o--o--o      |
 
|      \  /      |
 
|        \ /        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{(} a, b, c \texttt{)}</math>
 
| <math>\texttt{(} a, b, c \texttt{)}</math>
 
|
 
|
Line 1,018: Line 665:
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
|
+
| height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]]
<pre>
 
o-------------------o
 
|                   |
 
|      a  b  c      |
 
|      o  o  o      |
 
|      |  |  |      |
 
|      o--o--o      |
 
|      \  /      |
 
|        \ /        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
 
| <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
 
|
 
|
Line 1,050: Line 684:
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
|
+
| height="160px" | [[Image:Cactus ((A,B,C)) Big.jpg|65px]]
<pre>
 
o-------------------o
 
|                   |
 
|      a  b  c      |
 
|      o--o--o      |
 
|      \  /      |
 
|        \ /        |
 
|        o        |
 
|        |        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{((} a, b, c \texttt{))}</math>
 
| <math>\texttt{((} a, b, c \texttt{))}</math>
 
|
 
|
Line 1,082: Line 703:
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
|
+
| height="200px" | [[Image:Cactus (((A),(B),(C))) Big.jpg|65px]]
<pre>
 
o-------------------o
 
|                   |
 
|      a  b  c      |
 
|      o  o  o      |
 
|      |  |  |      |
 
|      o--o--o      |
 
|      \  /      |
 
|        \ /        |
 
|        o        |
 
|        |        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{(((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{)))}</math>
 
| <math>\texttt{(((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{)))}</math>
 
|
 
|
Line 1,116: Line 722:
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
|
+
| height="160px" | [[Image:Cactus (A,(B),(C)) Big.jpg|65px]]
<pre>
 
o-------------------o
 
|                   |
 
|        b  c      |
 
|        o  o      |
 
|      a  |  |      |
 
|      o--o--o      |
 
|      \  /      |
 
|        \ /        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
 
| <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
 
|
 
|
Line 1,139: Line 732:
 
| &nbsp;
 
| &nbsp;
 
|-
 
|-
|
+
| height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|65px]]
<pre>
 
o-------------------o
 
|                   |
 
|      a            |
 
|      o            |
 
|      |  b  c      |
 
|      o--o--o      |
 
|      \  /      |
 
|        \ /        |
 
|        o        |
 
|        |        |
 
|        @        |
 
|                  |
 
o-------------------o
 
</pre>
 
 
| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
 
| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
 
| &nbsp;
 
| &nbsp;
Line 1,164: Line 742:
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|}
 
|}
 +
 +
<br>
  
 
==Differential Logic==
 
==Differential Logic==
Line 2,582: Line 2,162:
 
<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"
Line 2,839: Line 2,419:
 
<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"
Line 3,125: Line 2,705:
 
<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"
Line 3,399: Line 2,979:
 
<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"
Line 3,663: Line 3,243:
 
<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"
Line 3,923: Line 3,503:
 
<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"
Line 4,187: Line 3,767:
 
<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"
Line 4,338: Line 3,918:
 
<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"
Line 4,595: Line 4,175:
 
<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"
Line 4,881: Line 4,461:
 
<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"
Line 5,155: Line 4,735:
 
<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"
Line 5,419: Line 4,999:
 
<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"
Line 5,679: Line 5,259:
 
<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"
Line 5,945: Line 5,525:
 
{| 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>
Line 5,984: Line 5,564:
 
{| 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>
Line 6,025: Line 5,605:
 
<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"
Line 6,087: Line 5,667:
 
<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"
Line 6,751: Line 6,331:
 
\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>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
 +
 +
<tr>
 +
<td style="border-right:2px solid black"><math>m_{1}</math></td>
 +
<td>&nbsp;</td>
 +
<td>Just false</td>
 +
<td>Nothing exists</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
 +
 +
<tr>
 +
<td style="border-right:2px solid black"><math>m_{2}</math></td>
 +
<td>&nbsp;</td>
 +
<td>Just not <math>x</math></td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
 +
 +
<tr>
 +
<td style="border-right:2px solid black"><math>m_{3}</math></td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>Nothing is <math>x</math></td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
 +
 +
<tr>
 +
<td style="border-right:2px solid black"><math>m_{4}</math></td>
 +
<td>&nbsp;</td>
 +
<td>Just <math>x</math></td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
 +
 +
<tr>
 +
<td style="border-right:2px solid black"><math>m_{5}</math></td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>Everything is <math>x</math></td>
 +
<td><math>f</math> is linear</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
 +
 +
<tr>
 +
<td style="border-right:2px solid black"><math>m_{6}</math></td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</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>&nbsp;</td>
 +
<td>Not just true</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
 +
 +
<tr>
 +
<td style="border-right:2px solid black"><math>m_{8}</math></td>
 +
<td>&nbsp;</td>
 +
<td>Just true</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
 +
 +
<tr>
 +
<td style="border-right:2px solid black"><math>m_{9}</math></td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</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>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>Something is not <math>x</math></td>
 +
<td><math>f</math> is not linear</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
 +
 +
<tr>
 +
<td style="border-right:2px solid black"><math>m_{11}</math></td>
 +
<td>&nbsp;</td>
 +
<td>Not just <math>x</math></td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
 +
 +
<tr>
 +
<td style="border-right:2px solid black"><math>m_{12}</math></td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>Something is <math>x</math></td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
 +
 +
<tr>
 +
<td style="border-right:2px solid black"><math>m_{13}</math></td>
 +
<td>&nbsp;</td>
 +
<td>Not just not <math>x</math></td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
 +
 +
<tr>
 +
<td style="border-right:2px solid black"><math>m_{14}</math></td>
 +
<td>&nbsp;</td>
 +
<td>Not just false</td>
 +
<td>Something exists</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
 +
 +
<tr>
 +
<td style="border-right:2px solid black"><math>m_{15}</math></td>
 +
<td>Anything happens</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</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>&nbsp;</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">&nbsp;</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>&nbsp;</td>
 +
<td>&nbsp;</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">&nbsp;</td>
 +
<td style="border-bottom:1px solid black">&nbsp;</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>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
 +
<td>&nbsp;</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">&nbsp;</td>
 +
<td style="border-bottom:1px solid black">&nbsp;</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">&nbsp;</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>&nbsp;</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>&nbsp;</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==
Line 8,624: Line 10,595:
 
