Difference between revisions of "User:Jon Awbrey/Figures and Tables 1"

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==Format Examples==
+
==Logic Syllabus==
  
−
<ul>
+
===Related articles===
−
<li><math>\rightsquigarrow</math></li>
 
  
−
<li><math>\leftrightsquigarrow</math></li>
+
<table style="border:none;width:120%;">
  
−
<li><math>\xrightarrow{\mathrm{Parse}}</math></li>
+
<tr>
 +
<td style="border:none;width:50%;">&nbsp;&#9726; [[Cactus Language]]</td>
 +
<td style="border:none;width:50%;">&nbsp;&#9726; [[Propositions As Types]]</td>
 +
</tr>
  
−
<li><math>\xrightarrow[\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}]{\mathrm{Parse}}</math></li>
+
<tr>
−
</ul>
+
<td style="border:none;width:50%;">&nbsp;&#9726; [[Futures Of Logical Graphs]]</td>
 +
<td style="border:none;width:50%;">&nbsp;&#9726; [[Propositional Equation Reasoning Systems]]</td>
 +
</tr>
  
−
==Font Size Examples==
+
<tr><td style="border:none;"></td></tr>
  
−
{| align="center" cellpadding="10" cellspacing="0"
+
<tr>
−
| align="left" width="20%" | Medium
+
<td style="border:none;width:50%;">&nbsp;&#9726; [[Correspondence Theory Of Truth]]</td>
−
| align="center" style="font-size:medium" | <math>\text{Algorithmic Translation Rules}</math>
+
<td style="border:none;width:50%;">&nbsp;&#9726; [[Pragmatic Theory Of Truth]]</td>
 +
</tr>
 +
 
 +
<tr><td style="border:none;"></td></tr>
 +
 
 +
<tr>
 +
<td style="border:none;width:50%;">&nbsp;&#9726; [[Differential Logic &bull; Introduction]]</td>
 +
<td style="border:none;width:50%;">&nbsp;&#9726; [[Introduction to Inquiry Driven Systems]]</td>
 +
</tr>
 +
 
 +
<tr>
 +
<td style="border:none;width:50%;">&nbsp;&#9726; [[Differential Propositional Calculus]]</td>
 +
<td style="border:none;width:50%;">&nbsp;&#9726; [[Prospects for Inquiry Driven Systems]]</td>
 +
</tr>
 +
 
 +
<tr>
 +
<td style="border:none;width:50%;">&nbsp;&#9726; [[Differential Logic and Dynamic Systems]]</td>
 +
<td style="border:none;width:50%;">&nbsp;&#9726; [[Inquiry Driven Systems &bull; Inquiry Into Inquiry]]</td>
 +
</tr>
 +
 
 +
</table>
 +
 
 +
==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>
 
|-
 
|-
−
| Larger
+
| height="120px" | [[File:Cactus (A) Big.jpg|20px]]
−
| align="center" style="font-size:larger" | <math>\text{Algorithmic Translation Rules}</math>
+
| <math>\texttt{(} a \texttt{)}</math>
 +
|
 +
<math>\begin{matrix}
 +
\tilde{a}
 +
\\[2pt]
 +
a^\prime
 +
\\[2pt]
 +
\lnot a
 +
\\[2pt]
 +
\mathrm{not}~ a
 +
\end{matrix}</math>
 
|-
 
|-
−
| 125%
+
| height="100px" | [[File:Cactus ABC Big.jpg|50px]]
−
| align="center" style="font-size:125%" | <math>\text{Algorithmic Translation Rules}</math>
+
| <math>a~b~c</math>
 +
|
 +
<math>\begin{matrix}
 +
a \land b \land c
 +
\\[6pt]
 +
a ~\mathrm{and}~ b ~\mathrm{and}~ c
 +
\end{matrix}</math>
 
|-
 
|-
−
| Large
+
| height="160px" | [[File:Cactus ((A)(B)(C)) Big.jpg|65px]]
−
| align="center" style="font-size:large" | <math>\text{Algorithmic Translation Rules}</math>
+
| <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>
 
|-
 
|-
−
| 150%
+
| height="120px" | [[File:Cactus (A(B)) Big.jpg|60px]]
−
| align="center" style="font-size:150%" | <math>\text{Algorithmic Translation Rules}</math>
+
| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
−
|}
+
|
−
 
