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Directory:Jon Awbrey/Papers/Differential Logic : Introduction (view source)
Revision as of 13:52, 2 July 2009
, 13:52, 2 July 2009→Cactus Language for Propositional Logic: sub [a, b, c / x, y, z]
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<math>\begin{matrix}
<math>\begin{matrix}
a + b
& = &
& = &
\texttt{(} x, y \texttt{)}
\texttt{(} a, b \texttt{)}
\end{matrix}</math>
\end{matrix}</math>
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<math>\begin{matrix}
<math>\begin{matrix}
a + b + c
& = &
& = &
\texttt{((} x, y \texttt{)}, z \texttt{)}
\texttt{((} a, b \texttt{)}, c \texttt{)}
& = &
& = &
\texttt{(} x, \texttt{(} y, z \texttt{))}
\texttt{(} a, \texttt{(} b, c \texttt{))}
\end{matrix}</math>
\end{matrix}</math>
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It is important to note that the last expressions are not equivalent to the triple bracket <math>\texttt{(} x, y, z \texttt{)}.</math>
It is important to note that the last expressions are not equivalent to the triple bracket <math>\texttt{(} a, b, c \texttt{)}.</math>
<br>
<br>
| <math>0\!</math>
| <math>0\!</math>
|-
|-
| <math>x\!</math>
| <math>a\!</math>
| <math>x\!</math>
| <math>a\!</math>
| <math>x\!</math>
| <math>a\!</math>
|-
|-
| <math>(x)\!</math>
| <math>(a)\!</math>
| <math>\operatorname{Not}\ x</math>
| <math>\operatorname{Not}\ a</math>
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<math>\begin{matrix}
<math>\begin{matrix}
a' \\
\tilde{x} \\
\tilde{a} \\
\lnot x \\
\lnot a \\
\end{matrix}</math>
\end{matrix}</math>
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|-
| <math>x\ y\ z</math>
| <math>a\ b\ c</math>
| <math>x\ \operatorname{and}\ y\ \operatorname{and}\ z</math>
| <math>a\ \operatorname{and}\ b\ \operatorname{and}\ c</math>
| <math>x \land y \land z</math>
| <math>a \land b \land c</math>
|-
|-
| <math>((x)(y)(z))\!</math>
| <math>((a)(b)(c))\!</math>
| <math>x\ \operatorname{or}\ y\ \operatorname{or}\ z</math>
| <math>a\ \operatorname{or}\ b\ \operatorname{or}\ c</math>
| <math>x \lor y \lor z</math>
| <math>a \lor b \lor c</math>
|-
|-
| <math>(x\ (y))\!</math>
| <math>(a\ (b))\!</math>
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<math>\begin{matrix}
<math>\begin{matrix}
a\ \operatorname{implies}\ b \\
\operatorname{If}\ x\ \operatorname{then}\ y \\
\operatorname{If}\ a\ \operatorname{then}\ b \\
\end{matrix}</math>
\end{matrix}</math>
| <math>x \Rightarrow y\!</math>
| <math>a \Rightarrow b\!</math>
|-
|-
| <math>(x, y)\!</math>
| <math>(a, b)\!</math>
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<math>\begin{matrix}
<math>\begin{matrix}
a\ \operatorname{not~equal~to}\ b \\
a\ \operatorname{exclusive~or}\ b \\
\end{matrix}</math>
\end{matrix}</math>
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<math>\begin{matrix}
<math>\begin{matrix}
a \neq b \\
a + b \\
\end{matrix}</math>
\end{matrix}</math>
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| <math>((x, y))\!</math>
| <math>((a, b))\!</math>
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<math>\begin{matrix}
<math>\begin{matrix}
a\ \operatorname{is~equal~to}\ b \\
a\ \operatorname{if~and~only~if}\ b \\
\end{matrix}</math>
\end{matrix}</math>
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<math>\begin{matrix}
<math>\begin{matrix}
a = b \\
a \Leftrightarrow b \\
\end{matrix}</math>
\end{matrix}</math>
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| <math>(x, y, z)\!</math>
| <math>(a, b, c)\!</math>
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<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{Just~one~of} \\
\operatorname{Just~one~of} \\
a, b, c \\
\operatorname{is~false}. \\
\operatorname{is~false}. \\
\end{matrix}</math>
\end{matrix}</math>
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<math>\begin{matrix}
<math>\begin{matrix}
a'b~c~ & \lor \\
a~b'c~ & \lor \\
a~b~c' & \\
\end{matrix}</math>
\end{matrix}</math>
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| <math>((x),(y),(z))\!</math>
| <math>((a),(b),(c))\!</math>
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<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{Just~one~of} \\
\operatorname{Just~one~of} \\
a, b, c \\
\operatorname{is~true}. \\
\operatorname{is~true}. \\
& \\
& \\
\operatorname{Partition~all} \\
\operatorname{Partition~all} \\
\operatorname{into}\ x, y, z. \\
\operatorname{into}\ a, b, c. \\
\end{matrix}</math>
\end{matrix}</math>
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<math>\begin{matrix}
<math>\begin{matrix}
a~b'c' & \lor \\
a'b~c' & \lor \\
a'b'c~ & \\
\end{matrix}</math>
\end{matrix}</math>
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<math>\begin{matrix}
<math>\begin{matrix}
((x, y), z) \\
((a, b), c) \\
& \\
& \\
(x, (y, z)) \\
(a, (b, c)) \\
\end{matrix}</math>
\end{matrix}</math>
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<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{Oddly~many~of} \\
\operatorname{Oddly~many~of} \\
a, b, c \\
\operatorname{are~true}. \\
\operatorname{are~true}. \\
\end{matrix}</math>
\end{matrix}</math>
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<p><math>x + y + z\!</math></p>
<p><math>a + b + c\!</math></p>
<br>
<br>
<p><math>\begin{matrix}
<p><math>\begin{matrix}
a~b~c~ & \lor \\
a~b'c' & \lor \\
a'b~c' & \lor \\
a'b'c~ & \\
\end{matrix}</math></p>
\end{matrix}</math></p>
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| <math>(w, (x),(y),(z))\!</math>
| <math>(x, (a),(b),(c))\!</math>
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<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{Partition}\ w \\
\operatorname{Partition}\ x \\
\operatorname{into}\ x, y, z. \\
\operatorname{into}\ a, b, c. \\
& \\
& \\
\operatorname{Genus}\ w\ \operatorname{comprises} \\
\operatorname{Genus}\ x\ \operatorname{comprises} \\
\operatorname{species}\ x, y, z. \\
\operatorname{species}\ a, b, c. \\
\end{matrix}</math>
\end{matrix}</math>
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<math>\begin{matrix}
<math>\begin{matrix}
x'a'b'c' & \lor \\
x~a~b'c' & \lor \\
x~a'b~c' & \lor \\
x~a'b'c~ & \\
\end{matrix}</math>
\end{matrix}</math>
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