|-
 
|-
 
|
 
|
{| 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>
Line 8,654: Line 10,625:
 
|-
 
|-
 
|
 
|
{| 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>

Latest revision as of 03:22, 26 April 2012

Cactus Language

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|       o o o       |
|        \|/        |
|         o         |
|         |         |
|         @         |
|                   |
o-------------------o
|                   |
|         a   b     |
|         o---o     |
|         |         |
|         @         |
|                   |
o-------------------o
|                   |
|       a   b       |
|       o---o       |
|        \ /        |
|         @         |
|                   |
o-------------------o
|                   |
|       a   b       |
|       o---o       |
|        \ /        |
|         o         |
|         |         |
|         @         |
|                   |
o-------------------o
|                   |
|      a  b  c      |
|      o--o--o      |
|       \   /       |
|        \ /        |
|         @         |
|                   |
o-------------------o
|                   |
|      a  b  c      |
|      o  o  o      |
|      |  |  |      |
|      o--o--o      |
|       \   /       |
|        \ /        |
|         @         |
|                   |
o-------------------o
|                   |
|         b  c      |
|         o  o      |
|      a  |  |      |
|      o--o--o      |
|       \   /       |
|        \ /        |
|         @         |
|                   |
o-------------------o
Table 13.  The Existential Interpretation
o----o-------------------o-------------------o-------------------o
| Ex |   Cactus Graph    | Cactus Expression |    Existential    |
|    |                   |                   |  Interpretation   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|  1 |         @         |        " "        |       true.       |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  2 |         @         |        ( )        |      untrue.      |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         a         |                   |                   |
|  3 |         @         |         a         |         a.        |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         a         |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  4 |         @         |        (a)        |       not a.      |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a b c       |                   |                   |
|  5 |         @         |       a b c       |   a and b and c.  |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a b c       |                   |                   |
|    |       o o o       |                   |                   |
|    |        \|/        |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  6 |         @         |    ((a)(b)(c))    |    a or b or c.   |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |                   |                   |    a implies b.   |
|    |         a   b     |                   |                   |
|    |         o---o     |                   |    if a then b.   |
|    |         |         |                   |                   |
|  7 |         @         |     ( a (b))      |    no a sans b.   |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a   b       |                   |                   |
|    |       o---o       |                   | a exclusive-or b. |
|    |        \ /        |                   |                   |
|  8 |         @         |     ( a , b )     | a not equal to b. |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a   b       |                   |                   |
|    |       o---o       |                   |                   |
|    |        \ /        |                   |                   |
|    |         o         |                   | a if & only if b. |
|    |         |         |                   |                   |
|  9 |         @         |    (( a , b ))    | a equates with b. |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |      a  b  c      |                   |                   |
|    |      o--o--o      |                   |                   |
|    |       \   /       |                   |                   |
|    |        \ /        |                   |  just one false   |
| 10 |         @         |   ( a , b , c )   |  out of a, b, c.  |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |      a  b  c      |                   |                   |
|    |      o  o  o      |                   |                   |
|    |      |  |  |      |                   |                   |
|    |      o--o--o      |                   |                   |
|    |       \   /       |                   |                   |
|    |        \ /        |                   |   just one true   |
| 11 |         @         |   ((a),(b),(c))   |   among a, b, c.  |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |                   |                   |   genus a over    |
|    |         b  c      |                   |   species b, c.   |
|    |         o  o      |                   |                   |
|    |      a  |  |      |                   |   partition a     |
|    |      o--o--o      |                   |   among b & c.    |
|    |       \   /       |                   |                   |
|    |        \ /        |                   |   whole pie a:    |
| 12 |         @         |   ( a ,(b),(c))   |   slices b, c.    |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
Table 14.  The Entitative Interpretation
o----o-------------------o-------------------o-------------------o
| En |   Cactus Graph    | Cactus Expression |    Entitative     |
|    |                   |                   |  Interpretation   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|  1 |         @         |        " "        |      untrue.      |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  2 |         @         |        ( )        |       true.       |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         a         |                   |                   |
|  3 |         @         |         a         |         a.        |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         a         |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  4 |         @         |        (a)        |       not a.      |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a b c       |                   |                   |
|  5 |         @         |       a b c       |    a or b or c.   |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a b c       |                   |                   |
|    |       o o o       |                   |                   |
|    |        \|/        |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  6 |         @         |    ((a)(b)(c))    |   a and b and c.  |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |                   |                   |    a implies b.   |
|    |                   |                   |                   |
|    |         o a       |                   |    if a then b.   |
|    |         |         |                   |                   |
|  7 |         @ b       |      (a) b        |    not a, or b.   |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a   b       |                   |                   |
|    |       o---o       |                   | a if & only if b. |
|    |        \ /        |                   |                   |
|  8 |         @         |     ( a , b )     | a equates with b. |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a   b       |                   |                   |
|    |       o---o       |                   |                   |
|    |        \ /        |                   |                   |
|    |         o         |                   | a exclusive-or b. |
|    |         |         |                   |                   |
|  9 |         @         |    (( a , b ))    | a not equal to b. |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |      a  b  c      |                   |                   |
|    |      o--o--o      |                   |                   |
|    |       \   /       |                   |                   |
|    |        \ /        |                   | not just one true |
| 10 |         @         |   ( a , b , c )   | out of a, b, c.   |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |      a  b  c      |                   |                   |
|    |      o--o--o      |                   |                   |
|    |       \   /       |                   |                   |
|    |        \ /        |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |   just one true   |
| 11 |         @         |  (( a , b , c ))  |   among a, b, c.  |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |      a            |                   |                   |
|    |      o            |                   |   genus a over    |
|    |      |  b  c      |                   |   species b, c.   |
|    |      o--o--o      |                   |                   |
|    |       \   /       |                   |   partition a     |
|    |        \ /        |                   |   among b & c.    |
|    |         o         |                   |                   |
|    |         |         |                   |   whole pie a:    |
| 12 |         @         |  (((a), b , c ))  |   slices b, c.    |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
Table 15.  Existential & Entitative Interpretations of Cactus Structures
o-----------------o-----------------o-----------------o-----------------o
|  Cactus Graph   |  Cactus String  |  Existential    |   Entitative    |
|                 |                 | Interpretation  | Interpretation  |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|        @        |       " "       |      true       |      false      |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|        o        |                 |                 |                 |
|        |        |                 |                 |                 |
|        @        |       ( )       |      false      |      true       |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|   C_1 ... C_k   |                 |                 |                 |
|        @        |   C_1 ... C_k   | C_1 & ... & C_k | C_1 v ... v C_k |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|  C_1 C_2   C_k  |                 |  Just one       |  Not just one   |
|   o---o-...-o   |                 |                 |                 |
|    \       /    |                 |  of the C_j,    |  of the C_j,    |
|     \     /     |                 |                 |                 |
|      \   /      |                 |  j = 1 to k,    |  j = 1 to k,    |
|       \ /       |                 |                 |                 |
|        @        | (C_1, ..., C_k) |  is not true.   |  is true.       |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o

