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==Cactus Language Stretching Exercises Display 1==
+
==Boolean Functions on Two Variables==
  
−
===PNG===
+
===Old Table===
  
−
===LaTeX===
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
 +
|+ style="height:30px" | <math>\text{Table 7.} ~~ \text{Propositional Forms on Two Variables}</math>
 +
|- style="height:40px; background:ghostwhite"
 +
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math>
 +
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math>
 +
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math>
 +
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math>
 +
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math>
 +
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math>
 +
|- style="background:ghostwhite"
 +
| &nbsp;
 +
| align="right" | <math>x\colon</math>
 +
| <math>1~1~0~0</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|- style="background:ghostwhite"
 +
| &nbsp;
 +
| align="right" | <math>y\colon</math>
 +
| <math>1~0~1~0</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|-
 +
| <math>f_{0}</math>
 +
| <math>f_{0000}</math>
 +
| <math>0~0~0~0</math>
 +
| <math>(~)</math>
 +
| <math>\text{false}</math>
 +
| <math>0</math>
 +
|-
 +
| <math>f_{1}</math>
 +
| <math>f_{0001}</math>
 +
| <math>0~0~0~1</math>
 +
| <math>(x)(y)</math>
 +
| <math>\text{neither}~ x ~\text{nor}~ y</math>
 +
| <math>\lnot x \land \lnot y</math>
 +
|-
 +
| <math>f_{2}</math>
 +
| <math>f_{0010}</math>
 +
| <math>0~0~1~0</math>
 +
| <math>(x)\ y</math>
 +
| <math>y ~\text{without}~ x</math>
 +
| <math>\lnot x \land y</math>
 +
|-
 +
| <math>f_{3}</math>
 +
| <math>f_{0011}</math>
 +
| <math>0~0~1~1</math>
 +
| <math>(x)</math>
 +
| <math>\text{not}~ x</math>
 +
| <math>\lnot x</math>
 +
|-
 +
| <math>f_{4}</math>
 +
| <math>f_{0100}</math>
 +
| <math>0~1~0~0</math>
 +
| <math>x\ (y)</math>
 +
| <math>x ~\text{without}~ y</math>
 +
| <math>x \land \lnot y</math>
 +
|-
 +
| <math>f_{5}</math>
 +
| <math>f_{0101}</math>
 +
| <math>0~1~0~1</math>
 +
| <math>(y)</math>
 +
| <math>\text{not}~ y</math>
 +
| <math>\lnot y</math>
 +
|-
 +
| <math>f_{6}</math>
 +
| <math>f_{0110}</math>
 +
| <math>0~1~1~0</math>
 +
| <math>(x, y)</math>
 +
| <math>x ~\text{not equal to}~ y</math>
 +
| <math>x \ne y</math>
 +
|-
 +
| <math>f_{7}</math>
 +
| <math>f_{0111}</math>
 +
| <math>0~1~1~1</math>
 +
| <math>(x\ y)</math>
 +
| <math>\text{not both}~ x ~\text{and}~ y</math>
 +
| <math>\lnot x \lor \lnot y</math>
 +
|-
 +
| <math>f_{8}</math>
 +
| <math>f_{1000}</math>
 +
| <math>1~0~0~0</math>
 +
| <math>x\ y</math>
 +
| <math>x ~\text{and}~ y</math>
 +
| <math>x \land y</math>
 +
|-
 +
| <math>f_{9}</math>
 +
| <math>f_{1001}</math>
 +
| <math>1~0~0~1</math>
 +
| <math>((x, y))</math>
 +
| <math>x ~\text{equal to}~ y</math>
 +
| <math>x = y</math>
 +
|-
 +
| <math>f_{10}</math>
 +
| <math>f_{1010}</math>
 +
| <math>1~0~1~0</math>
 +
| <math>y</math>
 +
| <math>y</math>
 +
| <math>y</math>
 +
|-
 +
| <math>f_{11}</math>
 +
| <math>f_{1011}</math>
 +
| <math>1~0~1~1</math>
 +
| <math>(x\ (y))</math>
 +
| <math>\text{not}~ x ~\text{without}~ y</math>
 +
| <math>x \Rightarrow y</math>
 +
|-
 +
| <math>f_{12}</math>
 +
| <math>f_{1100}</math>
 +
| <math>1~1~0~0</math>
 +
| <math>x</math>
 +
| <math>x</math>
 +
| <math>x</math>
 +
|-
 +
| <math>f_{13}</math>
 +
| <math>f_{1101}</math>
 +
| <math>1~1~0~1</math>
 +
| <math>((x)\ y)</math>
 +
| <math>\text{not}~ y ~\text{without}~ x</math>
 +
| <math>x \Leftarrow y</math>
 +
|-
 +
| <math>f_{14}</math>
 +
| <math>f_{1110}</math>
 +
| <math>1~1~1~0</math>
 +
| <math>((x)(y))</math>
 +
| <math>x ~\text{or}~ y</math>
 +
| <math>x \lor y</math>
 +
|-
 +
| <math>f_{15}</math>
 +
| <math>f_{1111}</math>
 +
| <math>1~1~1~1</math>
 +
| <math>((~))</math>
 +
| <math>\text{true}</math>
 +
| <math>1</math>
 +
|}
 +
 
 +
===New Tables===
 +
 
 +
====Template====
 +
 
 +
<table align="center" cellpadding="10px" cellspacing ="0" style="border:1px solid black; font-size:medium; text-align:center;">
 +
 
 +
<caption style="height:2em;">
 +
<math>\text{Boolean Functions on Two Variables}</math></caption>
 +
 
 +
<tr style="height:2em;">
 +
<th style="border-bottom:2px solid black;"><math>\text{Boolean Function}</math></th>
 +
<th style="border-bottom:2px solid black;"><math>\text{Linguistic Formula}</math></th>
 +
<th style="border-bottom:2px solid black;"><math>\text{Entitative Graph}</math></th>
 +
<th style="border-bottom:2px solid black;"><math>\text{Existential Graph}</math></th>
 +
</tr>
 +
 
 +
<tr style="height:100px;">
 +
<td style="vertical-align:middle;"><math>f_0</math></td>
 +
<td style="vertical-align:middle;"><math>\text{false}</math></td>
 +
<td style="vertical-align:middle;">[[File:Cactus Patch Root.jpg|32px]]</td>
 +
<td style="vertical-align:middle;">[[File:Cactus Patch Stem.jpg|32px]]</td>
 +
</tr>
 +
 
 +
<tr style="height:100px;">
 +
<td style="vertical-align:middle;"><math>f_1</math></td>
 +
<td style="vertical-align:middle;"><math>\text{neither}~ x ~\text{nor}~ y</math></td>
 +
<td style="vertical-align:middle;"><math>f_1</math></td>
 +
<td style="vertical-align:middle;"><math>f_1</math></td>
 +
</tr>
 +
 
 +
<tr style="height:100px;">
 +
<td style="vertical-align:middle;"><math>f_2</math></td>
 +
<td style="vertical-align:middle;"><math>y ~\text{without}~ x</math></td>
 +
<td style="vertical-align:middle;"><math>f_2</math></td>
 +
<td style="vertical-align:middle;"><math>f_2</math></td>
 +
</tr>
 +
 
 +
<tr style="height:100px;">
 +
<td style="vertical-align:middle;"><math>f_3</math></td>
 +
<td style="vertical-align:middle;"><math>\text{not}~ x</math></td>
 +
<td style="vertical-align:middle;">[[File:Cactus Patch (X).jpg|32px]]</td>
 +
<td style="vertical-align:middle;">[[File:Cactus Patch (X).jpg|32px]]</td>
 +
</tr>
 +
 
 +
<tr style="height:100px;">
 +
<td style="vertical-align:middle;"><math>f_4</math></td>
 +
<td style="vertical-align:middle;"><math>x ~\text{without}~ y</math></td>
 +
<td style="vertical-align:middle;"><math>f_4</math></td>
 +
<td style="vertical-align:middle;"><math>f_4</math></td>
 +
</tr>
 +
 
