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==Logic Syllabus==
===Related articles===
<math>\xrightarrow{\mathrm{Parse}}</math>
<table style="border:none;width:120%;">
<math>\rightsquigarrow</math>
<tr>
<td style="border:none;width:50%;"> ◾ [[Cactus Language]]</td>
<td style="border:none;width:50%;"> ◾ [[Propositions As Types]]</td>
</tr>
<math>\leftrightsquigarrow</math>
<tr>
<td style="border:none;width:50%;"> ◾ [[Futures Of Logical Graphs]]</td>
<td style="border:none;width:50%;"> ◾ [[Propositional Equation Reasoning Systems]]</td>
</tr>
<tr><td style="border:none;"></td></tr>
<tr>
<td style="border:none;width:50%;"> ◾ [[Correspondence Theory Of Truth]]</td>
<td style="border:none;width:50%;"> ◾ [[Pragmatic Theory Of Truth]]</td>
</tr>
<tr><td style="border:none;"></td></tr>
<tr>
<td style="border:none;width:50%;"> ◾ [[Differential Logic • Introduction]]</td>
<td style="border:none;width:50%;"> ◾ [[Introduction to Inquiry Driven Systems]]</td>
</tr>
<tr>
<td style="border:none;width:50%;"> ◾ [[Differential Propositional Calculus]]</td>
<td style="border:none;width:50%;"> ◾ [[Prospects for Inquiry Driven Systems]]</td>
</tr>
<tr>
<td style="border:none;width:50%;"> ◾ [[Differential Logic and Dynamic Systems]]</td>
<td style="border:none;width:50%;"> ◾ [[Inquiry Driven Systems • Inquiry Into Inquiry]]</td>
</tr>
</table>
==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.
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 13.} ~~ \text{Existential Interpretation}</math>
|- style="height:40px; background:#f0f0ff"
| <math>\text{Cactus Graph}</math>
| <math>\text{Cactus Expression}</math>
| <math>\text{Interpretation}</math>
|-
| height="100px" | [[File:Cactus Graph Node Big.jpg|24px]]
| <math>\mathrm{~}</math>
| <math>\mathrm{true}</math>
|-
| height="100px" | [[File:Cactus Graph Spike Big.jpg|24px]]
| <math>\texttt{(} ~ \texttt{)}</math>
| <math>\mathrm{false}</math>
|-
| height="100px" | [[File:Cactus A Big.jpg|20px]]
| <math>a</math>
| <math>a</math>
|-
| height="120px" | [[File:Cactus (A) Big.jpg|20px]]
| <math>\texttt{(} a \texttt{)}</math>
|
<math>\begin{matrix}
\tilde{a}
\\[2pt]
a^\prime
\\[2pt]
\lnot a
\\[2pt]
\mathrm{not}~ a
\end{matrix}</math>
|-
| height="100px" | [[File:Cactus ABC Big.jpg|50px]]
| <math>a~b~c</math>
|
<math>\begin{matrix}
a \land b \land c
\\[6pt]
a ~\mathrm{and}~ b ~\mathrm{and}~ c
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus ((A)(B)(C)) Big.jpg|65px]]
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
|
<math>\begin{matrix}
a \lor b \lor c
\\[6pt]
a ~\mathrm{or}~ b ~\mathrm{or}~ c
\end{matrix}</math>
|-
| height="120px" | [[File:Cactus (A(B)) Big.jpg|60px]]
| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
|
<math>\begin{matrix}
a \Rightarrow b
\\[2pt]
a ~\mathrm{implies}~ b
\\[2pt]
\mathrm{if}~ a ~\mathrm{then}~ b
\\[2pt]
\mathrm{not}~ a ~\mathrm{without}~ b
\end{matrix}</math>
|-
| height="120px" | [[File:Cactus (A,B) Big.jpg|65px]]
| <math>\texttt{(} a, b \texttt{)}</math>
|
<math>\begin{matrix}
a + b
\\[2pt]
a \neq b
\\[2pt]
a ~\mathrm{exclusive~or}~ b
\\[2pt]
a ~\mathrm{not~equal~to}~ b
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus ((A,B)) Big.jpg|65px]]
| <math>\texttt{((} a, b \texttt{))}</math>
|
<math>\begin{matrix}
a = b
\\[2pt]
a \iff b
\\[2pt]
a ~\mathrm{equals}~ b
\\[2pt]
a ~\mathrm{if~and~only~if}~ b
\end{matrix}</math>
|-
| height="120px" | [[File:Cactus (A,B,C) Big.jpg|65px]]
| <math>\texttt{(} a, b, c \texttt{)}</math>
|
<math>\begin{matrix}
\mathrm{just~one~of}
\\
a, b, c
\\
\mathrm{is~false}
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus ((A),(B),(C)) Big.jpg|65px]]
| <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
|
<math>\begin{matrix}
\mathrm{just~one~of}
\\
a, b, c
\\
\mathrm{is~true}
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus (A,(B),(C)) Big.jpg|65px]]
| <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
|
<math>\begin{matrix}
\mathrm{genus}~ a ~\mathrm{of~species}~ b, c
\\[6pt]
\mathrm{partition}~ a ~\mathrm{into}~ b, c
\\[6pt]
\mathrm{pie}~ a ~\mathrm{of~slices}~ b, c
\end{matrix}</math>
|}
===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.
