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Revision as of 15:39, 9 September 2013
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{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"
|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
|+ style="height:30px" | <math>\text{Table A1.} ~~ \text{Propositional Forms on Two Variables}\!</math>
|- style="background:#f0f0ff"
|- style="background:ghostwhite"
| width="15%" |
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math>
| width="15%" |
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math>
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math>
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|- style="background:ghostwhite"
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| width="25%" |
| width="15%" |
|- style="background:#f0f0ff"
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| align="right" | <math>x\colon\!</math>
| align="right" | <math>x\colon\!</math>
| <math>1~1~0~0\!</math>
| <math>1~1~0~0\!</math>
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|- style="background:ghostwhite"
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|- style="background:#f0f0ff"
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| align="right" | <math>y\colon\!</math>
| align="right" | <math>y\colon\!</math>
| <math>1~0~1~0\!</math>
| <math>1~0~1~0\!</math>
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<math>\begin{matrix}
<math>\begin{matrix}
f_{0}\\f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7}
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\end{matrix}</math>
\\[4pt]
\\[4pt]
\\[4pt]
\\[4pt]
\\[4pt]
\\[4pt]
\end{matrix}\!</math>
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<math>\begin{matrix}
<math>\begin{matrix}
f_{0000}
f_{0000}\\f_{0001}\\f_{0010}\\f_{0011}\\f_{0100}\\f_{0101}\\f_{0110}\\f_{0111}
\\[4pt]
\end{matrix}</math>
f_{0001}
\\[4pt]
f_{0010}
\\[4pt]
f_{0011}
\\[4pt]
f_{0100}
\\[4pt]
f_{0101}
\\[4pt]
f_{0110}
\\[4pt]
f_{0111}
\end{matrix}\!</math>
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<math>\begin{matrix}
<math>\begin{matrix}
0~0~0~0
0~0~0~0\\0~0~0~1\\0~0~1~0\\0~0~1~1\\0~1~0~0\\0~1~0~1\\0~1~1~0\\0~1~1~1
\\[4pt]
0~0~0~1
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0~0~1~0
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0~0~1~1
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0~1~0~0
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0~1~0~1
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0~1~1~0
\\[4pt]
0~1~1~1
\end{matrix}\!</math>
\end{matrix}\!</math>
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<math>\begin{matrix}
<math>\begin{matrix}
(~)
\texttt{(~)}
\\[4pt]
\\
(x)(y)
\texttt{(} x \texttt{)(} y \texttt{)}
\\[4pt]
\\
(x)~y~
\texttt{(} x \texttt{)~} y \texttt{~}
\\[4pt]
\\
(x)~~~
\texttt{(} x \texttt{)~~~}
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~x~(y)
\texttt{~} x \texttt{~(} y \texttt{)}
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~~~(y)
\texttt{~~~(} y \texttt{)}
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\\
(x,~y)
\texttt{(} x \texttt{,~} y \texttt{)}
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\\
(x~~y)
\texttt{(} x \texttt{~~} y \texttt{)}
\end{matrix}\!</math>
\end{matrix}</math>
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<math>\begin{matrix}
<math>\begin{matrix}
\text{false}
\text{false}
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\\
\text{neither}~ x ~\text{nor}~ y
\text{neither}~ x ~\text{nor}~ y
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\\
y ~\text{without}~ x
y ~\text{without}~ x
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\\
\text{not}~ x
\text{not}~ x
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\\
x ~\text{without}~ y
x ~\text{without}~ y
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\\
\text{not}~ y
\text{not}~ y
\\[4pt]
\\
x ~\text{not equal to}~ y