 +
<tr style="height:100px;">
 +
<td style="vertical-align:middle;"><math>f_5</math></td>
 +
<td style="vertical-align:middle;"><math>\text{not}~ y</math></td>
 +
<td style="vertical-align:middle;">[[File:Cactus Patch (Y).jpg|32px]]</td>
 +
<td style="vertical-align:middle;">[[File:Cactus Patch (Y).jpg|32px]]</td>
 +
</tr>
 +
 
 +
<tr style="height:100px;">
 +
<td style="vertical-align:middle;"><math>f_6</math></td>
 +
<td style="vertical-align:middle;"><math>x ~\text{not equal to}~ y</math></td>
 +
<td style="vertical-align:middle;"><math>f_6</math></td>
 +
<td style="vertical-align:middle;">[[File:Cactus Patch (X,Y).jpg|64px]]</td>
 +
</tr>
 +
 
 +
<tr style="height:100px;">
 +
<td style="vertical-align:middle;border-bottom:1px solid black;">
 +
<math>f_7</math></td>
 +
<td style="vertical-align:middle;border-bottom:1px solid black;">
 +
<math>\text{not both}~ x ~\text{and}~ y</math></td>
 +
<td style="vertical-align:middle;border-bottom:1px solid black;">
 +
<math>f_7</math></td>
 +
<td style="vertical-align:middle;border-bottom:1px solid black;">
 +
<math>f_7</math></td>
 +
</tr>
 +
 
 +
<tr style="height:100px;">
 +
<td style="vertical-align:middle;"><math>f_8</math></td>
 +
<td style="vertical-align:middle;"><math>x ~\text{and}~ y</math></td>
 +
<td style="vertical-align:middle;"><math>f_8</math></td>
 +
<td style="vertical-align:middle;">[[File:Cactus Patch XY.jpg|32px]]</td>
 +
</tr>
 +
 
 +
<tr style="height:100px;">
 +
<td style="vertical-align:middle;"><math>f_9</math></td>
 +
<td style="vertical-align:middle;"><math>x ~\text{equal to}~ y</math></td>
 +
<td style="vertical-align:middle;">[[File:Cactus Patch (X,Y).jpg|64px]]</td>
 +
<td style="vertical-align:middle;"><math>f_9</math></td>
 +
</tr>
 +
 
 +
<tr style="height:100px;">
 +
<td style="vertical-align:middle;"><math>f_{10}</math></td>
 +
<td style="vertical-align:middle;"><math>y</math></td>
 +
<td style="vertical-align:middle;">[[File:Cactus Patch Y.jpg|32px]]</td>
 +
<td style="vertical-align:middle;">[[File:Cactus Patch Y.jpg|32px]]</td>
 +
</tr>
 +
 
 +
<tr style="height:100px;">
 +
<td style="vertical-align:middle;"><math>f_{11}</math></td>
 +
<td style="vertical-align:middle;"><math>\text{not}~ x ~\text{without}~ y</math></td>
 +
<td style="vertical-align:middle;"><math>f_{11}</math></td>
 +
<td style="vertical-align:middle;"><math>f_{11}</math></td>
 +
</tr>
 +
 
 +
<tr style="height:100px;">
 +
<td style="vertical-align:middle;"><math>f_{12}</math></td>
 +
<td style="vertical-align:middle;"><math>x</math></td>
 +
<td style="vertical-align:middle;">[[File:Cactus Patch X.jpg|32px]]</td>
 +
<td style="vertical-align:middle;">[[File:Cactus Patch X.jpg|32px]]</td>
 +
</tr>
 +
 
 +
<tr style="height:100px;">
 +
<td style="vertical-align:middle;"><math>f_{13}</math></td>
 +
<td style="vertical-align:middle;"><math>\text{not}~ y ~\text{without}~ x</math></td>
 +
<td style="vertical-align:middle;"><math>f_{13}</math></td>
 +
<td style="vertical-align:middle;"><math>f_{13}</math></td>
 +
</tr>
 +
 
 +
<tr style="height:100px;">
 +
<td style="vertical-align:middle;"><math>f_{14}</math></td>
 +
<td style="vertical-align:middle;"><math>x ~\text{or}~ y</math></td>
 +
<td style="vertical-align:middle;"><math>f_{14}</math></td>
 +
<td style="vertical-align:middle;"><math>f_{14}</math></td>
 +
</tr>
 +
 
 +
<tr style="height:100px;">
 +
<td style="vertical-align:middle;"><math>f_{15}</math></td>
 +
<td style="vertical-align:middle;"><math>\text{true}</math></td>
 +
<td style="vertical-align:middle;">[[File:Cactus Patch Stem.jpg|32px]]</td>
 +
<td style="vertical-align:middle;">[[File:Cactus Patch Root.jpg|32px]]</td>
 +
</tr>
 +
 
 +
</table>
 +
 
 +
====Entitative Interpretation====
 +
 
 +
====Existential Interpretation====
 +
 
 +
==Logical Cacti &bull; Theme One Exposition==
 +
 
 +
Up till now we've been working to hammer out a two-edged sword of syntax, honing the syntax of ''painted and rooted cacti and expressions'' (PARCAE), and turning it to use in taming the syntax of two-level formal languages.
 +
 
 +
But the purpose of a logical syntax is to support a logical semantics, which means, for starters, to bear interpretation as sentential signs that can denote objective propositions about some universe of objects.
 +
 
 +
One of the difficulties that we face in this discussion is that the words ''interpretation'', ''meaning'', ''semantics'', and so on will have so many different meanings from one moment to the next of their use.  A dedicated neologician might be able to think up distinctive names for all of the aspects of meaning and all of the approaches to them that will concern us here, but I will just have to do the best that I can with the common lot of ambiguous terms, leaving it to context and the intelligent interpreter to sort it out as much as possible.
 +
 
 +
As it happens, the language of cacti is so abstract that it can bear at least two different interpretations as logical sentences denoting logical propositions.  The two interpretations that I know about are descended from the ones that C.S. Peirce called the ''entitative'' and the ''existential'' interpretations of his systems of graphical logics.  For our present aims, I shall briefly introduce the alternatives and then quickly move to the existential interpretation of logical cacti.
 +
 
