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Directory talk:Jon Awbrey/Papers/Differential Logic : Introduction (view source)
Revision as of 13:38, 30 June 2009
, 13:38, 30 June 2009→Logical Cacti: add equivalents
Table A illustrates the existential interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.
Table A illustrates the existential interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.
<br>
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
</pre>
</pre>
| <math>\texttt{(} a \texttt{)}</math>
| <math>\texttt{(} a \texttt{)}</math>
| <math>\operatorname{not}~ a.</math>
|
<math>\begin{matrix}
\tilde{a}
\\[6pt]
a^\prime
\\[6pt]
\lnot a
\\[6pt]
\operatorname{not}~ a.
\end{matrix}</math>
|-
|-
|
|
</pre>
</pre>
| <math>a~b~c</math>
| <math>a~b~c</math>
| <math>a ~\operatorname{and}~ b ~\operatorname{and}~ c.</math>
|
<math>\begin{matrix}
a \land b \land c
\\[6pt]
a ~\operatorname{and}~ b ~\operatorname{and}~ c.
\end{matrix}</math>
|-
|-
|
|
</pre>
</pre>
| <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>\begin{matrix}
a \lor b \lor c
\\[6pt]
a ~\operatorname{or}~ b ~\operatorname{or}~ c.
\end{matrix}</math>
|-
|-
|
|
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a \Rightarrow b
\\[6pt]
a ~\operatorname{implies}~ b.
a ~\operatorname{implies}~ b.
\\[6pt]
\\[6pt]
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a + b
\\[6pt]
a \neq b
\\[6pt]
a ~\operatorname{exclusive-or}~ b.
a ~\operatorname{exclusive-or}~ b.
\\[6pt]
\\[6pt]
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a = b
\\[6pt]
a \iff b
\\[6pt]
a ~\operatorname{equals}~ 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}
\operatorname{genus}~ a ~\operatorname{with~species}~ b, c.
\operatorname{genus}~ a ~\operatorname{of~species}~ b, c.
\\[4pt]
\\[6pt]
\operatorname{partition}~ a ~\operatorname{among}~ b, c.
\operatorname{partition}~ a ~\operatorname{into}~ b, c.
\\[4pt]
\\[6pt]
\operatorname{pie}~ a ~\operatorname{with~slices}~ b, c.
\operatorname{pie}~ a ~\operatorname{of~slices}~ b, c.
\end{matrix}</math>
\end{matrix}</math>
|}
|}
<br>
Table 14 illustrates the entitative interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.
Table 14 illustrates the entitative interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.