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Directory:Jon Awbrey/Papers/Differential Logic : Introduction (view source)
Revision as of 14:16, 2 July 2009
, 14:16, 2 July 2009→Cactus Language for Propositional Logic: reformat table
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{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
|+ <math>\text{Table 1.}~~\text{Syntax and Semantics of a Calculus for Propositional Logic}</math>
|+ <math>\text{Table 1.}~~\text{Syntax and Semantics of a Calculus for Propositional Logic}</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
|-
|-
| <math>~</math>
| <math>~</math>
| <math>\operatorname{True}</math>
| <math>\operatorname{true}</math>
| <math>1\!</math>
| <math>1\!</math>
|-
|-
| <math>(~)</math>
| <math>\texttt{(~)}</math>
| <math>\operatorname{False}</math>
| <math>\operatorname{false}</math>
| <math>0\!</math>
| <math>0\!</math>
|-
|-
| <math>a\!</math>
| <math>a\!</math>
|-
|-
| <math>(a)\!</math>
| <math>\texttt{(} a \texttt{)}</math>
| <math>\operatorname{Not}\ a</math>
| <math>\operatorname{not}~ a</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a' \\
a^\prime
\tilde{a} \\
\\
\lnot a \\
\tilde{a}
\\
\lnot a
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| <math>a\ b\ c</math>
| <math>a ~ b ~ c</math>
| <math>a\ \operatorname{and}\ b\ \operatorname{and}\ c</math>
| <math>a ~\operatorname{and}~ b ~\operatorname{and}~ c</math>
| <math>a \land b \land c</math>
| <math>a \land b \land c</math>
|-
|-
| <math>((a)(b)(c))\!</math>
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
| <math>a\ \operatorname{or}\ b\ \operatorname{or}\ c</math>
| <math>a ~\operatorname{or}~ b ~\operatorname{or}~ c</math>
| <math>a \lor b \lor c</math>
| <math>a \lor b \lor c</math>
|-
|-
| <math>(a\ (b))\!</math>
| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a\ \operatorname{implies}\ b \\
a ~\operatorname{implies}~ b
\operatorname{If}\ a\ \operatorname{then}\ b \\
\\
\operatorname{if}~ a ~\operatorname{then}~ b
\end{matrix}</math>
\end{matrix}</math>
| <math>a \Rightarrow b\!</math>
| <math>a \Rightarrow b</math>
|-
|-
| <math>(a, b)\!</math>
| <math>\texttt{(} a, b \texttt{)}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a\ \operatorname{not~equal~to}\ b \\
a ~\operatorname{not~equal~to}~ b
a\ \operatorname{exclusive~or}\ b \\
\\
a ~\operatorname{exclusive~or}~ b
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a \neq b \\
a \neq b
a + b \\
\\
a + b
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| <math>((a, b))\!</math>
| <math>\texttt{((} a, b \texttt{))}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a\ \operatorname{is~equal~to}\ b \\
a ~\operatorname{is~equal~to}~ b
a\ \operatorname{if~and~only~if}\ b \\
\\
a ~\operatorname{if~and~only~if}~ b
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a = b \\
a = b
a \Leftrightarrow b \\
\\
a \Leftrightarrow b
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| <math>(a, b, c)\!</math>
| <math>\texttt{(} a, b, c \texttt{)}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{Just~one~of} \\
\operatorname{just~one~of}
a, b, c \\
\\
\operatorname{is~false}. \\
a, b, c
\\
\operatorname{is~false}.
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a'b~c~ & \lor \\
a'b~c~ & \lor
a~b'c~ & \lor \\
\\
a~b~c' & \\
a~b'c~ & \lor
\\
a~b~c' &
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| <math>((a),(b),(c))\!</math>
| <math>\texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{Just~one~of} \\
\operatorname{just~one~of}
a, b, c \\
\\
\operatorname{is~true}. \\
a, b, c
\\
\operatorname{Partition~all} \\
\operatorname{is~true}.
\operatorname{into}\ a, b, c. \\
\\[6pt]
\operatorname{partition~all}
\\
\operatorname{into}~ a, b, c.
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a~b'c' & \lor \\
a~b'c' & \lor
a'b~c' & \lor \\
\\
a'b'c~ & \\
a'b~c' & \lor
\\
a'b'c~ &
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
|
<math>\begin{matrix}
<math>\begin{matrix}
((a, b), c) \\
\texttt{((} a, b \texttt{)}, c \texttt{)}
\\[6pt]
(a, (b, c)) \\
\texttt{(} a, \texttt{(} b, c \texttt{))}
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{Oddly~many~of} \\
\operatorname{oddly~many~of}
a, b, c \\
\\
\operatorname{are~true}. \\
a, b, c
\\
\operatorname{are~true}.
\end{matrix}</math>
\end{matrix}</math>
|
|
<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' & \lor \\
a~b'c' & \lor
a'b'c~ & \\
\\
a'b~c' & \lor
\\
a'b'c~ &
\end{matrix}</math></p>
\end{matrix}</math></p>
|-
|-
| <math>(x, (a),(b),(c))\!</math>
| <math>\texttt{(} x, \texttt{(} a \texttt{),(} b \texttt{),(} c \texttt{))}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{Partition}\ x \\
\operatorname{partition}~ x
\operatorname{into}\ a, b, c. \\
\\
\operatorname{into}~ a, b, c.
\operatorname{Genus}\ x\ \operatorname{comprises} \\
\\[6pt]
\operatorname{species}\ a, b, c. \\
\operatorname{genus}~ x ~\operatorname{comprises}
\\
\operatorname{species}~ a, b, c.
\end{matrix}</math>
\end{matrix}</math>
|
|
<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' & \lor \\
x~a~b'c' & \lor
x~a'b'c~ & \\
\\
x~a'b~c' & \lor
\\
x~a'b'c~ &
\end{matrix}</math>
\end{matrix}</math>
|}
|}