 +
===Existential Interpretation===
 +
 
 +
Table&nbsp;13 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.
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
 +
|+ style="height:30px" |
 +
<math>\text{Table 13.} ~~ \text{Existential Interpretation}</math>
 +
|- style="height:40px; background:#f0f0ff"
 +
| <math>\text{Cactus Graph}</math>
 +
| <math>\text{Cactus Expression}</math>
 +
| <math>\text{Interpretation}</math>
 +
|-
 +
| height="100px" | [[File:Cactus Graph Node Big.jpg|24px]]
 +
| <math>\mathrm{~}</math>
 +
| <math>\mathrm{true}</math>
 +
|-
 +
| height="100px" | [[File:Cactus Graph Spike Big.jpg|24px]]
 +
| <math>\texttt{(} ~ \texttt{)}</math>
 +
| <math>\mathrm{false}</math>
 +
|-
 +
| height="100px" | [[File:Cactus A Big.jpg|20px]]
 +
| <math>a</math>
 +
| <math>a</math>
 +
|-
 +
| height="120px" | [[File:Cactus (A) Big.jpg|20px]]
 +
| <math>\texttt{(} a \texttt{)}</math>
 +
|
 +
<math>\begin{matrix}
 +
\tilde{a}
 +
\\[2pt]
 +
a^\prime
 +
\\[2pt]
 +
\lnot a
 +
\\[2pt]
 +
\mathrm{not}~ a
 +
\end{matrix}</math>
 +
|-
 +
| height="100px" | [[File:Cactus ABC Big.jpg|50px]]
 +
| <math>a~b~c</math>
 +
|
 +
<math>\begin{matrix}
 +
a \land b \land c
 +
\\[6pt]
 +
a ~\mathrm{and}~ b ~\mathrm{and}~ c
 +
\end{matrix}</math>
 +
|-
 +
| height="160px" | [[File:Cactus ((A)(B)(C)) Big.jpg|65px]]
 +
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
 +
|
 +
<math>\begin{matrix}
 +
a \lor b \lor c
 +
\\[6pt]
 +
a ~\mathrm{or}~ b ~\mathrm{or}~ c
 +
\end{matrix}</math>
 +
|-
 +
| height="120px" | [[File:Cactus (A(B)) Big.jpg|60px]]
 +
| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
 +
|
 +
<math>\begin{matrix}
 +
a \Rightarrow b
 +
\\[2pt]
 +
a ~\mathrm{implies}~ b
 +
\\[2pt]
 +
\mathrm{if}~ a ~\mathrm{then}~ b
 +
\\[2pt]
 +
\mathrm{not}~ a ~\mathrm{without}~ b
 +
\end{matrix}</math>
 +
|-
 +
| height="120px" | [[File:Cactus (A,B) Big.jpg|65px]]
 +
| <math>\texttt{(} a, b \texttt{)}</math>
 +
|
 +
<math>\begin{matrix}
 +
a + b
 +
\\[2pt]
 +
a \neq b
 +
\\[2pt]
 +
a ~\mathrm{exclusive~or}~ b
 +
\\[2pt]
 +
a ~\mathrm{not~equal~to}~ b
 +
\end{matrix}</math>
 +
|-
 +
| height="160px" | [[File:Cactus ((A,B)) Big.jpg|65px]]
 +
| <math>\texttt{((} a, b \texttt{))}</math>
 +
|
 +
<math>\begin{matrix}
 +
a = b
 +
\\[2pt]
 +
a \iff b
 +
\\[2pt]
 +
a ~\mathrm{equals}~ b
 +
\\[2pt]
 +
a ~\mathrm{if~and~only~if}~ b
 +
\end{matrix}</math>
 +
|-
 +
| height="120px" | [[File:Cactus (A,B,C) Big.jpg|65px]]
 +
| <math>\texttt{(} a, b, c \texttt{)}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{just~one~of}
 +
\\
 +
a, b, c
 +
\\
 +
\mathrm{is~false}
 +
\end{matrix}</math>
 +
|-
 +
| height="160px" | [[File:Cactus ((A),(B),(C)) Big.jpg|65px]]
 +
| <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{just~one~of}
 +
\\
 +
a, b, c
 +
\\
 +
\mathrm{is~true}
 +
\end{matrix}</math>
 +
|-
 +
| height="160px" | [[File:Cactus (A,(B),(C)) Big.jpg|65px]]
 +
| <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{genus}~ a ~\mathrm{of~species}~ b, c
 +
\\[6pt]
 +
\mathrm{partition}~ a ~\mathrm{into}~ b, c
 +
\\[6pt]
 +
\mathrm{pie}~ a ~\mathrm{of~slices}~ b, c
 +
\end{matrix}</math>
 +
|}
  
−
{| align="center" cellpadding="8"
+
===Entitative Interpretation===
−
| <math>F(x, y) ~=~ F_{6}^{(2)} (x, y) ~=~ \texttt{(} ~x~ \texttt{,} ~y~ \texttt{)}</math>
+
 
 +
Table&nbsp;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.
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
 +
|+ style="height:30px" |
 +
<math>\text{Table 14.} ~~ \text{Entitative Interpretation}</math>
 +
|- style="height:40px; background:#f0f0ff"
 +
| <math>\text{Cactus Graph}</math>
 +
| <math>\text{Cactus Expression}</math>
 +
| <math>\text{Interpretation}</math>
 +
|-
 +
| height="100px" | [[File:Cactus Graph Node Big.jpg|24px]]
 +
| <math>\mathrm{~}</math>
 +
| <math>\mathrm{false}</math>
 +
|-
 +
| height="100px" | [[File:Cactus Graph Spike Big.jpg|24px]]
 +
| <math>\texttt{(} ~ \texttt{)}</math>
 +
| <math>\mathrm{true}</math>
 +
|-
 +
| height="100px" | [[File:Cactus A Big.jpg|20px]]
 +
| <math>a</math>
 +
| <math>a</math>
 +
|-
 +
| height="120px" | [[File:Cactus (A) Big.jpg|20px]]
 +
| <math>\texttt{(} a \texttt{)}</math>
 +
|
 +
<math>\begin{matrix}
 +
\tilde{a}
 +
\\[2pt]
 +
a^\prime
 +
\\[2pt]
 +
\lnot a
 +
\\[2pt]
 +
\mathrm{not}~ a
 +
\end{matrix}</math>
 +
|-
 +
| height="100px" | [[File:Cactus ABC Big.jpg|50px]]
 +
| <math>a~b~c</math>
 +
|
 +
<math>\begin{matrix}
 +
a \lor b \lor c
 +
\\[6pt]
 +
a ~\mathrm{or}~ b ~\mathrm{or}~ c
 +
\end{matrix}</math>
 +
|-
 +
| height="160px" | [[File:Cactus ((A)(B)(C)) Big.jpg|65px]]
 +
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
 +
|
 +
<math>\begin{matrix}
 +
a \land b \land c
 +
\\[6pt]
 +
a ~\mathrm{and}~ b ~\mathrm{and}~ c
 +
\end{matrix}</math>
 +
|-
 +
| height="120px" | [[File:Cactus (A)B Big.jpg|35px]]
 +
| <math>\texttt{(} a \texttt{)} b</math>
 +
|
 +
<math>\begin{matrix}
 +
a \Rightarrow b
 +
\\[2pt]
 +
a ~\mathrm{implies}~ b
 +
\\[2pt]
 +
\mathrm{if}~ a ~\mathrm{then}~ b
 +
\\[2pt]
 +
\mathrm{not}~ a, ~\mathrm{or}~ b
 +
\end{matrix}</math>
 +
|-
 +
| height="120px" | [[File:Cactus (A,B) Big.jpg|65px]]
 +
| <math>\texttt{(} a, b \texttt{)}</math>
 +
|
 +
<math>\begin{matrix}
 +
a = b
 +
\\[2pt]
 +
a \iff b
 +
\\[2pt]
 +
a ~\mathrm{equals}~ b
 +
\\[2pt]
 +
a ~\mathrm{if~and~only~if}~ b
 +
\end{matrix}</math>
 +
|-
 +
| height="160px" | [[File:Cactus ((A,B)) Big.jpg|65px]]
 +
| <math>\texttt{((} a, b \texttt{))}</math>
 +
|
 +
<math>\begin{matrix}
 +
a + b
 +
\\[2pt]
 +
a \neq b
 +
\\[2pt]
 +
a ~\mathrm{exclusive~or}~ b
 +
\\[2pt]
 +
a ~\mathrm{not~equal~to}~ b
 +
\end{matrix}</math>
 +
|-
 +
| height="120px" | [[File:Cactus (A,B,C) Big.jpg|65px]]
 +
| <math>\texttt{(} a, b, c \texttt{)}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{not~just~one~of}
 +
\\
 +
a, b, c
 +
\\
 +
\mathrm{is~true}
 +
\end{matrix}</math>
 +
|-
 +
| height="160px" | [[File:Cactus ((A,B,C)) Big.jpg|65px]]
 +
| <math>\texttt{((} a, b, c \texttt{))}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{just~one~of}
 +
\\
 +
a, b, c
 +
\\
 +
\mathrm{is~true}
 +
\end{matrix}</math>
 +
|-
 +
| height="200px" | [[File:Cactus (((A),B,C)) Big.jpg|65px]]
 +
| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{genus}~ a ~\mathrm{of~species}~ b, c
 +
\\[6pt]
 +
\mathrm{partition}~ a ~\mathrm{into}~ b, c
 +
\\[6pt]
 +
\mathrm{pie}~ a ~\mathrm{of~slices}~ b, c
 +
\end{matrix}</math>
 
|}
 
|}
  
−
==Cactus Language Stretching Exercises Display 2==
+
==Logical Graphs==
  
−
===PNG===
+
===Old Versions===
  
−
===LaTeX===
+
====Example 1====
  
−
{| align="center" cellpadding="8"
+
{| align="center" cellpadding="10"
 
|
 
|
−
<math>\begin{array}{lll}
+
<pre>
−
[| \downharpoonleft s \downharpoonright |]
+
o-----------------------------------------------------------o
−
& = & [| F |]
+
|                                                          |
−
\\[4pt]
+
|  o o o            o o              o                  |
−
& = & F^{-1} (1)
+
|   \| |            | |              |                  |
−
\\[4pt]
+
|     o o o          o o o            o o            o  |
−
& = & \{~ (x, y) \in \mathbb{B}^2 ~:~ s ~\}
+
|      \|/            \|/              |/              |  |
−
\\[4pt]
+
|      @      =      @      =      @      =       @  |
−
& = & \{~ (x, y) \in \mathbb{B}^2 ~:~ F(x, y) = 1 ~\}
+
|                                                          |
−
\\[4pt]
+
o-----------------------------------------------------------o
−
& = & \{~ (x, y) \in \mathbb{B}^2 ~:~ F(x, y) ~\}
+
|                                                          |
−
\\[4pt]
+
| (()())(())() =   (())(())() =     (())()   =     ( ) |
−
& = & \{~ (x, y) \in \mathbb{B}^2 ~:~ \texttt{(}~x~,~y~\texttt{)} = 1 ~\}
+
|                                                          |
−
\\[4pt]
+
o-----------------------------------------------------------o
−
& = & \{~ (x, y) \in \mathbb{B}^2 ~:~ \texttt{(}~x~,~y~\texttt{)} ~\}
+
</pre>
−
\\[4pt]
 