+
<math>\begin{matrix}
−
==Algorithmic Translation Rules==
+
a \Rightarrow b
−
 
+
\\[2pt]
−
===PNG===
+
a ~\mathrm{implies}~ b
−
 
+
\\[2pt]
−
{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
+
\mathrm{if}~ a ~\mathrm{then}~ b
−
| style="height:25px; font-size:large" | <math>\text{Algorithmic Translation Rules}</math>
+
\\[2pt]
 +
\mathrm{not}~ a ~\mathrm{without}~ b
 +
\end{matrix}</math>
 
|-
 
|-
−
| [[File:Cactus Language Algorithmic Translation Rules.png|600px]]
+
| height="120px" | [[File:Cactus (A,B) Big.jpg|65px]]
−
|}
+
| <math>\texttt{(} a, b \texttt{)}</math>
−
 
 
−
===LaTeX===
 
−
 
 
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
 
−
|+ style="height:30px; font-size:large" | <math>\text{Algorithmic Translation Rules}</math>
 
−
|- style="height:40px; background:ghostwhite"
 
 
|
 
|
−
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; text-align:center; width:100%"
+
<math>\begin{matrix}
−
| width="33%" | <math>\text{Sentence in PARCE}</math>
+
a + b
−
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}</math>
+
\\[2pt]
−
| width="33%" | <math>\text{Graph in PARC}</math>
+
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>
 
|
 
|
−
{| align="center" border="0" cellpadding="8" cellspacing="0" style="text-align:center; width:100%"
+
<math>\begin{matrix}
−
| width="33%" | <math>\mathrm{Conc}^0</math>
+
a = b
−
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}</math>
+
\\[2pt]
−
| width="33%" | <math>\mathrm{Node}^0</math>
+
a \iff b
 +
\\[2pt]
 +
a ~\mathrm{equals}~ b
 +
\\[2pt]
 +
a ~\mathrm{if~and~only~if}~ b
 +
\end{matrix}</math>
 
|-
 
|-
−
| width="33%" | <math>\mathrm{Conc}_{j=1}^k s_j</math>
+
| height="120px" | [[File:Cactus (A,B,C) Big.jpg|65px]]
−
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}</math>
+
| <math>\texttt{(} a, b, c \texttt{)}</math>
−
| width="33%" | <math>\mathrm{Node}_{j=1}^k \mathrm{Parse} (s_j)</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>
 
|
 
|
−
{| align="center" border="0" cellpadding="8" cellspacing="0" style="text-align:center; width:100%"
+
<math>\begin{matrix}
−
| width="33%" | <math>\mathrm{Surc}^0</math>
+
\mathrm{just~one~of}
−
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}</math>
+
\\
−
| width="33%" | <math>\mathrm{Lobe}^0</math>
+
a, b, c
 +
\\
 +
\mathrm{is~true}
 +
\end{matrix}</math>
 
|-
 
|-
−
| width="33%" | <math>\mathrm{Surc}_{j=1}^k s_j</math>
+
| height="160px" | [[File:Cactus (A,(B),(C)) Big.jpg|65px]]
−
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}</math>
+
| <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
−
| width="33%" | <math>\mathrm{Lobe}_{j=1}^k \mathrm{Parse} (s_j)</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>
 
|}
 
|}
  
−
<br>
+
===Entitative Interpretation===
  
−
==Semantic Translation &bull; Functional Form==
+
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.
  
