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Directory talk:Jon Awbrey/Papers/Differential Logic : Introduction (view source)

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We may understand the enlarged proposition <math>\mathrm{E}f\!</math> as telling us all the different ways to reach a model of the proposition <math>f\!</math> from each point of the universe <math>X.\!</math>
 
We may understand the enlarged proposition <math>\mathrm{E}f\!</math> as telling us all the different ways to reach a model of the proposition <math>f\!</math> from each point of the universe <math>X.\!</math>
   −
==Logical Cacti==
+
==Propositional Forms on Two Variables==
   −
; Theme One Program &mdash; Logical Cacti
+
To broaden our experience with simple examples, let us examine the sixteen functions of concrete type <math>P \times Q \to \mathbb{B}\!</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.\!</math>  A&nbsp;few Tables are set here that detail the actions of <math>\mathrm{E}\!</math> and <math>\mathrm{D}\!</math> on each of these functions, allowing us to view the results in several different ways.
−
: http://stderr.org/pipermail/inquiry/2005-February/thread.html#2348
  −
: http://stderr.org/pipermail/inquiry/2005-February/002360.html
  −
: http://stderr.org/pipermail/inquiry/2005-February/002361.html
     −
===Original Version===
+
Tables&nbsp;A1 and A2 show two ways of arranging the 16 boolean functions on two variables, giving equivalent expressions for each function in several different systems of notation.
   −
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.
+
<br>
   −
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.
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
−
 
+
|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}\!</math>
−
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.
+
|- style="background:#f0f0ff"
−
 
+
| width="15%" |
−
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 Charles Sanders 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.
+
<p><math>\mathcal{L}_1\!</math></p>
−
 
+
<p><math>\text{Decimal}\!</math></p>
−
Table&nbsp;A 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.
+
| width="15%" |
 +
<p><math>\mathcal{L}_2\!</math></p>
 +
<p><math>\text{Binary}\!</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_3\!</math></p>
 +
<p><math>\text{Vector}\!</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_4\!</math></p>
 +
<p><math>\text{Cactus}\!</math></p>
 +
| width="25%" |
 +
<p><math>\mathcal{L}_5\!</math></p>
 +
<p><math>\text{English}\!</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_6~\!</math></p>
 +
<p><math>\text{Ordinary}\!</math></p>
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>p\colon\!</math>
 +
| <math>1~1~0~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>q\colon\!</math>
 +
| <math>1~0~1~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_0
 +
\\[4pt]
 +
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_3
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_5
 +
\\[4pt]
 +
f_6
 +
\\[4pt]
 +
f_7
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0000}
 +
\\[4pt]
 +
f_{0001}
 +
\\[4pt]
 +
f_{0010}
 +
\\[4pt]
 +
f_{0011}
 +
\\[4pt]
 +
f_{0100}
 +
\\[4pt]
 +
f_{0101}
 +
\\[4pt]
 +
f_{0110}
 +
\\[4pt]
 +
f_{0111}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
0~0~0~0
 +
\\[4pt]
 +
0~0~0~1
 +
\\[4pt]
 +
0~0~1~0
 +
\\[4pt]
 +
0~0~1~1
 +
\\[4pt]
 +
0~1~0~0
 +
\\[4pt]
 +
0~1~0~1
 +
\\[4pt]
 +
0~1~1~0
 +
\\[4pt]
 +
0~1~1~1
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
(p)[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])(q)
 +
\\[4pt]
 +
(p,~q)
 +
\\[4pt]
 +
(p~~q)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{false}
 +
\\[4pt]
 +
\text{neither}~ p ~\text{nor}~ q
 +
\\[4pt]
 +
q ~\text{without}~ p
 +
\\[4pt]
 +
\text{not}~ p
 +
\\[4pt]
 +
p ~\text{without}~ q
 +
\\[4pt]
 +
\text{not}~ q
 +
\\[4pt]
 +
p ~\text{not equal to}~ q
 +
\\[4pt]
 +
\text{not both}~ p ~\text{and}~ q
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
0
 +
\\[4pt]
 +
\lnot p \land \lnot q
 +
\\[4pt]
 +
\lnot p \land q
 +
\\[4pt]
 +
\lnot p
 +
\\[4pt]
 +
p \land \lnot q
 +
\\[4pt]
 +
\lnot q
 +
\\[4pt]
 +
p \ne q
 +
\\[4pt]
 +
\lnot p \lor \lnot q
 +
\end{matrix}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_8
 +
\\[4pt]
 +
f_9
 +
\\[4pt]
 +
f_{10}
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{12}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\\[4pt]
 +
f_{15}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{1000}
 +
\\[4pt]
 +
f_{1001}
 +
\\[4pt]
 +
f_{1010}
 +
\\[4pt]
 +
f_{1011}
 +
\\[4pt]
 +
f_{1100}
 +
\\[4pt]
 +
f_{1101}
 +
\\[4pt]
 +
f_{1110}
 +
\\[4pt]
 +
f_{1111}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
1~0~0~0
 +
\\[4pt]
 +
1~0~0~1
 +
\\[4pt]
 +
1~0~1~0
 +
\\[4pt]
 +
1~0~1~1
 +
\\[4pt]
 +
1~1~0~0
 +
\\[4pt]
 +
1~1~0~1
 +
\\[4pt]
 +
1~1~1~0
 +
\\[4pt]
 +
1~1~1~1
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~~p~~q~~
 +
\\[4pt]
 +
((p,~q))
 +
\\[4pt]
 +
18:32, 29 November 2015 (UTC)q~~
 +
\\[4pt]
 +
~(p~(q))
 +
\\[4pt]
 +
~~p18:32, 29 November 2015 (UTC)
 +
\\[4pt]
 +
((p)~q)~
 +
\\[4pt]
 +
((p)(q))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
p ~\text{and}~ q
 +
\\[4pt]
 +
p ~\text{equal to}~ q
 +
\\[4pt]
 +
q
 +
\\[4pt]
 +
\text{not}~ p ~\text{without}~ q
 +
\\[4pt]
 +
p
 +
\\[4pt]
 +
\text{not}~ q ~\text{without}~ p
 +
\\[4pt]
 +
p ~\text{or}~ q
 +
\\[4pt]
 +
\text{true}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
p \land q
 +
\\[4pt]
 +
p = q
 +
\\[4pt]
 +
q
 +
\\[4pt]
 +
p \Rightarrow q
 +
\\[4pt]
 +
p
 +
\\[4pt]
 +
p \Leftarrow q
 +
\\[4pt]
 +
p \lor q
 +
\\[4pt]
 +
1
 +
\end{matrix}\!</math>
 +
|}
    