−
& = & \{~ (x, y) \in \mathbb{B}^2 ~:~ x ~\mathrm{exclusive~or}~ y ~\}
 
−
\\[4pt]
 
−
& = & \{~ (x, y) \in \mathbb{B}^2 ~:~ \mathrm{just~one~true~of}~ x, y ~\}
 
−
\\[4pt]
 
−
& = & \{~ (x, y) \in \mathbb{B}^2 ~:~ x ~\mathrm{not~equal~to}~ y ~\}
 
−
\\[4pt]
 
−
& = & \{~ (x, y) \in \mathbb{B}^2 ~:~ x \nLeftrightarrow y ~\}
 
−
\\[4pt]
 
−
& = & \{~ (x, y) \in \mathbb{B}^2 ~:~ x \neq y ~\}
 
−
\\[4pt]
 
−
& = & \{~ (x, y) \in \mathbb{B}^2 ~:~ x + y ~\}.
 
−
\end{array}</math>
 
 
|}
 
|}
  
−
==Cactus Language Stretching Exercises Display 3==
+
====Example 2====
  
−
===PNG===
+
{| align="center" cellpadding="10"
 +
|
 +
<pre>
 +
o-------------------o-------------------o-------------------o
 +
| Object            | Sign              | Interpretant      |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
| Falsity          | "(()())(())()"    | "(())(())()"      |
 +
|                  |                  |                  |
 +
| Falsity          | "(())(())()"      | "(())()"          |
 +
|                  |                  |                  |
 +
| Falsity          | "(())()"          | "()"              |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
</pre>
 +
|}
  
−
===LaTeX===
+
====Example 3====
  
 +
{| align="center" cellpadding="10"
 +
|
 +
<pre>
 +
o-------------------o-------------------o-------------------o
 +
| Object            | Sign              | Interpretant      |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
| Falsity          | "(()())(())()"    | "(()())(())()"    |
 +
|                  |                  |                  |
 +
| Falsity          | "(()())(())()"    | "(())(())()"      |
 +
|                  |                  |                  |
 +
| Falsity          | "(()())(())()"    | "(())()"          |
 +
|                  |                  |                  |
 +
| Falsity          | "(()())(())()"    | "()"              |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
| Falsity          | "(())(())()"      | "(()())(())()"    |
 +
|                  |                  |                  |
 +
| Falsity          | "(())(())()"      | "(())(())()"      |
 +
|                  |                  |                  |
 +
| Falsity          | "(())(())()"      | "(())()"          |
 +
|                  |                  |                  |
 +
| Falsity          | "(())(())()"      | "()"              |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
| Falsity          | "(())()"          | "(()())(())()"    |
 +
|                  |                  |                  |
 +
| Falsity          | "(())()"          | "(())(())()"      |
 +
|                  |                  |                  |
 +
| Falsity          | "(())()"          | "(())()"          |
 +
|                  |                  |                  |
 +
| Falsity          | "(())()"          | "()"              |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
| Falsity          | "()"              | "(()())(())()"    |
 +
|                  |                  |                  |
 +
| Falsity          | "()"              | "(())(())()"      |
 +
|                  |                  |                  |
 +
| Falsity          | "()"              | "(())()"          |
 +
|                  |                  |                  |
 +
| Falsity          | "()"              | "()"              |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
</pre>
 +
|}
  
−
{| align="center" cellpadding="8"
+
====Example 4====
 +
 
 +
{| align="center" cellpadding="10"
 
|
 
|
−
<math>\begin{matrix}
+
<pre>
−
p & = & \upharpoonleft P \upharpoonright & : & X \to \mathbb{B}
+
o-------------------o-------------------o-------------------o
−
\\[4pt]
+
|        a        |        b        |      (a , b)      |
−
q & = & \upharpoonleft Q \upharpoonright & : & X \to \mathbb{B}
+
o-------------------o-------------------o-------------------o
−
\\[4pt]
+
|                  |                  |                  |
−
(p, q) & = & (\upharpoonleft P \upharpoonright, \upharpoonleft Q \upharpoonright) & : & (X \to \mathbb{B})^2
+
|      blank      |      blank      |      cross      |
−
\end{matrix}</math>
+
|                  |                  |                  |
 +
|      blank      |      cross      |      blank      |
 +
|                  |                  |                  |
 +
|      cross      |      blank      |      blank      |
 +
|                  |                  |                  |
 +
|      cross      |      cross      |      cross      |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
</pre>
 
|}
 
|}
  
−
==Cactus Language Stretching Exercises Display 4==
+
====Example 5====
  
−
===PNG===
+
{| align="center" cellpadding="10"
 +
|
 +
<pre>
 +
o-------o-------o-------o-----------o
 +
|  a  |  b  |  c  | (a, b, c) |
 +
o-------o-------o-------o-----------o
 +
|      |      |      |          |
 +
| blank | blank | blank |  cross  |
 +
|      |      |      |          |
 +
| blank | blank | cross |  blank  |
 +
|      |      |      |          |
 +
| blank | cross | blank |  blank  |
 +
|      |      |      |          |
 +
| blank | cross | cross |  cross  |
 +
|      |      |      |          |
 +
| cross | blank | blank |  blank  |
 +
|      |      |      |          |
 +
| cross | blank | cross |  cross  |
 +
|      |      |      |          |
 +
| cross | cross | blank |  cross  |
 +
|      |      |      |          |
 +
| cross | cross | cross |  cross  |
 +
|      |      |      |          |
 +
o-------o-------o-------o-----------o
 +
</pre>
 +
|}
  
−
===LaTeX===
+
====Example 6====
  
−
{| align="center" cellpadding="8"
+
{| align="center" cellpadding="10"
 
|
 
|
−
<math>\begin{array}{ccccl}
+
<pre>
−
F^\$ & = & \underline{(} \ldots, \ldots \underline{)}^\$ & : & (X \to \mathbb{B})^2 \to (X \to \mathbb{B})
+
o-------o-------o-------o-----------o
−
\\[4pt]
+
|  a  |  b  |  c  | (a, b, c) |
−
F^\$ (p, q) & = & \underline{(}~p~,~q~\underline{)}^\$ & : & X \to \mathbb{B}
+
o-------o-------o-------o-----------o
 +
|      |      |      |          |
 +
|  o  |  o  |  o  |    |    |
 +
|      |      |      |          |
 +
|  o  |  o  |  |  |    o    |
 +
|      |      |      |          |
 +
|  o  |  |  |  o  |    o    |
 +
|      |      |      |          |
 +
|  o  |  |  |  |  |    |    |
 +
|      |      |      |          |
 +
|  |  |  o  |  o  |    o    |
 +
|      |      |      |          |
 +
|  |  |  o  |  |  |    |    |
 +
|      |      |      |          |
 +
|  |  |  |  |  o  |    |    |
 +
|      |      |      |          |
 +
|  |  |  |  |  |  |    |    |
 +
|      |      |      |          |
 +
o-------o-------o-------o-----------o
 +
</pre>
 +
|}
 +
 