Wiki TeX Tables


\(\text{Table A.}~~\text{Existential Interpretation}\)
\(\text{Cactus Graph}\!\) \(\text{Cactus Expression}\!\) \(\text{Interpretation}\!\)
Cactus Node Big Fat.jpg \({}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}\) \(\operatorname{true}.\)
Cactus Spike Big Fat.jpg \(\texttt{(~)}\) \(\operatorname{false}.\)
Cactus A Big.jpg \(a\!\) \(a.\!\)
Cactus (A) Big.jpg \(\texttt{(} a \texttt{)}\)

\(\begin{matrix} \tilde{a} \'"`UNIQ-MathJax1-QINU`"' '''Generalized''' or '''n-ary''' XOR is true when the number of 1-bits is odd. '"`UNIQ--pre-0000001A-QINU`"' '"`UNIQ--pre-0000001B-QINU`"' '"`UNIQ--pre-0000001C-QINU`"' '"`UNIQ-MathJax2-QINU`"' ===='"`UNIQ--h-39--QINU`"'[[Logical implication]]==== The '''material conditional''' and '''logical implication''' are both associated with an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if the first operand is true and the second operand is false. The [[truth table]] associated with the material conditional '''if p then q''' (symbolized as '''p → q''') and the logical implication '''p implies q''' (symbolized as '''p ⇒ q''') is as follows: {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" |+ '''Logical Implication''' |- style="background:aliceblue" ! style="width:15%" | p ! style="width:15%" | q ! style="width:15%" | p ⇒ q |- | F || F || T |- | F || T || T |- | T || F || F |- | T || T || T |} <br> ===='"`UNIQ--h-40--QINU`"'[[Logical NAND]]==== The '''NAND operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if both of its operands are true. In other words, it produces a value of ''true'' if and only if at least one of its operands is false. The [[truth table]] of '''p NAND q''' (also written as '''p | q''' or '''p ↑ q''') is as follows: {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" |+ '''Logical NAND''' |- style="background:aliceblue" ! style="width:15%" | p ! style="width:15%" | q ! style="width:15%" | p ↑ q |- | F || F || T |- | F || T || T |- | T || F || T |- | T || T || F |} <br> ===='"`UNIQ--h-41--QINU`"'[[Logical NNOR]]==== The '''NNOR operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both of its operands are false. In other words, it produces a value of ''false'' if and only if at least one of its operands is true. The [[truth table]] of '''p NNOR q''' (also written as '''p ⊥ q''' or '''p ↓ q''') is as follows: {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" |+ '''Logical NOR''' |- style="background:aliceblue" ! style="width:15%" | p ! style="width:15%" | q ! style="width:15%" | p ↓ q |- | F || F || T |- | F || T || F |- | T || F || F |- | T || T || F |} <br> =='"`UNIQ--h-42--QINU`"'Relational Tables== ==='"`UNIQ--h-43--QINU`"'Factorization=== {| align="center" style="text-align:center; width:60%" | {| align="center" style="text-align:center; width:100%" | \(\text{Table 7. Plural Denotation}\!\)

|- |

\(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\)