−
===PNG===
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
−
 
+
|+ style="height:30px" |
−
{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
+
<math>\text{Table 14.} ~~ \text{Entitative Interpretation}</math>
−
| style="height:25px; font-size:large" | <math>\text{Semantic Translation}</math> &bull; <math>\text{Functional Form}</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>
 
|-
 
|-
−
| [[File:Cactus Language Semantic Translation Functional Form.png|600px]]
+
| height="120px" | [[File:Cactus (A)B Big.jpg|35px]]
−
|}
+
| <math>\texttt{(} a \texttt{)} b</math>
−
 
 
−
===LaTeX===
 
−
 
 
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
 
−
|+ style="height:30px; font-size:large" | <math>\text{Semantic Translation}</math> &bull; <math>\text{Functional Form}</math>
 
−
|- style="height:40px; background:ghostwhite"
 
 
|
 
|
−
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; width:100%"
+
<math>\begin{matrix}
−
| width="20%" | <math>\mathrm{Sentence}</math>
+
a \Rightarrow b
−
| width="20%" | <math>\xrightarrow[\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}]{\mathrm{Parse}}</math>
+
\\[2pt]
−
| width="20%" | <math>\mathrm{Graph}</math>
+
a ~\mathrm{implies}~ b
−
| width="20%" | <math>\xrightarrow[\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}]{\mathrm{Denotation}}</math>
+
\\[2pt]
−
| width="20%" | <math>\mathrm{Proposition}</math>
+
\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>
 
|
 
|
−
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
+
<math>\begin{matrix}
−
| width="20%" | <math>s_j</math>
+
a = b
−
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
+
\\[2pt]
−
| width="20%" | <math>C_j</math>
+
a \iff b
−
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
+
\\[2pt]
−
| width="20%" | <math>q_j</math>
+
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>
 
|
 
|
−
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
+
<math>\begin{matrix}
−
| width="20%" | <math>\mathrm{Conc}^0</math>
+
a + b
−
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
+
\\[2pt]
−
| width="20%" | <math>\mathrm{Node}^0</math>
+
a \neq b
−
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
+
\\[2pt]
−
| width="20%" | <math>1</math>
+
a ~\mathrm{exclusive~or}~ b
 +
\\[2pt]
 +
a ~\mathrm{not~equal~to}~ b
 +
\end{matrix}</math>
 
|-
 
|-
−
| width="20%" | <math>\mathrm{Conc}^k_j s_j</math>
+
| height="120px" | [[File:Cactus (A,B,C) Big.jpg|65px]]
−
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
+
| <math>\texttt{(} a, b, c \texttt{)}</math>
−
| width="20%" | <math>\mathrm{Node}^k_j C_j</math>
+
|
−
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
+
<math>\begin{matrix}
−
| width="20%" | <math>\mathrm{Conj}^k_j q_j</math>
+
\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>
 
|
 
|
−
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
+
<math>\begin{matrix}
−
| width="20%" | <math>\mathrm{Surc}^0</math>
+
\mathrm{just~one~of}
−
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
+
\\
−
| width="20%" | <math>\mathrm{Lobe}^0</math>
+
a, b, c
−
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
+
\\
−
| width="20%" | <math>0</math>
+
\mathrm{is~true}
 +
\end{matrix}</math>
 
|-
 
|-
−
| width="20%" | <math>\mathrm{Surc}^k_j s_j</math>
+
| height="200px" | [[File:Cactus (((A),B,C)) Big.jpg|65px]]
−
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
+
| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
−
| width="20%" | <math>\mathrm{Lobe}^k_j C_j</math>
+
|
−
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
+
<math>\begin{matrix}
−
| width="20%" | <math>\mathrm{Surj}^k_j q_j</math>
+
\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>
 