 +
===New Versions===
 +
 
 +
====Example 1====
 +
 
 +
&hellip;
 +
 
 +
====Example 2a====
 +
 
 +
{| align="center" border="1" cellpadding="12" cellspacing="0" width="50%"
 +
|- style="background:ghostwhite; height:40px"
 +
| width="33%" | <math>\text{Object}</math>
 +
| width="33%" | <math>\text{Sign}</math>
 +
| width="33%" | <math>\text{Interpretant}</math>
 +
|-
 +
| <math>\mathrm{Falsity}</math>
 +
| <math>{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}</math>
 +
| <math>{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}</math>
 +
|-
 +
| <math>\mathrm{Falsity}</math>
 +
| <math>{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}</math>
 +
| <math>{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}</math>
 +
|-
 +
| <math>\mathrm{Falsity}</math>
 +
| <math>{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}</math>
 +
| <math>{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}</math>
 +
|}
 +
 
 +
====Example 2b====
 +
 
 +
{| align="center" border="1" cellpadding="12" cellspacing="0" width="50%"
 +
|- style="background:ghostwhite; height:40px"
 +
| width="33%" | <math>\text{Object}</math>
 +
| width="33%" | <math>\text{Sign}</math>
 +
| width="33%" | <math>\text{Interpretant}</math>
 +
|-
 +
| valign="bottom" |
 +
<math>\begin{array}{l}
 +
\mathrm{Falsity}
 +
\\[6pt]
 +
\mathrm{Falsity}
 +
\\[6pt]
 +
\mathrm{Falsity}
 +
\end{array}</math>
 +
| valign="bottom" |
 +
<math>\begin{array}{l}
 +
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
 +
\end{array}</math>
 +
| valign="bottom" |
 +
<math>\begin{array}{l}
 +
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
 
\end{array}</math>
 
\end{array}</math>
 
|}
 
|}
  
−
==Cactus Language Stretching Exercises Display 5==
+
====Example 3====
  
−
===PNG===
+
{| align="center" border="1" cellpadding="12" cellspacing="0" width="50%"
 +
|- style="background:ghostwhite; height:40px"
 +
| width="33%" | <math>\text{Object}</math>
 +
| width="33%" | <math>\text{Sign}</math>
 +
| width="33%" | <math>\text{Interpretant}</math>
 +
|-
 +
| valign="bottom" |
 +
<math>\begin{array}{l}
 +
\mathrm{Falsity}
 +
\\[6pt]
 +
\mathrm{Falsity}
 +
\\[6pt]
 +
\mathrm{Falsity}
 +
\\[6pt]
 +
\mathrm{Falsity}
 +
\end{array}</math>
 +
| valign="bottom" |
 +
<math>\begin{array}{l}
 +
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
 +
\end{array}</math>
 +
| valign="bottom" |
 +
<math>\begin{array}{l}
 +
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
 +
\end{array}</math>
 +
|-
 +
| valign="bottom" |
 +
<math>\begin{array}{l}
 +
\mathrm{Falsity}
 +
\\[6pt]
 +
\mathrm{Falsity}
 +
\\[6pt]
 +
\mathrm{Falsity}
 +
\\[6pt]
 +
\mathrm{Falsity}
 +
\end{array}</math>
 +
| valign="bottom" |
 +
<math>\begin{array}{l}
 +
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
 +
\end{array}</math>
 +
| valign="bottom" |
 +
<math>\begin{array}{l}
 +
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
 +
\end{array}</math>
 +
|-
 +
| valign="bottom" |
 +
<math>\begin{array}{l}
 +
\mathrm{Falsity}
 +
\\[6pt]
 +
\mathrm{Falsity}
 +
\\[6pt]
 +
\mathrm{Falsity}
 +
\\[6pt]
 +
\mathrm{Falsity}
 +
\end{array}</math>
 +
| valign="bottom" |
 +
<math>\begin{array}{l}
 +
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
 +
\end{array}</math>
 +
| valign="bottom" |
 +
<math>\begin{array}{l}
 +
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
 +
\end{array}</math>
 +
|-
 +
| valign="bottom" |
 +
<math>\begin{array}{l}
 +
\mathrm{Falsity}
 +
\\[6pt]
 +
\mathrm{Falsity}
 +
\\[6pt]
 +
\mathrm{Falsity}
 +
\\[6pt]
 +
\mathrm{Falsity}
 +
\end{array}</math>
 +
| valign="bottom" |
 +
<math>\begin{array}{l}
 +
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
 +
\end{array}</math>
 +
| valign="bottom" |
 +
<math>\begin{array}{l}
 +
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
 +
\\[6pt]
 +
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
 +
\end{array}</math>
 +
|}
  
−
===LaTeX===
+
====Example 4====
  
−
{| align="center" cellpadding="8"
+
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"
−
|
+
|- style="background:ghostwhite; height:40px"
 +
| width="33%" | <math>a</math>
 +
| width="33%" | <math>b</math>
 +
| width="33%" | <math>\texttt{(} a \texttt{,} b \texttt{)}</math>
 +
|-
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
\texttt{Blank}
 +
\\[6pt]
 +
\texttt{Blank}
 +
\\[6pt]
 +
\texttt{Cross}
 +
\\[6pt]
 +
\texttt{Cross}
 +
\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
\texttt{Blank}
 +
\\[6pt]
 +
\texttt{Cross}
 +
\\[6pt]
 +
\texttt{Blank}
 +
\\[6pt]
 +
\texttt{Cross}
 +
\end{matrix}</math>
 +
| valign="bottom" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
F^\$ (p, q)(x) & = & \underline{(}~p~,~q~\underline{)}^\$ (x) & \in & \mathbb{B}
+
\texttt{Cross}
−
\\[4pt]
+
\\[6pt]
−
\Updownarrow  &  & \Updownarrow
+
\texttt{Blank}
−
\\[4pt]
+
\\[6pt]
−
F(p(x), q(x))  & = & \underline{(}~p(x)~,~q(x)~\underline{)}  & \in & \mathbb{B}
+
\texttt{Blank}
 +
\\[6pt]
 +
\texttt{Cross}
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|}
 
|}
  
−
==Cactus Language Stretching Exercises Display 6==
+
====Example 5====
  
−
===PNG===
+
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"
 +
|- style="background:ghostwhite; height:40px"
 +
| width="25%" | <math>a</math>
 +
| width="25%" | <math>b</math>
 +
| width="25%" | <math>c</math>
 +
| width="25%" | <math>\texttt{(} a \texttt{,} b \texttt{,} \texttt{c} \texttt{)}</math>
 +
|-
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
\texttt{Blank}
 +
\\[6pt]
 +
\texttt{Blank}
 +
\\[6pt]
 +
\texttt{Blank}
 +
\\[6pt]
 +
\texttt{Blank}
 +
\\[6pt]
 +
\texttt{Cross}
 +
\\[6pt]
 +
\texttt{Cross}
 +
\\[6pt]
 +
\texttt{Cross}
 +
\\[6pt]
 +
\texttt{Cross}
 +
\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
\texttt{Blank}
 +
\\[6pt]
 +
\texttt{Blank}
 +
\\[6pt]
 +
\texttt{Cross}
 +
\\[6pt]
 +
\texttt{Cross}
 +
\\[6pt]
 +
\texttt{Blank}
 +
\\[6pt]
 +
\texttt{Blank}
 +
\\[6pt]
 +
\texttt{Cross}
 +
\\[6pt]
 +
\texttt{Cross}
 +
\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
\texttt{Blank}
 +
\\[6pt]
 +
\texttt{Cross}
 +
\\[6pt]
 +
\texttt{Blank}
 +
\\[6pt]
 +
\texttt{Cross}
 +
\\[6pt]
 +
\texttt{Blank}
 +
\\[6pt]
 +
\texttt{Cross}
 +
\\[6pt]
 +
\texttt{Blank}
 +
\\[6pt]
 +
\texttt{Cross}
 +
\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
\texttt{Cross}
 +
\\[6pt]
 +
\texttt{Blank}
 +
\\[6pt]
 +
\texttt{Blank}
 +
\\[6pt]
 +
\texttt{Cross}
 +
\\[6pt]
 +
\texttt{Blank}
 +
\\[6pt]
 +
\texttt{Cross}
 +
\\[6pt]
 +
\texttt{Cross}
 +
\\[6pt]
 +
\texttt{Cross}
 +
\end{matrix}</math>
 +
|}
  
