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Directory:Jon Awbrey/Papers/Differential Logic : Introduction (view source)
Revision as of 19:46, 2 July 2009
, 19:46, 2 July 2009→Cactus Language for Propositional Logic: new table with cactus graphs and existential interpretation
|+ <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>\text{Graph}\!</math>
| <math>\text{Expression}\!</math>
| <math>\text{Expression}\!</math>
| <math>\text{Interpretation}\!</math>
| <math>\text{Interpretation}\!</math>
| <math>\text{Other Notations}\!</math>
| <math>\text{Other Notations}\!</math>
|-
|-
| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
| <math>~</math>
| <math>~</math>
| <math>\operatorname{true}</math>
| <math>\operatorname{true}</math>
| <math>1\!</math>
| <math>1\!</math>
|-
|-
| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
| <math>\texttt{(~)}</math>
| <math>\texttt{(~)}</math>
| <math>\operatorname{false}</math>
| <math>\operatorname{false}</math>
| <math>0\!</math>
| <math>0\!</math>
|-
|-
| height="100px" | [[Image:Cactus A Big.jpg|20px]]
| <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]]
| <math>\texttt{(} a \texttt{)}</math>
| <math>\texttt{(} a \texttt{)}</math>
| <math>\operatorname{not}~ a</math>
| <math>\operatorname{not}~ a</math>
|
| <math>\lnot a \quad \bar{a} \quad \tilde{a} \quad a^\prime</math>
<math>\begin{matrix}
\lnot a & \bar{a} & \tilde{a} & a^\prime
|-
|-
| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
| <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>
|-
|-
| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</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>
|-
|-
| height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]]
| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a ~\operatorname{implies}~ b
a ~\operatorname{implies}~ b
\\
\\[6pt]
\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>
|-
|-
| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
| <math>\texttt{(} a, b \texttt{)}</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
\\
\\[6pt]
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
\\
\\[6pt]
a + b
a + b
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
| <math>\texttt{((} a, b \texttt{))}</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
\\
\\[6pt]
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
\\
\\[6pt]
a \Leftrightarrow b
a \Leftrightarrow b
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
| <math>\texttt{(} a, b, c \texttt{)}</math>
| <math>\texttt{(} a, b, c \texttt{)}</math>
|
|
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]]
| <math>\texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}</math>
| <math>\texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}</math>
|
|
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="160px" | [[Image:Cactus (A,(B,C)) Big.jpg|90px]]
| <math>\texttt{(} a, \texttt{(} b, c \texttt{))}</math>
\texttt{(} a, \texttt{(} b, c \texttt{))}
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\end{matrix}</math></p>
\end{matrix}</math></p>
|-
|-
| height="160px" | [[Image:Cactus (X,(A),(B),(C)) Big.jpg|90px]]
| <math>\texttt{(} x, \texttt{(} a \texttt{),(} b \texttt{),(} c \texttt{))}</math>
| <math>\texttt{(} x, \texttt{(} a \texttt{),(} b \texttt{),(} c \texttt{))}</math>
|
|