|}
 
|}
  
−
<br>
+
==Logical Graphs==
  
−
==Semantic Translation &bull; Equational Form==
+
===Old Versions===
  
−
===PNG===
+
====Example 1====
  
−
{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
+
{| align="center" cellpadding="10"
−
| style="height:25px; font-size:large" | <math>\text{Semantic Translation}</math> &bull; <math>\text{Equational Form}</math>
+
|
−
|-
+
<pre>
−
| [[File:Cactus Language Semantic Translation Equational Form.png|600px]]
+
o-----------------------------------------------------------o
 +
|                                                          |
 +
|  o o o            o o              o                  |
 +
|    \| |            | |              |                  |
 +
|     o o o          o o o            o o            o  |
 +
|     \|/             \|/              |/             |  |
 +
|      @      =      @      =      @      =      @  |
 +
|                                                           |
 +
o-----------------------------------------------------------o
 +
|                                                           |
 +
| (()())(())()  =  (())(())()  =    (())()    =      ( )  |
 +
|                                                          |
 +
o-----------------------------------------------------------o
 +
</pre>
 
|}
 
|}
  
−
===LaTeX===
+
====Example 2====
  
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
+
{| align="center" cellpadding="10"
−
|+ style="height:30px; font-size:large" | <math>\text{Semantic Translation}</math> &bull; <math>\text{Equational Form}</math>
 
−
|- style="height:40px; background:ghostwhite"
 
 
|
 
|
−
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; width:100%"
+
<pre>
−
| width="20%" | <math>\downharpoonleft \mathrm{Sentence} \downharpoonright</math>
+
o-------------------o-------------------o-------------------o
−
| width="20%" | <math>\stackrel{\mathrm{Parse}}{=}</math>
+
| Object            | Sign              | Interpretant      |
−
| width="20%" | <math>\downharpoonleft \mathrm{Graph} \downharpoonright</math>
+
o-------------------o-------------------o-------------------o
−
| width="20%" | <math>\stackrel{\mathrm{Denotation}}{=}</math>
+
|                  |                  |                  |
−
| width="20%" | <math>\mathrm{Proposition}</math>
+
| Falsity          | "(()())(())()"   | "(())(())()"     |
 +
|                  |                  |                  |
 +
| Falsity          | "(())(())()"     | "(())()"         |
 +
|                   |                  |                  |
 +
| Falsity          | "(())()"         | "()"             |
 +
|                   |                   |                   |
 +
o-------------------o-------------------o-------------------o
 +
</pre>
 
|}
 
|}
−
|-
+
 
 +
====Example 3====
 +
 
 +
{| align="center" cellpadding="10"
 
|
 
|
−
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
+
<pre>
−
| width="20%" | <math>\downharpoonleft s_j \downharpoonright</math>
+
o-------------------o-------------------o-------------------o
−
| width="20%" | <math>=</math>
+
| Object            | Sign              | Interpretant      |
−
| width="20%" | <math>\downharpoonleft C_j \downharpoonright</math>
+
o-------------------o-------------------o-------------------o
−
| width="20%" | <math>=</math>
+
|                  |                  |                  |
−
| width="20%" | <math>q_j</math>
+
| 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>
 
|}
 
|}
−
|-
+
 
 +
====Example 4====
 +
 
 +
{| align="center" cellpadding="10"
 
|
 
|
−
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
+
<pre>
−
| width="20%" | <math>\downharpoonleft \mathrm{Conc}^0 \downharpoonright</math>
+
o-------------------o-------------------o-------------------o
−
| width="20%" | <math>=</math>
+
|        a        |        b        |      (a , b)      |
−
| width="20%" | <math>\downharpoonleft \mathrm{Node}^0 \downharpoonright</math>
+
o-------------------o-------------------o-------------------o
−
| width="20%" | <math>=</math>
+
|                  |                  |                   |
−
| width="20%" | <math>1</math>
+
|       blank      |      blank      |      cross      |
−
|-
+
|                   |                  |                  |
−
| width="20%" | <math>\downharpoonleft \mathrm{Conc}^k_j s_j \downharpoonright</math>
+
|       blank      |      cross      |      blank      |
−
| width="20%" | <math>=</math>
+
|                   |                  |                  |
−
| width="20%" | <math>\downharpoonleft \mathrm{Node}^k_j C_j \downharpoonright</math>
+
|       cross      |      blank      |      blank      |
−
| width="20%" | <math>=</math>
+
|                   |                  |                  |
−
| width="20%" | <math>\mathrm{Conj}^k_j q_j</math>
+
|       cross      |       cross      |       cross      |
 +
|                   |                   |                   |
 +
o-------------------o-------------------o-------------------o
 +
</pre>
 