−
===LaTeX===
+
====Example 6====
  
−
{| align="center" cellpadding="8"
+
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"
−
|
+
|- style="background:ghostwhite; height:40px"
−
<math>\begin{array}{lll}
+
| width="25%" | <math>a</math>
−
[| F^\$ (p, q) |]
+
| width="25%" | <math>b</math>
−
& = & [| \underline{(}~p~,~q~\underline{)}^\$ |]
+
| width="25%" | <math>c</math>
−
\\[4pt]
+
| width="25%" | <math>\texttt{(} a \texttt{,} b \texttt{,} \texttt{c} \texttt{)}</math>
−
& = & (F^\$ (p, q))^{-1} (1)
+
|-
−
\\[4pt]
+
| valign="bottom" |
−
& = & \{~ x \in X ~:~ F^\$ (p, q)(x) ~\}
+
<math>\begin{matrix}
−
\\[4pt]
+
\texttt{o}
−
& = & \{~ x \in X ~:~ \underline{(}~p~,~q~\underline{)}^\$ (x) ~\}
+
\\[6pt]
−
\\[4pt]
+
\texttt{o}
−
& = & \{~ x \in X ~:~ \underline{(}~p(x)~,~q(x)~\underline{)} ~\}
+
\\[6pt]
−
\\[4pt]
+
\texttt{o}
−
& = & \{~ x \in X ~:~ p(x) + q(x) ~\}
+
\\[6pt]
−
\\[4pt]
+
\texttt{o}
−
& = & \{~ x \in X ~:~ p(x) \neq q(x) ~\}
+
\\[6pt]
−
\\[4pt]
+
\texttt{|}
−
& = & \{~ x \in X ~:~ \upharpoonleft P \upharpoonright (x) ~\neq~ \upharpoonleft Q \upharpoonright (x) ~\}
+
\\[6pt]
−
\\[4pt]
+
\texttt{|}
−
& = & \{~ x \in X ~:~ x \in P ~\nLeftrightarrow~ x \in Q ~\}
+
\\[6pt]
−
\\[4pt]
+
\texttt{|}
−
& = & \{~ x \in X ~:~ x \in P\!-\!Q ~\mathrm{or}~ x \in Q\!-\!P ~\}
+
\\[6pt]
−
\\[4pt]
+
\texttt{|}
−
& = & \{~ x \in X ~:~ x \in P\!-\!Q ~\cup~ Q\!-\!P ~\}
+
\end{matrix}</math>
−
\\[4pt]
+
| valign="bottom" |
−
& = & \{~ x \in X ~:~ x \in P + Q ~\}
+
<math>\begin{matrix}
−
\\[4pt]
+
\texttt{o}
−
& = & P + Q ~\subseteq~ X
+
\\[6pt]
−
\\[4pt]
+
\texttt{o}
−
& = & [|p|] + [|q|] ~\subseteq~ X
+
\\[6pt]
−
\end{array}</math>
+
\texttt{|}
 +
\\[6pt]
 +
\texttt{|}
 +
\\[6pt]
 +
\texttt{o}
 +
\\[6pt]
 +
\texttt{o}
 +
\\[6pt]
 +
\texttt{|}
 +
\\[6pt]
 +
\texttt{|}
 +
\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
\texttt{o}
 +
\\[6pt]
 +
\texttt{|}
 +
\\[6pt]
 +
\texttt{o}
 +
\\[6pt]
 +
\texttt{|}
 +
\\[6pt]
 +
\texttt{o}
 +
\\[6pt]
 +
\texttt{|}
 +
\\[6pt]
 +
\texttt{o}
 +
\\[6pt]
 +
\texttt{|}
 +
\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
\texttt{|}
 +
\\[6pt]
 +
\texttt{o}
 +
\\[6pt]
 +
\texttt{o}
 +
\\[6pt]
 +
\texttt{|}
 +
\\[6pt]
 +
\texttt{o}
 +
\\[6pt]
 +
\texttt{|}
 +
\\[6pt]
 +
\texttt{|}
 +
\\[6pt]
 +
\texttt{|}
 +
\end{matrix}</math>
 
|}
 
|}

Latest revision as of 13:58, 12 March 2026

Boolean Functions on Two Variables

Old Table

\(\text{Table 7.} ~~ \text{Propositional Forms on Two Variables}\)
\(\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}\) \(\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}\) \(\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}\) \(\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}\) \(\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}\) \(\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}\)
  \(x\colon\) \(1~1~0~0\)      
  \(y\colon\) \(1~0~1~0\)      
\(f_{0}\) \(f_{0000}\) \(0~0~0~0\) \((~)\) \(\text{false}\) \(0\)
\(f_{1}\) \(f_{0001}\) \(0~0~0~1\) \((x)(y)\) \(\text{neither}~ x ~\text{nor}~ y\) \(\lnot x \land \lnot y\)
\(f_{2}\) \(f_{0010}\) \(0~0~1~0\) \((x)\ y\) \(y ~\text{without}~ x\) \(\lnot x \land y\)
\(f_{3}\) \(f_{0011}\) \(0~0~1~1\) \((x)\) \(\text{not}~ x\) \(\lnot x\)
\(f_{4}\) \(f_{0100}\) \(0~1~0~0\) \(x\ (y)\) \(x ~\text{without}~ y\) \(x \land \lnot y\)
\(f_{5}\) \(f_{0101}\) \(0~1~0~1\) \((y)\) \(\text{not}~ y\) \(\lnot y\)
\(f_{6}\) \(f_{0110}\) \(0~1~1~0\) \((x, y)\) \(x ~\text{not equal to}~ y\) \(x \ne y\)
\(f_{7}\) \(f_{0111}\) \(0~1~1~1\) \((x\ y)\) \(\text{not both}~ x ~\text{and}~ y\) \(\lnot x \lor \lnot y\)
\(f_{8}\) \(f_{1000}\) \(1~0~0~0\) \(x\ y\) \(x ~\text{and}~ y\) \(x \land y\)
\(f_{9}\) \(f_{1001}\) \(1~0~0~1\) \(((x, y))\) \(x ~\text{equal to}~ y\) \(x = y\)
\(f_{10}\) \(f_{1010}\) \(1~0~1~0\) \(y\) \(y\) \(y\)
\(f_{11}\) \(f_{1011}\) \(1~0~1~1\) \((x\ (y))\) \(\text{not}~ x ~\text{without}~ y\) \(x \Rightarrow y\)
\(f_{12}\) \(f_{1100}\) \(1~1~0~0\) \(x\) \(x\) \(x\)
\(f_{13}\) \(f_{1101}\) \(1~1~0~1\) \(((x)\ y)\) \(\text{not}~ y ~\text{without}~ x\) \(x \Leftarrow y\)
\(f_{14}\) \(f_{1110}\) \(1~1~1~0\) \(((x)(y))\) \(x ~\text{or}~ y\) \(x \lor y\)
\(f_{15}\) \(f_{1111}\) \(1~1~1~1\) \(((~))\) \(\text{true}\) \(1\)

New Tables

Template

\(\text{Boolean Functions on Two Variables}\)
\(\text{Boolean Function}\) \(\text{Linguistic Formula}\) \(\text{Entitative Graph}\) \(\text{Existential Graph}\)
\(f_0\) \(\text{false}\) 32px 32px
\(f_1\) \(\text{neither}~ x ~\text{nor}~ y\) \(f_1\) \(f_1\)
\(f_2\) \(y ~\text{without}~ x\) \(f_2\) \(f_2\)
\(f_3\) \(\text{not}~ x\) 32px 32px
\(f_4\) \(x ~\text{without}~ y\) \(f_4\) \(f_4\)
\(f_5\) \(\text{not}~ y\) 32px 32px
\(f_6\) \(x ~\text{not equal to}~ y\) \(f_6\) 64px
\(f_7\) \(\text{not both}~ x ~\text{and}~ y\) \(f_7\) \(f_7\)
\(f_8\) \(x ~\text{and}~ y\) \(f_8\) 32px
\(f_9\) \(x ~\text{equal to}~ y\) 64px \(f_9\)
\(f_{10}\) \(y\) 32px 32px
\(f_{11}\) \(\text{not}~ x ~\text{without}~ y\) \(f_{11}\) \(f_{11}\)
\(f_{12}\) \(x\) 32px 32px
\(f_{13}\) \(\text{not}~ y ~\text{without}~ x\) \(f_{13}\) \(f_{13}\)
\(f_{14}\) \(x ~\text{or}~ y\) \(f_{14}\) \(f_{14}\)
\(f_{15}\) \(\text{true}\) 32px 32px

Entitative Interpretation

Existential Interpretation

Logical Cacti • Theme One Exposition

Up till now we've been working to hammer out a two-edged sword of syntax, honing the syntax of painted and rooted cacti and expressions (PARCAE), and turning it to use in taming the syntax of two-level formal languages.

