Changes

→‎Logical Cacti: convert graphics
Line 31: Line 31:  
| <math>\text{Interpretation}\!</math>
 
| <math>\text{Interpretation}\!</math>
 
|-
 
|-
|
+
| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <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]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|        o        |
  −
|        |        |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <math>\texttt{(~)}</math>
 
| <math>\texttt{(~)}</math>
 
| <math>\operatorname{false}.</math>
 
| <math>\operatorname{false}.</math>
 
|-
 
|-
|
+
| height="100px" | [[Image:Cactus A Big.jpg|20px]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|        a        |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <math>a\!</math>
 
| <math>a\!</math>
 
| <math>a.\!</math>
 
| <math>a.\!</math>
 
|-
 
|-
|
+
| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|        a        |
  −
|        o        |
  −
|        |        |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <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]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|      a b c      |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <math>a~b~c</math>
 
| <math>a~b~c</math>
 
|
 
|
Line 107: Line 65:  
\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
|
+
| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|70px]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|      a b c      |
  −
|      o o o      |
  −
|        \|/        |
  −
|        o        |
  −
|        |        |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
 
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
 
|
 
|
Line 128: Line 74:  
\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
|
+
| height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|        a  b    |
  −
|        o---o    |
  −
|        |        |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <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]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|      a  b      |
  −
|      o---o      |
  −
|        \ /        |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <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]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|      a  b      |
  −
|      o---o      |
  −
|        \ /        |
  −
|        o        |
  −
|        |        |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <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]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|      a  b  c      |
  −
|      o--o--o      |
  −
|      \  /      |
  −
|        \ /        |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <math>\texttt{(} a, b, c \texttt{)}</math>
 
| <math>\texttt{(} a, b, c \texttt{)}</math>
 
|
 
|
Line 221: Line 124:  
\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
|
+
| height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|70px]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|      a  b  c      |
  −
|      o  o  o      |
  −
|      |  |  |      |
  −
|      o--o--o      |
  −
|      \  /      |
  −
|        \ /        |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
 
| <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
 
|
 
|
Line 245: Line 135:  
\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
|
+
| height="160px" | [[Image:Cactus (A,(B),(C)) Big.jpg|70px]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|        b  c      |
  −
|        o  o      |
  −
|      a  |  |      |
  −
|      o--o--o      |
  −
|      \  /      |
  −
|        \ /        |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
 
| <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
 
|
 
|
Line 273: Line 150:     
Table&nbsp;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&nbsp;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%"
Line 281: Line 160:  
| <math>\text{Interpretation}\!</math>
 
| <math>\text{Interpretation}\!</math>
 
|-
 
|-
|
+
| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <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]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|        o        |
  −
|        |        |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <math>\texttt{(~)}</math>
 
| <math>\texttt{(~)}</math>
 
| <math>\operatorname{true}.</math>
 
| <math>\operatorname{true}.</math>
 
|-
 
|-
|
+
| height="100px" | [[Image:Cactus A Big.jpg|20px]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|        a        |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <math>a\!</math>
 
| <math>a\!</math>
 
| <math>a.\!</math>
 
| <math>a.\!</math>
 
|-
 
|-
|
+
| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|        a        |
  −
|        o        |
  −
|        |        |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <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]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|      a b c      |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <math>a~b~c</math>
 
| <math>a~b~c</math>
 
|
 
|
Line 357: Line 194:  
\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
|
+
| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|70px]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|      a b c      |
  −
|      o o o      |
  −
|        \|/        |
  −
|        o        |
  −
|        |        |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
 
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
 
|
 
|
Line 378: Line 203:  
\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
|
+
| height="120px" | [[Image:Cactus (A)B Big.jpg|35px]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|        o a      |
  −
|        |        |
  −
|        @ b      |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <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]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|      a  b      |
  −
|      o---o      |
  −
|        \ /        |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <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]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|      a  b      |
  −
|      o---o      |
  −
|        \ /        |
  −
|        o        |
  −
|        |        |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <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]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|      a  b  c      |
  −
|      o--o--o      |
  −
|      \  /      |
  −
|        \ /        |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <math>\texttt{(} a, b, c \texttt{)}</math>
 
| <math>\texttt{(} a, b, c \texttt{)}</math>
 
|
 
|
Line 470: Line 253:  
\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
|
+
| height="160px" | [[Image:Cactus ((A,B,C)) Big.jpg|70px]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|      a  b  c      |
  −
|      o--o--o      |
  −
|      \  /      |
  −
|        \ /        |
  −
|        o        |
  −
|        |        |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <math>\texttt{((} a, b, c \texttt{))}</math>
 
| <math>\texttt{((} a, b, c \texttt{))}</math>
 
|
 
|
Line 494: Line 264:  
\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
|
+
| height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|70px]]
<pre>
  −
o-------------------o
  −
|                   |
  −
|      a            |
  −
|      o            |
  −
|      |  b  c      |
  −
|      o--o--o      |
  −
|      \  /      |
  −
|        \ /        |
  −
|        o        |
  −
|        |        |
  −
|        @        |
  −
|                  |
  −
o-------------------o
  −
</pre>
   
| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
 
| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
 
|
 
|
Line 525: Line 280:  
For the time being, the main things to take away from Tables&nbsp;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&nbsp;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:
 +
|-
 +
| &nbsp;
 +
|
 
<pre>
 
<pre>
1.  The "node connective" joins a number of
  −
    component cacti C_1, ..., C_k at a node:
  −
   
     C_1 ... C_k
 
     C_1 ... C_k
 
         @
 
         @
 
+
</pre>
2. The "lobe connective" joins a number of
+
|-
    component cacti C_1, ..., C_k to a lobe:
+
| valign="top" | 2.
 
+
| The ''lobe connective'' joins a number of component cacti <math>C_1, \ldots, C_k</math> to a lobe:
 +
|-
 +
| &nbsp;
 +
|
 +
<pre>
 
     C_1 C_2  C_k
 
     C_1 C_2  C_k
 
     o---o-...-o
 
     o---o-...-o
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           @
 
           @
 
</pre>
 
</pre>
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Table&nbsp;15 summarizes the existential and entitative interpretations of the primitive cactus structures, in effect, the graphical constants and connectives.
 
Table&nbsp;15 summarizes the existential and entitative interpretations of the primitive cactus structures, in effect, the graphical constants and connectives.
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<pre>
 
<pre>
 
Table 15.  Existential & Entitative Interpretations of Cactus Structures
 
Table 15.  Existential & Entitative Interpretations of Cactus Structures
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o-----------------o-----------------o-----------------o-----------------o
 
o-----------------o-----------------o-----------------o-----------------o
 
</pre>
 
</pre>
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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.
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Table&nbsp;16 schematizes the two types of basic reductions in a purely formal, interpretation-independent fashion.
 
Table&nbsp;16 schematizes the two types of basic reductions in a purely formal, interpretation-independent fashion.
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<pre>
 
<pre>
 
Table 16.  Basic Reductions
 
Table 16.  Basic Reductions
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o---------------------------------------o
 
o---------------------------------------o
 
</pre>
 
</pre>
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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.
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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''.
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<pre>
 
<pre>
 
o---------------------------------------o
 
o---------------------------------------o
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The rule of reduction for a lobe is:
 
The rule of reduction for a lobe is:
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<pre>
 
<pre>
 
o---------------------------------------o
 
o---------------------------------------o
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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):
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<pre>
 
<pre>
 
o---------------------------------------o
 
o---------------------------------------o
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<pre>
 
<pre>
 
o---------------------------------------o
 
o---------------------------------------o
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