|}
 
|}
−
|-
+
 
 +
====Example 5====
 +
 
 +
{| align="center" cellpadding="10"
 
|
 
|
−
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
+
<pre>
−
| width="20%" | <math>\downharpoonleft \mathrm{Surc}^0 \downharpoonright</math>
+
o-------o-------o-------o-----------o
−
| width="20%" | <math>=</math>
+
|  a  |  b  |  c  | (a, b, c) |
−
| width="20%" | <math>\downharpoonleft \mathrm{Lobe}^0 \downharpoonright</math>
+
o-------o-------o-------o-----------o
−
| width="20%" | <math>=</math>
+
|      |      |      |          |
−
| width="20%" | <math>0</math>
+
| blank | blank | blank |  cross  |
−
|-
+
|      |      |      |          |
−
| width="20%" | <math>\downharpoonleft \mathrm{Surc}^k_j s_j \downharpoonright</math>
+
| blank | blank | cross |  blank  |
−
| width="20%" | <math>=</math>
+
|      |      |      |          |
−
| width="20%" | <math>\downharpoonleft \mathrm{Lobe}^k_j C_j \downharpoonright</math>
+
| blank | cross | blank |  blank  |
−
| width="20%" | <math>=</math>
+
|       |      |      |          |
−
| width="20%" | <math>\mathrm{Surj}^k_j q_j</math>
+
| 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>
 
|}
 
|}
 +
 +
====Example 6====
 +
 +
{| align="center" cellpadding="10"
 +
|
 +
<pre>
 +
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
 +
</pre>
 
|}
 
|}
  
−
<br>
+
===New Versions===
 +
 
 +
====Example 1====
  
−
==Boolean Functions on Zero Variables==
+
&hellip;
  
−
===PNG===
+
====Example 2a====
  
−
{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
+
{| align="center" border="1" cellpadding="12" cellspacing="0" width="50%"
−
|+ style="height:25px; font-size:large" | <math>\text{Boolean Functions on Zero Variables}</math>
+
|- 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>
 
|-
 
|-
−
| [[File:Boolean Functions on Zero Variables.png|600px]]
+
| <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>
 
|}
 
|}
  
−
===LaTeX===
+
====Example 2b====
  
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
+
{| align="center" border="1" cellpadding="12" cellspacing="0" width="50%"
−
|+ style="height:30px; font-size:large" | <math>\text{Boolean Functions on Zero Variables}</math>
+
|- style="background:ghostwhite; height:40px"
−
|- style="height:40px; background:ghostwhite"
+
| width="33%" | <math>\text{Object}</math>
−
| width="14%" | <math>F</math>
+
| width="33%" | <math>\text{Sign}</math>
−
| width="14%" | <math>F</math>
+
| width="33%" | <math>\text{Interpretant}</math>
−
| width="48%" | <math>F()</math>
 