\(\begin{matrix} o_1 \\ o_2 \\ o_3 \\ \ldots \\ o_k \\ \ldots \end{matrix}\)

\(\begin{matrix} s \\ s \\ s \\ \ldots \\ s \\ \ldots \end{matrix}\)

\(\begin{matrix} \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \end{matrix}\)

|}


\(\text{Table 8. Sign Relation}~ L\)
\(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\)

\(\begin{matrix} o_1 \\ o_2 \\ o_3 \end{matrix}\)

\(\begin{matrix} s \\ s \\ s \end{matrix}\)

\(\begin{matrix} \ldots \\ \ldots \\ \ldots \end{matrix}\)

Sign Relations

  O = Object Domain
  S = Sign Domain
  I = Interpretant Domain


  O = {Ann, Bob} = {A, B}
  S = {"Ann", "Bob", "I", "You"} = {"A", "B", "i", "u"}
  I = {"Ann", "Bob", "I", "You"} = {"A", "B", "i", "u"}


LA = Sign Relation of Interpreter A
Object Sign Interpretant
A "A" "A"
A "A" "i"
A "i" "A"
A "i" "i"
B "B" "B"
B "B" "u"
B "u" "B"
B "u" "u"


LB = Sign Relation of Interpreter B
Object Sign Interpretant
A "A" "A"
A "A" "u"
A "u" "A"
A "u" "u"
B "B" "B"
B "B" "i"
B "i" "B"
B "i" "i"


Triadic Relations

Algebraic Examples

L0 = {(x, y, z) ∈ B3 : x + y + z = 0}
X Y Z
0 0 0
0 1 1
1 0 1
1 1 0


L1 = {(x, y, z) ∈ B3 : x + y + z = 1}
X Y Z
0 0 1
0 1 0
1 0 0
1 1 1


Semiotic Examples

LA = Sign Relation of Interpreter A
Object Sign Interpretant
A "A" "A"
A "A" "i"
A "i" "A"
A "i" "i"
B "B" "B"
B "B" "u"
B "u" "B"
B "u" "u"


LB = Sign Relation of Interpreter B
Object Sign Interpretant
A "A" "A"
A "A" "u"
A "u" "A"
A "u" "u"
B "B" "B"
B "B" "i"
B "i" "B"
B "i" "i"


Dyadic Projections

  LOS = projOS(L) = { (o, s) ∈ O × S : (o, s, i) ∈ L for some iI }
  LSO = projSO(L) = { (s, o) ∈ S × O : (o, s, i) ∈ L for some iI }
  LIS = projIS(L) = { (i, s) ∈ I × S : (o, s, i) ∈ L for some oO }
  LSI = projSI(L) = { (s, i) ∈ S × I : (o, s, i) ∈ L for some oO }
  LOI = projOI(L) = { (o, i) ∈ O × I : (o, s, i) ∈ L for some sS }
  LIO = projIO(L) = { (i, o) ∈ I × O : (o, s, i) ∈ L for some sS }


Method 1 : Subtitles as Captions

projOS(LA)
Object Sign
A "A"
A "i"
B "B"
B "u"
projOS(LB)
Object Sign
A "A"
A "u"
B "B"
B "i"


projSI(LA)
Sign Interpretant
"A" "A"
"A" "i"
"i" "A"
"i" "i"
"B" "B"
"B" "u"
"u" "B"
"u" "u"
projSI(LB)
Sign Interpretant
"A" "A"
"A" "u"
"u" "A"
"u" "u"
"B" "B"
"B" "i"
"i" "B"
"i" "i"


projOI(LA)
Object Interpretant
A "A"
A "i"
B "B"
B "u"
projOI(LB)
Object Interpretant
A "A"
A "u"
B "B"
B "i"


Method 2 : Subtitles as Top Rows

projOS(LA)
Object Sign
A "A"
A "i"
B "B"
B "u"
projOS(LB)
Object Sign
A "A"
A "u"
B "B"
B "i"


projSI(LA)
Sign Interpretant
"A" "A"
"A" "i"
"i" "A"
"i" "i"
"B" "B"
"B" "u"
"u" "B"
"u" "u"
projSI(LB)
Sign Interpretant
"A" "A"
"A" "u"
"u" "A"
"u" "u"
"B" "B"
"B" "i"
"i" "B"
"i" "i"


projOI(LA)
Object Interpretant
A "A"
A "i"
B "B"
B "u"
projOI(LB)
Object Interpretant
A "A"
A "u"
B "B"
B "i"


Relation Reduction

Method 1 : Subtitles as Captions

L0 = {(x, y, z) ∈ B3 : x + y + z = 0}
X Y Z
0 0 0
0 1 1
1 0 1
1 1 0


L1 = {(x, y, z) ∈ B3 : x + y + z = 1}
X Y Z
0 0 1
0 1 0
1 0 0
1 1 1


projXY(L0)
X Y
0 0
0 1
1 0
1 1
projXZ(L0)
X Z
0 0
0 1
1 1
1 0
projYZ(L0)
Y Z
0 0
1 1
0 1
1 0


projXY(L1)
X Y
0 0
0 1
1 0
1 1
projXZ(L1)
X Z
0 1
0 0
1 0
1 1
projYZ(L1)
Y Z
0 1
1 0
0 0
1 1


projXY(L0) = projXY(L1) projXZ(L0) = projXZ(L1) projYZ(L0) = projYZ(L1)