But the purpose of a logical syntax is to support a logical semantics, which means, for starters, to bear interpretation as sentential signs that can denote objective propositions about some universe of objects.

One of the difficulties that we face in this discussion is that the words interpretation, meaning, semantics, and so on will have so many different meanings from one moment to the next of their use. A dedicated neologician might be able to think up distinctive names for all of the aspects of meaning and all of the approaches to them that will concern us here, but I will just have to do the best that I can with the common lot of ambiguous terms, leaving it to context and the intelligent interpreter to sort it out as much as possible.

As it happens, the language of cacti is so abstract that it can bear at least two different interpretations as logical sentences denoting logical propositions. The two interpretations that I know about are descended from the ones that C.S. Peirce called the entitative and the existential interpretations of his systems of graphical logics. For our present aims, I shall briefly introduce the alternatives and then quickly move to the existential interpretation of logical cacti.

Existential Interpretation

Table 13 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.

\(\text{Table 13.} ~~ \text{Existential Interpretation}\)
\(\text{Cactus Graph}\) \(\text{Cactus Expression}\) \(\text{Interpretation}\)
24px \(\mathrm{~}\) \(\mathrm{true}\)
24px \(\texttt{(} ~ \texttt{)}\) \(\mathrm{false}\)
Cactus A Big.jpg \(a\) \(a\)
Cactus (A) Big.jpg \(\texttt{(} a \texttt{)}\)

\(\begin{matrix} \tilde{a} \\[2pt] a^\prime \\[2pt] \lnot a \\[2pt] \mathrm{not}~ a \end{matrix}\)

Cactus ABC Big.jpg \(a~b~c\)

\(\begin{matrix} a \land b \land c \\[6pt] a ~\mathrm{and}~ b ~\mathrm{and}~ c \end{matrix}\)

Cactus ((A)(B)(C)) Big.jpg \(\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}\)

\(\begin{matrix} a \lor b \lor c \\[6pt] a ~\mathrm{or}~ b ~\mathrm{or}~ c \end{matrix}\)

Cactus (A(B)) Big.jpg \(\texttt{(} a \texttt{(} b \texttt{))}\)

\(\begin{matrix} a \Rightarrow b \\[2pt] a ~\mathrm{implies}~ b \\[2pt] \mathrm{if}~ a ~\mathrm{then}~ b \\[2pt] \mathrm{not}~ a ~\mathrm{without}~ b \end{matrix}\)

Cactus (A,B) Big.jpg \(\texttt{(} a, b \texttt{)}\)

\(\begin{matrix} a + b \\[2pt] a \neq b \\[2pt] a ~\mathrm{exclusive~or}~ b \\[2pt] a ~\mathrm{not~equal~to}~ b \end{matrix}\)

Cactus ((A,B)) Big.jpg \(\texttt{((} a, b \texttt{))}\)

\(\begin{matrix} a = b \\[2pt] a \iff b \\[2pt] a ~\mathrm{equals}~ b \\[2pt] a ~\mathrm{if~and~only~if}~ b \end{matrix}\)

Cactus (A,B,C) Big.jpg \(\texttt{(} a, b, c \texttt{)}\)

\(\begin{matrix} \mathrm{just~one~of} \\ a, b, c \\ \mathrm{is~false} \end{matrix}\)

Cactus ((A),(B),(C)) Big.jpg \(\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}\)

\(\begin{matrix} \mathrm{just~one~of} \\ a, b, c \\ \mathrm{is~true} \end{matrix}\)

Cactus (A,(B),(C)) Big.jpg \(\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}\)

\(\begin{matrix} \mathrm{genus}~ a ~\mathrm{of~species}~ b, c \\[6pt] \mathrm{partition}~ a ~\mathrm{into}~ b, c \\[6pt] \mathrm{pie}~ a ~\mathrm{of~slices}~ b, c \end{matrix}\)

Entitative Interpretation

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.

\(\text{Table 14.} ~~ \text{Entitative Interpretation}\)
\(\text{Cactus Graph}\) \(\text{Cactus Expression}\) \(\text{Interpretation}\)
24px \(\mathrm{~}\) \(\mathrm{false}\)
24px \(\texttt{(} ~ \texttt{)}\) \(\mathrm{true}\)
Cactus A Big.jpg \(a\) \(a\)
Cactus (A) Big.jpg \(\texttt{(} a \texttt{)}\)

\(\begin{matrix} \tilde{a} \\[2pt] a^\prime \\[2pt] \lnot a \\[2pt] \mathrm{not}~ a \end{matrix}\)

Cactus ABC Big.jpg \(a~b~c\)

\(\begin{matrix} a \lor b \lor c \\[6pt] a ~\mathrm{or}~ b ~\mathrm{or}~ c \end{matrix}\)

Cactus ((A)(B)(C)) Big.jpg \(\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}\)

\(\begin{matrix} a \land b \land c \\[6pt] a ~\mathrm{and}~ b ~\mathrm{and}~ c \end{matrix}\)

Cactus (A)B Big.jpg \(\texttt{(} a \texttt{)} b\)

\(\begin{matrix} a \Rightarrow b \\[2pt] a ~\mathrm{implies}~ b \\[2pt] \mathrm{if}~ a ~\mathrm{then}~ b \\[2pt] \mathrm{not}~ a, ~\mathrm{or}~ b \end{matrix}\)

Cactus (A,B) Big.jpg \(\texttt{(} a, b \texttt{)}\)

\(\begin{matrix} a = b \\[2pt] a \iff b \\[2pt] a ~\mathrm{equals}~ b \\[2pt] a ~\mathrm{if~and~only~if}~ b \end{matrix}\)

Cactus ((A,B)) Big.jpg \(\texttt{((} a, b \texttt{))}\)

\(\begin{matrix} a + b \\[2pt] a \neq b \\[2pt] a ~\mathrm{exclusive~or}~ b \\[2pt] a ~\mathrm{not~equal~to}~ b \end{matrix}\)

Cactus (A,B,C) Big.jpg \(\texttt{(} a, b, c \texttt{)}\)

\(\begin{matrix} \mathrm{not~just~one~of} \\ a, b, c \\ \mathrm{is~true} \end{matrix}\)

Cactus ((A,B,C)) Big.jpg \(\texttt{((} a, b, c \texttt{))}\)

\(\begin{matrix} \mathrm{just~one~of} \\ a, b, c \\ \mathrm{is~true} \end{matrix}\)

Cactus (((A),B,C)) Big.jpg \(\texttt{(((} a \texttt{)}, b, c \texttt{))}\)

\(\begin{matrix} \mathrm{genus}~ a ~\mathrm{of~species}~ b, c \\[6pt] \mathrm{partition}~ a ~\mathrm{into}~ b, c \\[6pt] \mathrm{pie}~ a ~\mathrm{of~slices}~ b, c \end{matrix}\)

Logical Graphs

Old Versions

Example 1

o-----------------------------------------------------------o
|                                                           |
|   o o o             o o               o                   |
|    \| |             | |               |                   |
|     o o o           o o o             o o             o   |
|      \|/             \|/              |/              |   |
|       @       =       @       =       @       =       @   |
|                                                           |
o-----------------------------------------------------------o
|                                                           |
| (()())(())()  =   (())(())()  =     (())()    =      ( )  |
|                                                           |
o-----------------------------------------------------------o