−
| width="24%" | <math>F</math>
 
 
|-
 
|-
−
| <math>0</math>
+
| valign="bottom" |
−
| <math>F_0^{(0)}</math>
+
<math>\begin{array}{l}
−
| <math>0</math>
+
\mathrm{Falsity}
−
| <math>\texttt{( )}</math>
+
\\[6pt]
−
|-
+
\mathrm{Falsity}
−
| <math>1</math>
+
\\[6pt]
−
| <math>F_1^{(0)}</math>
+
\mathrm{Falsity}
−
| <math>1</math>
+
\end{array}</math>
−
| <math>\texttt{(( ))}</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>
 
|}
 
|}
  
−
==Boolean Functions on One Variable==
+
====Example 3====
  
−
===PNG===
+
{| align="center" border="1" cellpadding="12" cellspacing="0" width="50%"
−
 
+
|- style="background:ghostwhite; height:40px"
−
===LaTeX===
+
| width="33%" | <math>\text{Object}</math>
−
 
+
| width="33%" | <math>\text{Sign}</math>
−
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:60%"
+
| width="33%" | <math>\text{Interpretant}</math>
−
|+ style="height:30px; font-size:large" | <math>\text{Boolean Functions on One Variable}</math>
 
−
|- style="height:40px; background:ghostwhite"
 
−
| width="14%" | <math>F</math>
 
−
| width="14%" | <math>F</math>
 
−
| colspan="2" | <math>F(x)</math>
 
−
| width="24%" | <math>F</math>
 
−
|- style="height:40px; background:ghostwhite"
 
−
| width="14%" | &nbsp;
 
−
| width="14%" | &nbsp;
 
−
| width="24%" | <math>F(1)</math>
 
−
| width="24%" | <math>F(0)</math>
 
−
| width="24%" | &nbsp;
 
 
|-
 
|-
−
| <math>F_0^{(1)}</math>
+
| valign="bottom" |
−
| <math>F_{00}^{(1)}</math>
+
<math>\begin{array}{l}
−
| <math>0</math>
+
\mathrm{Falsity}
−
| <math>0</math>
+
\\[6pt]
−
| <math>\texttt{( )}</math>
+
\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>
 
|-
 
|-
−
| <math>F_1^{(1)}</math>
+
| valign="bottom" |
−
| <math>F_{01}^{(1)}</math>
+
<math>\begin{array}{l}
−
| <math>0</math>
+
\mathrm{Falsity}
−
| <math>1</math>
+
\\[6pt]
−
| <math>\texttt{(} x \texttt{)}</math>
+
\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>
 
|-
 
|-
−
| <math>F_2^{(1)}</math>
+
| valign="bottom" |
−
| <math>F_{10}^{(1)}</math>
+
<math>\begin{array}{l}
−
| <math>1</math>
+
\mathrm{Falsity}
−
| <math>0</math>
+
\\[6pt]
−
| <math>x</math>
+
\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>
 
|-
 
|-
−
| <math>F_3^{(1)}</math>
+
| valign="bottom" |
−
| <math>F_{11}^{(1)}</math>
+
<math>\begin{array}{l}
−
| <math>1</math>
+
\mathrm{Falsity}
−
| <math>1</math>
+
\\[6pt]
−
| <math>\texttt{(( ))}</math>
+
\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>
 
|}
 
|}
  
−
<br>
+
====Example 4====
 +
 
 +
{| 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}
 +
\texttt{Cross}
 +
\\[6pt]
 +
\texttt{Blank}
 +
\\[6pt]
 +
\texttt{Blank}
 +
\\[6pt]
 +
\texttt{Cross}
 +
\end{matrix}</math>
 +
|}
  
−
==Boolean Functions on Two Variables==
+
====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" border="1" cellpadding="4" cellspacing="0" style="text-align:center; width:60%"
+
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"
−
|+ style="height:30px; font-size:large" | <math>\text{Boolean Functions on Two Variables}</math>
+
|- style="background:ghostwhite; height:40px"
−
|- style="height:40px; background:ghostwhite"
+
| width="25%" | <math>a</math>
−
| width="14%" | <math>F</math>
+
| width="25%" | <math>b</math>
−
| width="14%" | <math>F</math>
+
| width="25%" | <math>c</math>
−
| colspan="4" | <math>F(x, y)</math>
+
| width="25%" | <math>\texttt{(} a \texttt{,} b \texttt{,} \texttt{c} \texttt{)}</math>
−
| width="24%" | <math>F</math>
 