LA = Sign Relation of Interpreter A
Object Sign Interpretant
A "A" "A"
A "A" "i"
A "i" "A"
A "i" "i"
B "B" "B"
B "B" "u"
B "u" "B"
B "u" "u"


LB = Sign Relation of Interpreter B
Object Sign Interpretant
A "A" "A"
A "A" "u"
A "u" "A"
A "u" "u"
B "B" "B"
B "B" "i"
B "i" "B"
B "i" "i"


projXY(LA)
Object Sign
A "A"
A "i"
B "B"
B "u"
projXZ(LA)
Object Interpretant
A "A"
A "i"
B "B"
B "u"
projYZ(LA)
Sign Interpretant
"A" "A"
"A" "i"
"i" "A"
"i" "i"
"B" "B"
"B" "u"
"u" "B"
"u" "u"


projXY(LB)
Object Sign
A "A"
A "u"
B "B"
B "i"
projXZ(LB)
Object Interpretant
A "A"
A "u"
B "B"
B "i"
projYZ(LB)
Sign Interpretant
"A" "A"
"A" "u"
"u" "A"
"u" "u"
"B" "B"
"B" "i"
"i" "B"
"i" "i"


projXY(LA) ≠ projXY(LB) projXZ(LA) ≠ projXZ(LB) projYZ(LA) ≠ projYZ(LB)


Method 2 : Subtitles as Top Rows

L0 = {(x, y, z) ∈ B3 : x + y + z = 0}
X Y Z
0 0 0
0 1 1
1 0 1
1 1 0


L1 = {(x, y, z) ∈ B3 : x + y + z = 1}
X Y Z
0 0 1
0 1 0
1 0 0
1 1 1


projXY(L0)
X Y
0 0
0 1
1 0
1 1
projXZ(L0)
X Z
0 0
0 1
1 1
1 0
projYZ(L0)
Y Z
0 0
1 1
0 1
1 0


projXY(L1)
X Y
0 0
0 1
1 0
1 1
projXZ(L1)
X Z
0 1
0 0
1 0
1 1
projYZ(L1)
Y Z
0 1
1 0
0 0
1 1


projXY(L0) = projXY(L1) projXZ(L0) = projXZ(L1) projYZ(L0) = projYZ(L1)


LA = Sign Relation of Interpreter A
Object Sign Interpretant
A "A" "A"
A "A" "i"
A "i" "A"
A "i" "i"
B "B" "B"
B "B" "u"
B "u" "B"
B "u" "u"


LB = Sign Relation of Interpreter B
Object Sign Interpretant
A "A" "A"
A "A" "u"
A "u" "A"
A "u" "u"
B "B" "B"
B "B" "i"
B "i" "B"
B "i" "i"


projXY(LA)
Object Sign
A "A"
A "i"
B "B"
B "u"
projXZ(LA)
Object Interpretant
A "A"
A "i"
B "B"
B "u"
projYZ(LA)
Sign Interpretant
"A" "A"
"A" "i"
"i" "A"
"i" "i"
"B" "B"
"B" "u"
"u" "B"
"u" "u"


projXY(LB)
Object Sign
A "A"
A "u"
B "B"
B "i"
projXZ(LB)
Object Interpretant
A "A"
A "u"
B "B"
B "i"
projYZ(LB)
Sign Interpretant
"A" "A"
"A" "u"
"u" "A"
"u" "u"
"B" "B"
"B" "i"
"i" "B"
"i" "i"


projXY(LA) ≠ projXY(LB) projXZ(LA) ≠ projXZ(LB) projYZ(LA) ≠ projYZ(LB)


Formatted Text Display

So in a triadic fact, say, the example
A gives B to C
we make no distinction in the ordinary logic of relations between the subject nominative, the direct object, and the indirect object. We say that the proposition has three logical subjects. We regard it as a mere affair of English grammar that there are six ways of expressing this:
A gives B to C A benefits C with B
B enriches C at expense of A C receives B from A
C thanks A for B B leaves A for C
These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, "The Categories Defended", MS 308 (1903), EP 2, 170-171).

Work Area

Binary Operations
x0 x1 2f0 2f1 2f2 2f3 2f4 2f5 2f6 2f7 2f8 2f9 2f10 2f11 2f12 2f13 2f14 2f15
0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1


Draft 1

TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2
Constants
0f0 0f1
0 1
    
Unary Operations
x0 1f0 1f1 1f2 1f3
0 0 1 0 1
1 0 0 1 1
    
Binary Operations
x0 x1 2f0 2f1 2f2 2f3 2f4 2f5 2f6 2f7 2f8 2f9 2f10 2f11 2f12 2f13 2f14 2f15
0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

Draft 2

TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2
Constants
0f0 0f1
0 1
    
Unary Operations
x0 1f0 1f1 1f2 1f3
0 0 1 0 1
1 0 0 1 1
    
Binary Operations
x0 x1 2f0 2f1 2f2 2f3 2f4 2f5 2f6 2f7 2f8 2f9 2f10 2f11 2f12 2f13 2f14 2f15
0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