Example 2

o-------------------o-------------------o-------------------o
| Object            | Sign              | Interpretant      |
o-------------------o-------------------o-------------------o
|                   |                   |                   |
| Falsity           | "(()())(())()"    | "(())(())()"      |
|                   |                   |                   |
| Falsity           | "(())(())()"      | "(())()"          |
|                   |                   |                   |
| Falsity           | "(())()"          | "()"              |
|                   |                   |                   |
o-------------------o-------------------o-------------------o

Example 3

o-------------------o-------------------o-------------------o
| Object            | Sign              | Interpretant      |
o-------------------o-------------------o-------------------o
|                   |                   |                   |
| Falsity           | "(()())(())()"    | "(()())(())()"    |
|                   |                   |                   |
| Falsity           | "(()())(())()"    | "(())(())()"      |
|                   |                   |                   |
| Falsity           | "(()())(())()"    | "(())()"          |
|                   |                   |                   |
| Falsity           | "(()())(())()"    | "()"              |
|                   |                   |                   |
o-------------------o-------------------o-------------------o
|                   |                   |                   |
| Falsity           | "(())(())()"      | "(()())(())()"    |
|                   |                   |                   |
| Falsity           | "(())(())()"      | "(())(())()"      |
|                   |                   |                   |
| Falsity           | "(())(())()"      | "(())()"          |
|                   |                   |                   |
| Falsity           | "(())(())()"      | "()"              |
|                   |                   |                   |
o-------------------o-------------------o-------------------o
|                   |                   |                   |
| Falsity           | "(())()"          | "(()())(())()"    |
|                   |                   |                   |
| Falsity           | "(())()"          | "(())(())()"      |
|                   |                   |                   |
| Falsity           | "(())()"          | "(())()"          |
|                   |                   |                   |
| Falsity           | "(())()"          | "()"              |
|                   |                   |                   |
o-------------------o-------------------o-------------------o
|                   |                   |                   |
| Falsity           | "()"              | "(()())(())()"    |
|                   |                   |                   |
| Falsity           | "()"              | "(())(())()"      |
|                   |                   |                   |
| Falsity           | "()"              | "(())()"          |
|                   |                   |                   |
| Falsity           | "()"              | "()"              |
|                   |                   |                   |
o-------------------o-------------------o-------------------o

Example 4

o-------------------o-------------------o-------------------o
|         a         |         b         |      (a , b)      |
o-------------------o-------------------o-------------------o
|                   |                   |                   |
|       blank       |       blank       |       cross       |
|                   |                   |                   |
|       blank       |       cross       |       blank       |
|                   |                   |                   |
|       cross       |       blank       |       blank       |
|                   |                   |                   |
|       cross       |       cross       |       cross       |
|                   |                   |                   |
o-------------------o-------------------o-------------------o

Example 5

o-------o-------o-------o-----------o
|   a   |   b   |   c   | (a, b, c) |
o-------o-------o-------o-----------o
|       |       |       |           |
| blank | blank | blank |   cross   |
|       |       |       |           |
| blank | blank | cross |   blank   |
|       |       |       |           |
| blank | cross | blank |   blank   |
|       |       |       |           |
| blank | cross | cross |   cross   |
|       |       |       |           |
| cross | blank | blank |   blank   |
|       |       |       |           |
| cross | blank | cross |   cross   |
|       |       |       |           |
| cross | cross | blank |   cross   |
|       |       |       |           |
| cross | cross | cross |   cross   |
|       |       |       |           |
o-------o-------o-------o-----------o

Example 6

o-------o-------o-------o-----------o
|   a   |   b   |   c   | (a, b, c) |
o-------o-------o-------o-----------o
|       |       |       |           |
|   o   |   o   |   o   |     |     |
|       |       |       |           |
|   o   |   o   |   |   |     o     |
|       |       |       |           |
|   o   |   |   |   o   |     o     |
|       |       |       |           |
|   o   |   |   |   |   |     |     |
|       |       |       |           |
|   |   |   o   |   o   |     o     |
|       |       |       |           |
|   |   |   o   |   |   |     |     |
|       |       |       |           |
|   |   |   |   |   o   |     |     |
|       |       |       |           |
|   |   |   |   |   |   |     |     |
|       |       |       |           |
o-------o-------o-------o-----------o

New Versions

Example 1

…

Example 2a

\(\text{Object}\) \(\text{Sign}\) \(\text{Interpretant}\)
\(\mathrm{Falsity}\) \({}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}\) \({}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}\)
\(\mathrm{Falsity}\) \({}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}\) \({}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}\)
\(\mathrm{Falsity}\) \({}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}\) \({}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}\)

Example 2b

\(\text{Object}\) \(\text{Sign}\) \(\text{Interpretant}\)

\(\begin{array}{l} \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \end{array}\)

\(\begin{array}{l} {}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime} \end{array}\)

\(\begin{array}{l} {}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{()} {}^{\prime\prime} \end{array}\)

Example 3

\(\text{Object}\) \(\text{Sign}\) \(\text{Interpretant}\)

\(\begin{array}{l} \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \end{array}\)

\(\begin{array}{l} {}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime} \end{array}\)

\(\begin{array}{l} {}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{()} {}^{\prime\prime} \end{array}\)

\(\begin{array}{l} \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \end{array}\)

\(\begin{array}{l} {}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime} \end{array}\)

\(\begin{array}{l} {}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{()} {}^{\prime\prime} \end{array}\)

\(\begin{array}{l} \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \end{array}\)

\(\begin{array}{l} {}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime} \end{array}\)

\(\begin{array}{l} {}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{()} {}^{\prime\prime} \end{array}\)

\(\begin{array}{l} \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \end{array}\)

\(\begin{array}{l} {}^{\backprime\backprime} \texttt{()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{()} {}^{\prime\prime} \end{array}\)

\(\begin{array}{l} {}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{()} {}^{\prime\prime} \end{array}\)

Example 4

\(a\) \(b\) \(\texttt{(} a \texttt{,} b \texttt{)}\)

\(\begin{matrix} \texttt{Blank} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Cross} \end{matrix}\)

\(\begin{matrix} \texttt{Blank} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Cross} \end{matrix}\)

\(\begin{matrix} \texttt{Cross} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Cross} \end{matrix}\)

Example 5

\(a\) \(b\) \(c\) \(\texttt{(} a \texttt{,} b \texttt{,} \texttt{c} \texttt{)}\)

\(\begin{matrix} \texttt{Blank} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Cross} \end{matrix}\)

\(\begin{matrix} \texttt{Blank} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Cross} \end{matrix}\)

\(\begin{matrix} \texttt{Blank} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Cross} \end{matrix}\)

\(\begin{matrix} \texttt{Cross} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Cross} \end{matrix}\)

Example 6

\(a\) \(b\) \(c\) \(\texttt{(} a \texttt{,} b \texttt{,} \texttt{c} \texttt{)}\)

\(\begin{matrix} \texttt{o} \\[6pt] \texttt{o} \\[6pt] \texttt{o} \\[6pt] \texttt{o} \\[6pt] \texttt{|} \\[6pt] \texttt{|} \\[6pt] \texttt{|} \\[6pt] \texttt{|} \end{matrix}\)

\(\begin{matrix} \texttt{o} \\[6pt] \texttt{o} \\[6pt] \texttt{|} \\[6pt] \texttt{|} \\[6pt] \texttt{o} \\[6pt] \texttt{o} \\[6pt] \texttt{|} \\[6pt] \texttt{|} \end{matrix}\)

\(\begin{matrix} \texttt{o} \\[6pt] \texttt{|} \\[6pt] \texttt{o} \\[6pt] \texttt{|} \\[6pt] \texttt{o} \\[6pt] \texttt{|} \\[6pt] \texttt{o} \\[6pt] \texttt{|} \end{matrix}\)

\(\begin{matrix} \texttt{|} \\[6pt] \texttt{o} \\[6pt] \texttt{o} \\[6pt] \texttt{|} \\[6pt] \texttt{o} \\[6pt] \texttt{|} \\[6pt] \texttt{|} \\[6pt] \texttt{|} \end{matrix}\)

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