−
|- style="height:40px; background:ghostwhite"
 
−
| width="14%" | &nbsp;
 
−
| width="14%" | &nbsp;
 
−
| width="12%" | <math>F(1, 1)</math>
 
−
| width="12%" | <math>F(1, 0)</math>
 
−
| width="12%" | <math>F(0, 1)</math>
 
−
| width="12%" | <math>F(0, 0)</math>
 
−
| width="24%" | &nbsp;
 
−
|-
 
−
| <math>F_{0}^{(2)}</math>
 
−
| <math>F_{0000}^{(2)}</math>
 
−
| <math>0</math>
 
−
| <math>0</math>
 
−
| <math>0</math>
 
−
| <math>0</math>
 
−
| <math>\texttt{( )}</math>
 
−
|-
 
−
| <math>F_{1}^{(2)}</math>
 
−
| <math>F_{0001}^{(2)}</math>
 
−
| <math>0</math>
 
−
| <math>0</math>
 
−
| <math>0</math>
 
−
| <math>1</math>
 
−
| <math>\texttt{(} x \texttt{)(} y \texttt{)}</math>
 
−
|-
 
−
| <math>F_{2}^{(2)}</math>
 
−
| <math>F_{0010}^{(2)}</math>
 
−
| <math>0</math>
 
−
| <math>0</math>
 
−
| <math>1</math>
 
−
| <math>0</math>
 
−
| <math>\texttt{(} x \texttt{)} y</math>
 
 
|-
 
|-
−
| <math>F_{3}^{(2)}</math>
+
| valign="bottom" |
−
| <math>F_{0011}^{(2)}</math>
+
<math>\begin{matrix}
−
| <math>0</math>
+
\texttt{o}
−
| <math>0</math>
+
\\[6pt]
−
| <math>1</math>
+
\texttt{o}
−
| <math>1</math>
+
\\[6pt]
−
| <math>\texttt{(} x \texttt{)}</math>
+
\texttt{o}
−
|-
+
\\[6pt]
−
| <math>F_{4}^{(2)}</math>
+
\texttt{o}
−
| <math>F_{0100}^{(2)}</math>
+
\\[6pt]
−
| <math>0</math>
+
\texttt{|}
−
| <math>1</math>
+
\\[6pt]
−
| <math>0</math>
+
\texttt{|}
−
| <math>0</math>
+
\\[6pt]
−
| <math>x \texttt{(} y \texttt{)}</math>
+
\texttt{|}
−
|-
+
\\[6pt]
−
| <math>F_{5}^{(2)}</math>
+
\texttt{|}
−
| <math>F_{0101}^{(2)}</math>
+
\end{matrix}</math>
−
| <math>0</math>
+
| valign="bottom" |
−
| <math>1</math>
+
<math>\begin{matrix}
−
| <math>0</math>
+
\texttt{o}
−
| <math>1</math>
+
\\[6pt]
−
| <math>\texttt{(} y \texttt{)}</math>
+
\texttt{o}
−
|-
+
\\[6pt]
−
| <math>F_{6}^{(2)}</math>
+
\texttt{|}
−
| <math>F_{0110}^{(2)}</math>
+
\\[6pt]
−
| <math>0</math>
+
\texttt{|}
−
| <math>1</math>
+
\\[6pt]
−
| <math>1</math>
+
\texttt{o}
−
| <math>0</math>
+
\\[6pt]
−
| <math>\texttt{(} x \texttt{,} y \texttt{)}</math>
+
\texttt{o}
−
|-
+
\\[6pt]
−
| <math>F_{7}^{(2)}</math>
+
\texttt{|}
−
| <math>F_{0111}^{(2)}</math>
+
\\[6pt]
−
| <math>0</math>
+
\texttt{|}
−
| <math>1</math>
+
\end{matrix}</math>
−
| <math>1</math>
+
| valign="bottom" |
−
| <math>1</math>
