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Revision as of 14:54, 30 June 2009
, 14:54, 30 June 2009markup
==Place for Discussion==
==Discussion Area==
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<pre>
<pre>
o-------------------o
o-------------------o
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| a b |
| a b |
</pre>
</pre>
| <math>\texttt{((} a, b \texttt{))}</math>
| <math>\texttt{((} a, b \texttt{))}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a = b
a = b
<pre>
<pre>
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| b c |
| b c |
<br>
<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 B 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.
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
|+ <math>\text{Table B.}~~\text{Entitative Interpretation}</math>
|- style="background:#f0f0ff"
| <math>\text{Cactus Graph}\!</math>
| <math>\text{Cactus Expression}\!</math>
| <math>\text{Interpretation}\!</math>
|-
|
<pre>
o-------------------o
| |
| @ |
| |
o-------------------o
</pre>
| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
| <math>\operatorname{false}.</math>
|-
|
<pre>
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| o |
| | |
| @ |
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o-------------------o
</pre>
| <math>\texttt{(~)}</math>
| <math>\operatorname{true}.</math>
|-
|
<pre>
o-------------------o
| |
| a |
| @ |
| |
o-------------------o
</pre>
| <math>a\!</math>
| <math>a.\!</math>
|-
|
<pre>
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| a |
| o |
| | |
| @ |
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o-------------------o
</pre>
| <math>\texttt{(} a \texttt{)}</math>
|
<math>\begin{matrix}
\tilde{a}
\\[6pt]
a^\prime
\\[6pt]
\lnot a
\\[6pt]
\operatorname{not}~ a.
\end{matrix}</math>
|-
|
<pre>
o-------------------o
| |
| a b c |
| @ |
| |
o-------------------o
</pre>
| <math>a~b~c</math>
|
<math>\begin{matrix}
a \lor b \lor c
\\[6pt]
a ~\operatorname{or}~ b ~\operatorname{or}~ c.
\end{matrix}</math>
|-
|
<pre>
o-------------------o
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| a b c |
| o o o |
| \|/ |
| o |
| | |
| @ |
| |
o-------------------o
</pre>
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
|
<math>\begin{matrix}
a \land b \land c
\\[6pt]
a ~\operatorname{and}~ b ~\operatorname{and}~ c.
\end{matrix}</math>
|-
|
<pre>
o-------------------o
| |
| o a |
| | |
| @ b |
| |
o-------------------o
</pre>
| <math>\texttt{(} a \texttt{)} b</math>
|
<math>\begin{matrix}
a \Rightarrow b
\\[6pt]
a ~\operatorname{implies}~ b.
\\[6pt]
\operatorname{if}~ a ~\operatorname{then}~ b.
\\[6pt]
\operatorname{not}~ a, ~\operatorname{or}~ b.
\end{matrix}</math>
|-
|
<pre>
o-------------------o
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| a b |
| o---o |
| \ / |
| @ |
| |
o-------------------o
</pre>
| <math>\texttt{(} a, b \texttt{)}</math>
|
<math>\begin{matrix}
a = b
\\[6pt]
a \iff b
\\[6pt]
a ~\operatorname{equals}~ b.
\\[6pt]
a ~\operatorname{if~and~only~if}~ b.
\end{matrix}</math>
|-
|
<pre>
<pre>
o-------------------o
| |
| a b |
| o---o |
| \ / |
| o |
| | |
| @ |
| |
o-------------------o
</pre>
</pre>
| <math>\texttt{((} a, b \texttt{))}</math>
|
<math>\begin{matrix}
a + b
\\[6pt]
a \neq b
\\[6pt]
a ~\operatorname{exclusive-or}~ b.
\\[6pt]
a ~\operatorname{not~equal~to}~ b.
\end{matrix}</math>
|-
|
<pre>
o-------------------o
| |
| a b c |
| o--o--o |
| \ / |
| \ / |
| @ |
| |
o-------------------o
</pre>
| <math>\texttt{(} a, b, c \texttt{)}</math>
|
<math>\begin{matrix}
\operatorname{not~just~one~of}
\\
a, b, c
\\
\operatorname{is~true}.
\end{matrix}</math>
|-
|
<pre>
o-------------------o
| |
| a b c |
| o--o--o |
| \ / |
| \ / |
| o |
| | |
| @ |
| |
o-------------------o
</pre>
| <math>\texttt{((} a, b, c \texttt{))}</math>
|
<math>\begin{matrix}
\operatorname{just~one~of}
\\
a, b, c
\\
\operatorname{is~true}.
\end{matrix}</math>
|-
|
<pre>
o-------------------o
| |
| a |
| o |
| | b c |
| o--o--o |
| \ / |
| \ / |
| o |
| | |
| @ |
| |
o-------------------o
</pre>
| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
|
<math>\begin{matrix}
\operatorname{genus}~ a ~\operatorname{of~species}~ b, c.
\\[6pt]
\operatorname{partition}~ a ~\operatorname{into}~ b, c.
\\[6pt]
\operatorname{pie}~ a ~\operatorname{of~slices}~ b, c.
\end{matrix}</math>
|}
<br>
For the time being, the main things to take away from Tables 13 and 14 are the ideas that the compositional structure of cactus graphs and expressions can be articulated in terms of two different kinds of connective operations, and that there are two distinct ways of mapping this compositional structure into the compositional structure of propositional sentences, say, in English:
For the time being, the main things to take away from Tables A and B are the ideas that the compositional structure of cactus graphs and expressions can be articulated in terms of two different kinds of connective operations, and that there are two distinct ways of mapping this compositional structure into the compositional structure of propositional sentences, say, in English:
<pre>
<pre>