x ~\text{not equal to}~ y
\\[4pt]
\\
\text{not both}~ x ~\text{and}~ y
\text{not both}~ x ~\text{and}~ y
\end{matrix}\!</math>
\end{matrix}</math>
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<math>\begin{matrix}
<math>\begin{matrix}
0
0
\\[4pt]
\\
\lnot x \land \lnot y
\lnot x \land \lnot y
\\[4pt]
\\
\lnot x \land y
\lnot x \land y
\\[4pt]
\\
\lnot x
\lnot x
\\[4pt]
\\
x \land \lnot y
x \land \lnot y
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\\
\lnot y
\lnot y
\\[4pt]
\\
x \ne y
x \ne y
\\[4pt]
\\
\lnot x \lor \lnot y
\lnot x \lor \lnot y
\end{matrix}\!</math>
\end{matrix}</math>
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<math>\begin{matrix}
<math>\begin{matrix}
f_{8}\\f_{9}\\f_{10}\\f_{11}\\f_{12}\\f_{13}\\f_{14}\\f_{15}
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\end{matrix}</math>
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f_{10}
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f_{11}
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f_{12}
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f_{13}
\\[4pt]
f_{14}
\\[4pt]
f_{15}
\end{matrix}\!</math>
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<math>\begin{matrix}
<math>\begin{matrix}
f_{1000}
f_{1000}\\f_{1001}\\f_{1010}\\f_{1011}\\f_{1100}\\f_{1101}\\f_{1110}\\f_{1111}
\\[4pt]
f_{1001}
\\[4pt]
f_{1010}
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f_{1011}
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f_{1100}
\\[4pt]
f_{1101}
\\[4pt]
f_{1110}
\\[4pt]
f_{1111}
\end{matrix}\!</math>
\end{matrix}\!</math>
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<math>\begin{matrix}
<math>\begin{matrix}
1~0~0~0
1~0~0~0\\1~0~0~1\\1~0~1~0\\1~0~1~1\\1~1~0~0\\1~1~0~1\\1~1~1~0\\1~1~1~1
\\[4pt]
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1~0~0~1
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1~0~1~0
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1~0~1~1
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1~1~0~0
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1~1~0~1
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1~1~1~0
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1~1~1~1
\end{matrix}\!</math>
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<math>\begin{matrix}
<math>\begin{matrix}
~~x~~y~~
\texttt{~~} x \texttt{~~} y \texttt{~~}
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\\
((x,~y))
\texttt{((} x \texttt{,~} y \texttt{))}
\\[4pt]
\\
~~~~~y~~
\texttt{~~~~~} y \texttt{~~}
\\[4pt]
\\
~(x~(y))
\texttt{~(} x \texttt{~(} y \texttt{))}
\\[4pt]
\\
~~x~~~~~
\texttt{~~} x \texttt{~~~~~}
\\[4pt]
\\
((x)~y)~
\texttt{((} x \texttt{)~} y \texttt{)~}
\\[4pt]
\\
((x)(y))
\texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
\\
((~))
\texttt{((~))}
\end{matrix}\!</math>
\end{matrix}</math>
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<math>\begin{matrix}
<math>\begin{matrix}
x ~\text{and}~ y
x ~\text{and}~ y
\\[4pt]
\\
x ~\text{equal to}~ y
x ~\text{equal to}~ y
\\[4pt]
\\
y
y
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\\
\text{not}~ x ~\text{without}~ y
\text{not}~ x ~\text{without}~ y
\\[4pt]
\\
x
x
\\[4pt]
\\
\text{not}~ y ~\text{without}~ x
\text{not}~ y ~\text{without}~ x
\\[4pt]
\\
x ~\text{or}~ y
x ~\text{or}~ y
\\[4pt]
\\
\text{true}
\text{true}
\end{matrix}\!</math>
\end{matrix}</math>
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<math>\begin{matrix}
<math>\begin{matrix}
x \land y
x \land y
\\[4pt]
\\
x = y
x = y
\\[4pt]
\\
y
y
\\[4pt]
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x \Rightarrow y
x \Rightarrow y
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\\
x
x
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x \Leftarrow y
x \Leftarrow y
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\\
x \lor y
x \lor y
\\[4pt]
\\
1
1
\end{matrix}\!</math>
\end{matrix}</math>
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