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Revision as of 16:08, 1 July 2009
, 16:08, 1 July 2009→Logical Cacti: convert graphics
| <math>\text{Interpretation}\!</math>
| <math>\text{Interpretation}\!</math>
|-
|-
|
| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
| |
| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
| <math>\operatorname{true}.</math>
| <math>\operatorname{true}.</math>
|-
|-
|
| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
| |
| <math>\texttt{(~)}</math>
| <math>\texttt{(~)}</math>
| <math>\operatorname{false}.</math>
| <math>\operatorname{false}.</math>
|-
|-
|
| height="100px" | [[Image:Cactus A Big.jpg|20px]]
| |
| <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>\begin{matrix}
<math>\begin{matrix}
\tilde{a}
\tilde{a}
\\[6pt]
\\[2pt]
a^\prime
a^\prime
\\[6pt]
\\[2pt]
\lnot a
\lnot a
\\[6pt]
\\[2pt]
\operatorname{not}~ a.
\operatorname{not}~ a.
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
| |
| <math>a~b~c</math>
| <math>a~b~c</math>
|
|
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|70px]]
| |
| <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="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 \Rightarrow b
a \Rightarrow b
\\[6pt]
\\[2pt]
a ~\operatorname{implies}~ b.
a ~\operatorname{implies}~ b.
\\[6pt]
\\[2pt]
\operatorname{if}~ a ~\operatorname{then}~ b.
\operatorname{if}~ a ~\operatorname{then}~ b.
\\[6pt]
\\[2pt]
\operatorname{not}~ a ~\operatorname{without}~ b.
\operatorname{not}~ a ~\operatorname{without}~ b.
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="120px" | [[Image:Cactus (A,B) Big.jpg|70px]]
| |
| <math>\texttt{(} a, b \texttt{)}</math>
| <math>\texttt{(} a, b \texttt{)}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a + b
a + b
\\[6pt]
\\[2pt]
a \neq b
a \neq b
\\[6pt]
\\[2pt]
a ~\operatorname{exclusive-or}~ b.
a ~\operatorname{exclusive-or}~ b.
\\[6pt]
\\[2pt]
a ~\operatorname{not~equal~to}~ b.
a ~\operatorname{not~equal~to}~ b.
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|70px]]
| |
| <math>\texttt{((} a, b \texttt{))}</math>
| <math>\texttt{((} a, b \texttt{))}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a = b
a = b
\\[6pt]
\\[2pt]
a \iff b
a \iff b
\\[6pt]
\\[2pt]
a ~\operatorname{equals}~ b.
a ~\operatorname{equals}~ b.
\\[6pt]
\\[2pt]
a ~\operatorname{if~and~only~if}~ b.
a ~\operatorname{if~and~only~if}~ b.
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|70px]]
| |
| <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|70px]]
| |
| <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
| <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
|
|
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="160px" | [[Image:Cactus (A,(B),(C)) Big.jpg|70px]]
| |
| <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
| <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
|
|
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.
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.
<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%"
| <math>\text{Interpretation}\!</math>
| <math>\text{Interpretation}\!</math>
|-
|-
|
| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
| |
| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
| <math>\operatorname{false}.</math>
| <math>\operatorname{false}.</math>
|-
|-
|
| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
| |
| <math>\texttt{(~)}</math>
| <math>\texttt{(~)}</math>
| <math>\operatorname{true}.</math>
| <math>\operatorname{true}.</math>
|-
|-
|
| height="100px" | [[Image:Cactus A Big.jpg|20px]]
| |
| <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>\begin{matrix}
<math>\begin{matrix}
\tilde{a}
\tilde{a}
\\[6pt]
\\[2pt]
a^\prime
a^\prime
\\[6pt]
\\[2pt]
\lnot a
\lnot a
\\[6pt]
\\[2pt]
\operatorname{not}~ a.
\operatorname{not}~ a.
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
| |
| <math>a~b~c</math>
| <math>a~b~c</math>
|
|
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|70px]]
| |
| <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="120px" | [[Image:Cactus (A)B Big.jpg|35px]]
| |
| <math>\texttt{(} a \texttt{)} b</math>
| <math>\texttt{(} a \texttt{)} b</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a \Rightarrow b
a \Rightarrow b
\\[6pt]
\\[2pt]
a ~\operatorname{implies}~ b.
a ~\operatorname{implies}~ b.
\\[6pt]
\\[2pt]
\operatorname{if}~ a ~\operatorname{then}~ b.
\operatorname{if}~ a ~\operatorname{then}~ b.
\\[6pt]
\\[2pt]
\operatorname{not}~ a, ~\operatorname{or}~ b.
\operatorname{not}~ a, ~\operatorname{or}~ b.
