12,080
edits
Changes
MyWikiBiz, Author Your Legacy — Friday March 14, 2025
Jump to navigationJump to searchDirectory: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
Line 46:
Line 46:
& \\+& \\+& \\+
<br>
<br>
−{| 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"
Line 54:
Line 54:
|-
|-
| <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>
|-
|-
Line 65:
Line 65:
| <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>
|
|
Line 161:
Line 179:
<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>
|}
|}