Inquiry and Analogy

Test Patterns

1 0 1 0 1 0 1 0
0 1 0 1 0 1 0 1


1 0 1 0 1 0 1 0
0 1 0 1 0 1 0 1


1 0 1 0 1 0 1 0
0 1 0 1 0 1 0 1


Table 10

Table 10. Higher Order Propositions (n = 1)
\(x\): 1 0 \(f\) \(m_0\) \(m_1\) \(m_2\) \(m_3\) \(m_4\) \(m_5\) \(m_6\) \(m_7\) \(m_8\) \(m_9\) \(m_{10}\) \(m_{11}\) \(m_{12}\) \(m_{13}\) \(m_{14}\) \(m_{15}\)
\(f_0\) 0 0 \(0\!\) 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
\(f_1\) 0 1 \((x)\!\) 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
\(f_2\) 1 0 \(x\!\) 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
\(f_3\) 1 1 \(1\!\) 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1


Table 10. Higher Order Propositions (n = 1)
\(x:\) 1 0 \(f\!\) \(m_0\) \(m_1\) \(m_2\) \(m_3\) \(m_4\) \(m_5\) \(m_6\) \(m_7\) \(m_8\) \(m_9\) \(m_{10}\) \(m_{11}\) \(m_{12}\) \(m_{13}\) \(m_{14}\) \(m_{15}\)
\(f_0\) 0 0 \(0\!\) 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
\(f_1\) 0 1 \((x)\!\) 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
\(f_2\) 1 0 \(x\!\) 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
\(f_3\) 1 1 \(1\!\) 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1


Table 11

Table 11. Interpretive Categories for Higher Order Propositions (n = 1)
Measure Happening Exactness Existence Linearity Uniformity Information
\(m_0\!\) Nothing happens          
\(m_1\!\)   Just false Nothing exists      
\(m_2\!\)   Just not \(x\!\)        
\(m_3\!\)     Nothing is \(x\!\)      
\(m_4\!\)   Just \(x\!\)        
\(m_5\!\)     Everything is \(x\!\) \(f\!\) is linear    
\(m_6\!\)         \(f\!\) is not uniform \(f\!\) is informed
\(m_7\!\)   Not just true        
\(m_8\!\)   Just true        
\(m_9\!\)         \(f\!\) is uniform \(f\!\) is not informed
\(m_{10}\!\)     Something is not \(x\!\) \(f\!\) is not linear    
\(m_{11}\!\)   Not just \(x\!\)        
\(m_{12}\!\)     Something is \(x\!\)      
\(m_{13}\!\)   Not just not \(x\!\)        
\(m_{14}\!\)   Not just false Something exists      
\(m_{15}\!\) Anything happens          


Table 12

Table 12. Higher Order Propositions (n = 2)
\(x:\)
\(y:\)
1100
1010
\(f\!\) \(m_0\) \(m_1\) \(m_2\) \(m_3\) \(m_4\) \(m_5\) \(m_6\) \(m_7\) \(m_8\) \(m_9\) \(m_{10}\) \(m_{11}\) \(m_{12}\) \(m_{13}\) \(m_{14}\) \(m_{15}\) \(m_{16}\) \(m_{17}\) \(m_{18}\) \(m_{19}\) \(m_{20}\) \(m_{21}\) \(m_{22}\) \(m_{23}\)
\(f_0\) 0000 \((~)\) 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
\(f_1\) 0001 \((x)(y)\!\)     1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
\(f_2\) 0010 \((x) y\!\)         1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
\(f_3\) 0011 \((x)\!\)                 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
\(f_4\) 0100 \(x (y)\!\)                                 1 1 1 1 1 1 1 1
\(f_5\) 0101 \((y)\!\)                                                
\(f_6\) 0110 \((x, y)\!\)                                                
\(f_7\) 0111 \((x y)\!\)                                                
\(f_8\) 1000 \(x y\!\)                                                
\(f_9\) 1001 \(((x, y))\!\)                                                
\(f_{10}\) 1010 \(y\!\)                                                
\(f_{11}\) 1011 \((x (y))\!\)                                                
\(f_{12}\) 1100 \(x\!\)                                                
\(f_{13}\) 1101 \(((x) y)\!\)                                                
\(f_{14}\) 1110 \(((x)(y))\!\)                                                
\(f_{15}\) 1111 \(((~))\!\)                                                


Table 12. Higher Order Propositions (n = 2)
\(u:\)
\(v:\)
1100
1010
\(f\!\) \(m_0\) \(m_1\) \(m_2\) \(m_3\) \(m_4\) \(m_5\) \(m_6\) \(m_7\) \(m_8\) \(m_9\) \(m_{10}\) \(m_{11}\) \(m_{12}\) \(m_{13}\) \(m_{14}\) \(m_{15}\) \(m_{16}\) \(m_{17}\) \(m_{18}\) \(m_{19}\) \(m_{20}\) \(m_{21}\) \(m_{22}\) \(m_{23}\)
\(f_0\) 0000 \((~)\) 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
\(f_1\) 0001 \((u)(v)\!\) 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
\(f_2\) 0010 \((u) v\!\) 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
\(f_3\) 0011 \((u)\!\) 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
\(f_4\) 0100 \(u (v)\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
\(f_5\) 0101 \((v)\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_6\) 0110 \((u, v)\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_7\) 0111 \((u v)\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_8\) 1000 \(u v\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_9\) 1001 \(((u, v))\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_{10}\) 1010 \(v\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_{11}\) 1011 \((u (v))\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_{12}\) 1100 \(u\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_{13}\) 1101 \(((u) v)\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_{14}\) 1110 \(((u)(v))\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_{15}\) 1111 \(((~))\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0