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="120px" | [[Image:Cactus (A,B) Big.jpg|70px]]
| |
| <math>\texttt{(} a, b \texttt{)}</math>
| <math>\texttt{(} a, b \texttt{)}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a = b
a = b
\\[6pt]
\\[2pt]
a \iff b
a \iff b
\\[6pt]
\\[2pt]
a ~\operatorname{equals}~ b.
a ~\operatorname{equals}~ b.
\\[6pt]
\\[2pt]
a ~\operatorname{if~and~only~if}~ b.
a ~\operatorname{if~and~only~if}~ b.
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|70px]]
| |
| <math>\texttt{((} a, b \texttt{))}</math>
| <math>\texttt{((} a, b \texttt{))}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a + b
a + b
\\[6pt]
\\[2pt]
a \neq b
a \neq b
\\[6pt]
\\[2pt]
a ~\operatorname{exclusive-or}~ b.
a ~\operatorname{exclusive-or}~ b.
\\[6pt]
\\[2pt]
a ~\operatorname{not~equal~to}~ b.
a ~\operatorname{not~equal~to}~ b.
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|70px]]
| |
| <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|70px]]
| |
| <math>\texttt{((} a, b, c \texttt{))}</math>
| <math>\texttt{((} a, b, c \texttt{))}</math>
|
|
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
| height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|70px]]
| |
| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
|
|
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:
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:
{| align="center" cellpadding="6" width="90%"
| valign="top" | 1.
| The ''node connective'' joins a number of component cacti <math>C_1, \ldots, C_k</math> at a node:
|-
|
|
<pre>
<pre>
C_1 ... C_k
C_1 ... C_k
@
@
</pre>
2. The "lobe connective" joins a number of
|-
| valign="top" | 2.
| The ''lobe connective'' joins a number of component cacti <math>C_1, \ldots, C_k</math> to a lobe:
|-
|
|
<pre>
C_1 C_2 C_k
C_1 C_2 C_k
o---o-...-o
o---o-...-o
@
@
</pre>
</pre>
|}
Table 15 summarizes the existential and entitative interpretations of the primitive cactus structures, in effect, the graphical constants and connectives.
Table 15 summarizes the existential and entitative interpretations of the primitive cactus structures, in effect, the graphical constants and connectives.
{| align="center" cellpadding="6" style="text-align:center; width:90%"
|
<pre>
<pre>
Table 15. Existential & Entitative Interpretations of Cactus Structures
Table 15. Existential & Entitative Interpretations of Cactus Structures
o-----------------o-----------------o-----------------o-----------------o
o-----------------o-----------------o-----------------o-----------------o
</pre>
</pre>
|}
It is possible to specify ''abstract rules of equivalence'' (AROEs) between cacti, rules for transforming one cactus into another that are ''formal'' in the sense of being indifferent to the above choices for logical or semantic interpretations, and that partition the set of cacti into formal equivalence classes.
It is possible to specify ''abstract rules of equivalence'' (AROEs) between cacti, rules for transforming one cactus into another that are ''formal'' in the sense of being indifferent to the above choices for logical or semantic interpretations, and that partition the set of cacti into formal equivalence classes.
Table 16 schematizes the two types of basic reductions in a purely formal, interpretation-independent fashion.
Table 16 schematizes the two types of basic reductions in a purely formal, interpretation-independent fashion.
{| align="center" cellpadding="6" style="text-align:center; width:90%"
|
<pre>
<pre>
Table 16. Basic Reductions
Table 16. Basic Reductions
o---------------------------------------o
o---------------------------------------o
</pre>
</pre>
|}
The careful reader will have noticed that we have begun to use graphical paints like "a", "b", "c" and schematic proxies like "C_1", "C_j", "C_k" in a variety of novel and unjustified ways.
The careful reader will have noticed that we have begun to use graphical paints like "a", "b", "c" and schematic proxies like "C_1", "C_j", "C_k" in a variety of novel and unjustified ways.
The cactus graph and the cactus expression shown here are both described as a ''spike''.
The cactus graph and the cactus expression shown here are both described as a ''spike''.
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="6" style="text-align:center; width:90%"
|
<pre>
<pre>
o---------------------------------------o
o---------------------------------------o
The rule of reduction for a lobe is:
The rule of reduction for a lobe is:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="6" style="text-align:center; width:90%"
|
<pre>
<pre>
o---------------------------------------o
o---------------------------------------o
they parse into a type of graph called a ''painted and rooted cactus'' (PARC):
they parse into a type of graph called a ''painted and rooted cactus'' (PARC):
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="6" style="text-align:center; width:90%"
|
<pre>
<pre>
o---------------------------------------o
o---------------------------------------o
|}
|}
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="6" style="text-align:center; width:90%"
|
<pre>
<pre>
o---------------------------------------o
o---------------------------------------o