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 14.} ~~ \text{Entitative Interpretation}</math>
|- style="height:40px; background:#f0f0ff"
| <math>\text{Cactus Graph}</math>
| <math>\text{Cactus Expression}</math>
| <math>\text{Interpretation}</math>
|-
| height="100px" | [[File:Cactus Graph Node Big.jpg|24px]]
| <math>\mathrm{~}</math>
| <math>\mathrm{false}</math>
|-
| height="100px" | [[File:Cactus Graph Spike Big.jpg|24px]]
| <math>\texttt{(} ~ \texttt{)}</math>
| <math>\mathrm{true}</math>
|-
| height="100px" | [[File:Cactus A Big.jpg|20px]]
| <math>a</math>
| <math>a</math>
|-
| height="120px" | [[File:Cactus (A) Big.jpg|20px]]
| <math>\texttt{(} a \texttt{)}</math>
|
<math>\begin{matrix}
\tilde{a}
\\[2pt]
a^\prime
\\[2pt]
\lnot a
\\[2pt]
\mathrm{not}~ a
\end{matrix}</math>
|-
| height="100px" | [[File:Cactus ABC Big.jpg|50px]]
| <math>a~b~c</math>
|
<math>\begin{matrix}
a \lor b \lor c
\\[6pt]
a ~\mathrm{or}~ b ~\mathrm{or}~ c
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus ((A)(B)(C)) Big.jpg|65px]]
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
|
<math>\begin{matrix}
a \land b \land c
\\[6pt]
a ~\mathrm{and}~ b ~\mathrm{and}~ c
\end{matrix}</math>
|-
| height="120px" | [[File:Cactus (A)B Big.jpg|35px]]
| <math>\texttt{(} a \texttt{)} b</math>
|
<math>\begin{matrix}
a \Rightarrow b
\\[2pt]
a ~\mathrm{implies}~ b
\\[2pt]
\mathrm{if}~ a ~\mathrm{then}~ b
\\[2pt]
\mathrm{not}~ a, ~\mathrm{or}~ b
\end{matrix}</math>
|-
| height="120px" | [[File:Cactus (A,B) Big.jpg|65px]]
| <math>\texttt{(} a, b \texttt{)}</math>
|
<math>\begin{matrix}
a = b
\\[2pt]
a \iff b
\\[2pt]
a ~\mathrm{equals}~ b
\\[2pt]
a ~\mathrm{if~and~only~if}~ b
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus ((A,B)) Big.jpg|65px]]
| <math>\texttt{((} a, b \texttt{))}</math>
|
<math>\begin{matrix}
a + b
\\[2pt]
a \neq b
\\[2pt]
a ~\mathrm{exclusive~or}~ b
\\[2pt]
a ~\mathrm{not~equal~to}~ b
\end{matrix}</math>
|-
| height="120px" | [[File:Cactus (A,B,C) Big.jpg|65px]]
| <math>\texttt{(} a, b, c \texttt{)}</math>
|
<math>\begin{matrix}
\mathrm{not~just~one~of}
\\
a, b, c
\\
\mathrm{is~true}
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus ((A,B,C)) Big.jpg|65px]]
| <math>\texttt{((} a, b, c \texttt{))}</math>
|
<math>\begin{matrix}
\mathrm{just~one~of}
\\
a, b, c
\\
\mathrm{is~true}
\end{matrix}</math>
|-
| height="200px" | [[File:Cactus (((A),B,C)) Big.jpg|65px]]
| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
|
<math>\begin{matrix}
\mathrm{genus}~ a ~\mathrm{of~species}~ b, c
\\[6pt]
\mathrm{partition}~ a ~\mathrm{into}~ b, c
\\[6pt]
\mathrm{pie}~ a ~\mathrm{of~slices}~ b, c
\end{matrix}</math>
|}
==Logical Graphs==
===Old Versions===
====Example 1====
{| align="center" cellpadding="10"
|
<pre>
o-----------------------------------------------------------o
| |
| o o o o o o |
| \| | | | | |
| o o o o o o o o o |
| \|/ \|/ |/ | |
| @ = @ = @ = @ |
| |
o-----------------------------------------------------------o
| |
| (()())(())() = (())(())() = (())() = ( ) |
| |
o-----------------------------------------------------------o
</pre>
|}
====Example 2====
{| align="center" cellpadding="10"
|
<pre>
o-------------------o-------------------o-------------------o
| Object | Sign | Interpretant |
o-------------------o-------------------o-------------------o
| | | |
| Falsity | "(()())(())()" | "(())(())()" |
| | | |
| Falsity | "(())(())()" | "(())()" |
| | | |
| Falsity | "(())()" | "()" |
| | | |
o-------------------o-------------------o-------------------o
</pre>
|}
====Example 3====
{| align="center" cellpadding="10"
|
<pre>
o-------------------o-------------------o-------------------o
| Object | Sign | Interpretant |
o-------------------o-------------------o-------------------o
| | | |