Table 13

Table 13. Qualifiers of Implication Ordering:  \(\alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f)\)
\(u:\)
\(v:\)
1100
1010
\(f\!\) \(\alpha_0\) \(\alpha_1\) \(\alpha_2\) \(\alpha_3\) \(\alpha_4\) \(\alpha_5\) \(\alpha_6\) \(\alpha_7\) \(\alpha_8\) \(\alpha_9\) \(\alpha_{10}\) \(\alpha_{11}\) \(\alpha_{12}\) \(\alpha_{13}\) \(\alpha_{14}\) \(\alpha_{15}\)
\(f_0\) 0000 \((~)\) 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_1\) 0001 \((u)(v)\!\) 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_2\) 0010 \((u) v\!\) 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_3\) 0011 \((u)\!\) 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
\(f_4\) 0100 \(u (v)\!\) 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
\(f_5\) 0101 \((v)\!\) 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0
\(f_6\) 0110 \((u, v)\!\) 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0
\(f_7\) 0111 \((u v)\!\) 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
\(f_8\) 1000 \(u v\!\) 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
\(f_9\) 1001 \(((u, v))\!\) 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0
\(f_{10}\) 1010 \(v\!\) 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0
\(f_{11}\) 1011 \((u (v))\!\) 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0
\(f_{12}\) 1100 \(u\!\) 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0
\(f_{13}\) 1101 \(((u) v)\!\) 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0
\(f_{14}\) 1110 \(((u)(v))\!\) 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
\(f_{15}\) 1111 \(((~))\) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1


Table 14

Table 14. Qualifiers of Implication Ordering:  \(\beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)\)
\(u:\)
\(v:\)
1100
1010
\(f\!\) \(\beta_0\) \(\beta_1\) \(\beta_2\) \(\beta_3\) \(\beta_4\) \(\beta_5\) \(\beta_6\) \(\beta_7\) \(\beta_8\) \(\beta_9\) \(\beta_{10}\) \(\beta_{11}\) \(\beta_{12}\) \(\beta_{13}\) \(\beta_{14}\) \(\beta_{15}\)
\(f_0\) 0000 \((~)\) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
\(f_1\) 0001 \((u)(v)\!\) 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
\(f_2\) 0010 \((u) v\!\) 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
\(f_3\) 0011 \((u)\!\) 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1
\(f_4\) 0100 \(u (v)\!\) 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
\(f_5\) 0101 \((v)\!\) 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1
\(f_6\) 0110 \((u, v)\!\) 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1
\(f_7\) 0111 \((u v)\!\) 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
\(f_8\) 1000 \(u v\!\) 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
\(f_9\) 1001 \(((u, v))\!\) 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1
\(f_{10}\) 1010 \(v\!\) 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1
\(f_{11}\) 1011 \((u (v))\!\) 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1
\(f_{12}\) 1100 \(u\!\) 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1
\(f_{13}\) 1101 \(((u) v)\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1
\(f_{14}\) 1110 \(((u)(v))\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1
\(f_{15}\) 1111 \(((~))\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1


Figure 15

Table 16

Table 16. Syllogistic Premisses as Higher Order Indicator Functions

\(\begin{array}{clcl} \mathrm{A} & \mathrm{Universal~Affirmative} & \mathrm{All}\ u\ \mathrm{is}\ v & \mathrm{Indicator~of}\ u (v) = 0 \\ \mathrm{E} & \mathrm{Universal~Negative} & \mathrm{All}\ u\ \mathrm{is}\ (v) & \mathrm{Indicator~of}\ u \cdot v = 0 \\ \mathrm{I} & \mathrm{Particular~Affirmative} & \mathrm{Some}\ u\ \mathrm{is}\ v & \mathrm{Indicator~of}\ u \cdot v = 1 \\ \mathrm{O} & \mathrm{Particular~Negative} & \mathrm{Some}\ u\ \mathrm{is}\ (v) & \mathrm{Indicator~of}\ u (v) = 1 \\ \end{array}\)


Table 17

Table 17. Simple Qualifiers of Propositions (Version 1)
\(u:\)
\(v:\)
1100
1010
\(f\!\) \((\ell_{11})\)
\(\text{No } u \)
\(\text{is } v \)
\((\ell_{10})\)
\(\text{No } u \)
\(\text{is }(v)\)
\((\ell_{01})\)
\(\text{No }(u)\)
\(\text{is } v \)
\((\ell_{00})\)
\(\text{No }(u)\)
\(\text{is }(v)\)
\( \ell_{00} \)
\(\text{Some }(u)\)
\(\text{is }(v)\)
\( \ell_{01} \)
\(\text{Some }(u)\)
\(\text{is } v \)
\( \ell_{10} \)
\(\text{Some } u \)
\(\text{is }(v)\)
\( \ell_{11} \)
\(\text{Some } u \)
\(\text{is } v \)
\(f_0\) 0000 \((~)\) 1 1 1 1 0 0 0 0
\(f_1\) 0001 \((u)(v)\!\) 1 1 1 0 1 0 0 0
\(f_2\) 0010 \((u) v\!\) 1 1 0 1 0 1 0 0
\(f_3\) 0011 \((u)\!\) 1 1 0 0 1 1 0 0
\(f_4\) 0100 \(u (v)\!\) 1 0 1 1 0 0 1 0
\(f_5\) 0101 \((v)\!\) 1 0 1 0 1 0 1 0
\(f_6\) 0110 \((u, v)\!\) 1 0 0 1 0 1 1 0
\(f_7\) 0111 \((u v)\!\) 1 0 0 0 1 1 1 0
\(f_8\) 1000 \(u v\!\) 0 1 1 1 0 0 0 1
\(f_9\) 1001 \(((u, v))\!\) 0 1 1 0 1 0 0 1
\(f_{10}\) 1010 \(v\!\) 0 1 0 1 0 1 0 1
\(f_{11}\) 1011 \((u (v))\!\) 0 1 0 0 1 1 0 1
\(f_{12}\) 1100 \(u\!\) 0 0 1 1 0 0 1 1
\(f_{13}\) 1101 \(((u) v)\!\) 0 0 1 0 1 0 1 1
\(f_{14}\) 1110 \(((u)(v))\!\) 0 0 0 1 0 1 1 1
\(f_{15}\) 1111 \(((~))\) 0 0 0 0 1 1 1 1