+
<math>\begin{matrix}
−
| <math>\texttt{(} x y \texttt{)}</math>
+
\texttt{o}
−
|-
+
\\[6pt]
−
| <math>F_{8}^{(2)}</math>
+
\texttt{|}
−
| <math>F_{1000}^{(2)}</math>
+
\\[6pt]
−
| <math>1</math>
+
\texttt{o}
−
| <math>0</math>
+
\\[6pt]
−
| <math>0</math>
+
\texttt{|}
−
| <math>0</math>
+
\\[6pt]
−
| <math>x y</math>
+
\texttt{o}
−
|-
+
\\[6pt]
−
| <math>F_{9}^{(2)}</math>
+
\texttt{|}
−
| <math>F_{1001}^{(2)}</math>
+
\\[6pt]
−
| <math>1</math>
+
\texttt{o}
−
| <math>0</math>
+
\\[6pt]
−
| <math>0</math>
+
\texttt{|}
−
| <math>1</math>
+
\end{matrix}</math>
−
| <math>\texttt{((} x \texttt{,} y \texttt{))}</math>
+
| valign="bottom" |
−
|-
+
<math>\begin{matrix}
−
| <math>F_{10}^{(2)}</math>
+
\texttt{|}
−
| <math>F_{1010}^{(2)}</math>
+
\\[6pt]
−
| <math>1</math>
+
\texttt{o}
−
| <math>0</math>
+
\\[6pt]
−
| <math>1</math>
+
\texttt{o}
−
| <math>0</math>
+
\\[6pt]
−
| <math>y</math>
+
\texttt{|}
−
|-
+
\\[6pt]
−
| <math>F_{11}^{(2)}</math>
+
\texttt{o}
−
| <math>F_{1011}^{(2)}</math>
+
\\[6pt]
−
| <math>1</math>
+
\texttt{|}
−
| <math>0</math>
+
\\[6pt]
−
| <math>1</math>
+
\texttt{|}
−
| <math>1</math>
+
\\[6pt]
−
| <math>\texttt{(} x \texttt{(} y \texttt{))}</math>
+
\texttt{|}
−
|-
+
\end{matrix}</math>
−
| <math>F_{12}^{(2)}</math>
 
−
| <math>F_{1100}^{(2)}</math>
 
−
| <math>1</math>
 
−
| <math>1</math>
 
−
| <math>0</math>
 
−
| <math>0</math>
 
−
| <math>x</math>
 
−
|-
 
−
| <math>F_{13}^{(2)}</math>
 
−
| <math>F_{1101}^{(2)}</math>
 
−
| <math>1</math>
 
−
| <math>1</math>
 
−
| <math>0</math>
 
−
| <math>1</math>
 
−
| <math>\texttt{((} x \texttt{)} y \texttt{)}</math>
 
−
|-
 
−
| <math>F_{14}^{(2)}</math>
 
−
| <math>F_{1110}^{(2)}</math>
 
−
| <math>1</math>
 
−
| <math>1</math>
 
−
| <math>1</math>
 
−
| <math>0</math>
 
−
| <math>\texttt{((} x \texttt{)(} y \texttt{))}</math>
 
−
|-
 
−
| <math>F_{15}^{(2)}</math>
 
−
| <math>F_{1111}^{(2)}</math>
 
−
| <math>1</math>
 
−
| <math>1</math>
 
−
| <math>1</math>
 
−
| <math>1</math>
 
−
| <math>\texttt{(( ))}</math>
 
 
|}
 
|}
−
 
−
<br>
 

Latest revision as of 13:54, 12 March 2026

Logic Syllabus

Related articles

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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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