| Falsity | "(()())(())()" | "(()())(())()" |
| | | |
| Falsity | "(()())(())()" | "(())(())()" |
| | | |
| Falsity | "(()())(())()" | "(())()" |
| | | |
| Falsity | "(()())(())()" | "()" |
| | | |
o-------------------o-------------------o-------------------o
| | | |
| Falsity | "(())(())()" | "(()())(())()" |
| | | |
| Falsity | "(())(())()" | "(())(())()" |
| | | |
| Falsity | "(())(())()" | "(())()" |
| | | |
| Falsity | "(())(())()" | "()" |
| | | |
o-------------------o-------------------o-------------------o
| | | |
| Falsity | "(())()" | "(()())(())()" |
| | | |
| Falsity | "(())()" | "(())(())()" |
| | | |
| Falsity | "(())()" | "(())()" |
| | | |
| Falsity | "(())()" | "()" |
| | | |
o-------------------o-------------------o-------------------o
| | | |
| Falsity | "()" | "(()())(())()" |
| | | |
| Falsity | "()" | "(())(())()" |
| | | |
| Falsity | "()" | "(())()" |
| | | |
| Falsity | "()" | "()" |
| | | |
o-------------------o-------------------o-------------------o
</pre>
|}
====Example 4====
{| align="center" cellpadding="10"
|
<pre>
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
</pre>
|}
====Example 5====
{| align="center" cellpadding="10"
|
<pre>
o-------o-------o-------o-----------o
| a | b | c | (a, b, c) |
o-------o-------o-------o-----------o
| | | | |
| blank | blank | blank | cross |
| | | | |
| blank | blank | cross | blank |
| | | | |
| blank | cross | blank | blank |
| | | | |
| blank | cross | cross | cross |
| | | | |
| cross | blank | blank | blank |
| | | | |
| cross | blank | cross | cross |
| | | | |
| cross | cross | blank | cross |
| | | | |
| cross | cross | cross | cross |
| | | | |
o-------o-------o-------o-----------o
</pre>
|}
====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>
|}
===New Versions===
====Example 1====
…
====Example 2a====
{| align="center" border="1" cellpadding="12" cellspacing="0" width="50%"
|- style="background:ghostwhite; height:40px"
| width="33%" | <math>\text{Object}</math>
| width="33%" | <math>\text{Sign}</math>
| width="33%" | <math>\text{Interpretant}</math>
|-
| <math>\mathrm{Falsity}</math>
| <math>{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}</math>
|-
| <math>\mathrm{Falsity}</math>
| <math>{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}</math>
|-
| <math>\mathrm{Falsity}</math>
| <math>{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}</math>
|}
====Example 2b====
{| align="center" border="1" cellpadding="12" cellspacing="0" width="50%"
|- style="background:ghostwhite; height:40px"
| width="33%" | <math>\text{Object}</math>
| width="33%" | <math>\text{Sign}</math>
| width="33%" | <math>\text{Interpretant}</math>
|-
| valign="bottom" |
<math>\begin{array}{l}
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}</math>
|}
====Example 3====
{| align="center" border="1" cellpadding="12" cellspacing="0" width="50%"
|- style="background:ghostwhite; height:40px"
| width="33%" | <math>\text{Object}</math>
| width="33%" | <math>\text{Sign}</math>
| width="33%" | <math>\text{Interpretant}</math>
|-
| valign="bottom" |
<math>\begin{array}{l}
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}</math>
|-
| valign="bottom" |
<math>\begin{array}{l}
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}</math>
|-
| valign="bottom" |
<math>\begin{array}{l}
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}</math>
|-
| valign="bottom" |
<math>\begin{array}{l}
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}</math>
|}
====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>
|}
====Example 5====
{| 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>
|}
====Example 6====
{| 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{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\end{matrix}</math>
|}