Table 18

Table 18. Simple Qualifiers of Propositions (Version 2)
\(u:\)
\(v:\)
1100
1010
\(f\!\) \((\ell_{11})\)
\(\text{No } u \)
\(\text{is } v \)
\((\ell_{10})\)
\(\text{No } u \)
\(\text{is }(v)\)
\((\ell_{01})\)
\(\text{No }(u)\)
\(\text{is } v \)
\((\ell_{00})\)
\(\text{No }(u)\)
\(\text{is }(v)\)
\( \ell_{00} \)
\(\text{Some }(u)\)
\(\text{is }(v)\)
\( \ell_{01} \)
\(\text{Some }(u)\)
\(\text{is } v \)
\( \ell_{10} \)
\(\text{Some } u \)
\(\text{is }(v)\)
\( \ell_{11} \)
\(\text{Some } u \)
\(\text{is } v \)
\(f_0\) 0000 \((~)\) 1 1 1 1 0 0 0 0
\(f_1\) 0001 \((u)(v)\!\) 1 1 1 0 1 0 0 0
\(f_2\) 0010 \((u) v\!\) 1 1 0 1 0 1 0 0
\(f_4\) 0100 \(u (v)\!\) 1 0 1 1 0 0 1 0
\(f_8\) 1000 \(u v\!\) 0 1 1 1 0 0 0 1
\(f_3\) 0011 \((u)\!\) 1 1 0 0 1 1 0 0
\(f_{12}\) 1100 \(u\!\) 0 0 1 1 0 0 1 1
\(f_6\) 0110 \((u, v)\!\) 1 0 0 1 0 1 1 0
\(f_9\) 1001 \(((u, v))\!\) 0 1 1 0 1 0 0 1
\(f_5\) 0101 \((v)\!\) 1 0 1 0 1 0 1 0
\(f_{10}\) 1010 \(v\!\) 0 1 0 1 0 1 0 1
\(f_7\) 0111 \((u v)\!\) 1 0 0 0 1 1 1 0
\(f_{11}\) 1011 \((u (v))\!\) 0 1 0 0 1 1 0 1
\(f_{13}\) 1101 \(((u) v)\!\) 0 0 1 0 1 0 1 1
\(f_{14}\) 1110 \(((u)(v))\!\) 0 0 0 1 0 1 1 1
\(f_{15}\) 1111 \(((~))\) 0 0 0 0 1 1 1 1


Table 19

Table 19. Relation of Quantifiers to Higher Order Propositions
\(\text{Mnemonic}\) \(\text{Category}\) \(\text{Classical Form}\) \(\text{Alternate Form}\) \(\text{Symmetric Form}\) \(\text{Operator}\)
\(\text{E}\!\)
\(\text{Exclusive}\)
\(\text{Universal}\)
\(\text{Negative}\)
\(\text{All}\ u\ \text{is}\ (v)\)   \(\text{No}\ u\ \text{is}\ v \) \((\ell_{11})\)
\(\text{A}\!\)
\(\text{Absolute}\)
\(\text{Universal}\)
\(\text{Affirmative}\)
\(\text{All}\ u\ \text{is}\ v \)   \(\text{No}\ u\ \text{is}\ (v)\) \((\ell_{10})\)
    \(\text{All}\ v\ \text{is}\ u \) \(\text{No}\ v\ \text{is}\ (u)\) \(\text{No}\ (u)\ \text{is}\ v \) \((\ell_{01})\)
    \(\text{All}\ (v)\ \text{is}\ u \) \(\text{No}\ (v)\ \text{is}\ (u)\) \(\text{No}\ (u)\ \text{is}\ (v)\) \((\ell_{00})\)
    \(\text{Some}\ (u)\ \text{is}\ (v)\)   \(\text{Some}\ (u)\ \text{is}\ (v)\) \(\ell_{00}\!\)
    \(\text{Some}\ (u)\ \text{is}\ v\)   \(\text{Some}\ (u)\ \text{is}\ v\) \(\ell_{01}\!\)
\(\text{O}\!\)
\(\text{Obtrusive}\)
\(\text{Particular}\)
\(\text{Negative}\)
\(\text{Some}\ u\ \text{is}\ (v)\)   \(\text{Some}\ u\ \text{is}\ (v)\) \(\ell_{10}\!\)
\(\text{I}\!\)
\(\text{Indefinite}\)
\(\text{Particular}\)
\(\text{Affirmative}\)
\(\text{Some}\ u\ \text{is}\ v\)   \(\text{Some}\ u\ \text{is}\ v\) \(\ell_{11}\!\)