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

MyWikiBiz, Author Your Legacy — Friday March 13, 2026
Jump to navigationJump to search
(sub original User:Jon Awbrey/Figures and Tables 1)
 
(7 intermediate revisions by the same user not shown)
Line 1: Line 1:
−
==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" ccellpadding="10" 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" ccellpadding="10" 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" ccellpadding="10" 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="10" 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

 ◾ Cactus Language  ◾ Propositions As Types
 ◾ Futures Of Logical Graphs  ◾ Propositional Equation Reasoning Systems
 ◾ Correspondence Theory Of Truth  ◾ Pragmatic Theory Of Truth
 ◾ Differential Logic • Introduction  ◾ Introduction to Inquiry Driven Systems
 ◾ Differential Propositional Calculus  ◾ Prospects for Inquiry Driven Systems
 ◾ Differential Logic and Dynamic Systems  ◾ Inquiry Driven Systems • Inquiry Into Inquiry

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

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

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

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

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

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

Retrieved from "https://mywikibiz.com/index.php?title=User:Jon_Awbrey/Figures_and_Tables_1&oldid=481302"