<br>
 
<br>
   −
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
−
|+ <math>\text{Table A.}~~\text{Existential Interpretation}</math>
+
|+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}\!</math>
 
|- style="background:#f0f0ff"
 
|- style="background:#f0f0ff"
−
| <math>\text{Cactus Graph}\!</math>
+
| width="15%" |
−
| <math>\text{Cactus Expression}\!</math>
+
<p><math>\mathcal{L}_1\!</math></p>
−
| <math>\text{Interpretation}\!</math>
+
<p><math>\text{Decimal}\!</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_2\!</math></p>
 +
<p><math>\text{Binary}\!</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_3\!</math></p>
 +
<p><math>\text{Vector}\!</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_4\!</math></p>
 +
<p><math>\text{Cactus}\!</math></p>
 +
| width="25%" |
 +
<p><math>\mathcal{L}_5\!</math></p>
 +
<p><math>\text{English}\!</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_6~\!</math></p>
 +
<p><math>\text{Ordinary}\!</math></p>
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>p\colon\!</math>
 +
| <math>1~1~0~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>q\colon\!</math>
 +
| <math>1~0~1~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 
|-
 
|-
−
| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
+
| <math>f_0\!</math>
−
| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
+
| <math>f_{0000}\!</math>
−
| <math>\operatorname{true}.</math>
+
| <math>0~0~0~0\!</math>
 +
| <math>(~)\!</math>
 +
| <math>\text{false}\!</math>
 +
| <math>0\!</math>
 
|-
 
|-
−
| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
+
|
−
| <math>\texttt{(~)}</math>
+
<math>\begin{matrix}
−
| <math>\operatorname{false}.</math>
+
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0001}
 +
\\[4pt]
 +
f_{0010}
 +
\\[4pt]
 +
f_{0100}
 +
\\[4pt]
 +
f_{1000}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
0~0~0~1
 +
\\[4pt]
 +
0~0~1~0
 +
\\[4pt]
 +
0~1~0~0
 +
\\[4pt]
 +
1~0~0~0
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{neither}~ p ~\text{nor}~ q
 +
\\[4pt]
 +
q ~\text{without}~ p
 +
\\[4pt]
 +
p ~\text{without}~ q
 +
\\[4pt]
 +
p ~\text{and}~ q
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot p \land \lnot q
 +
\\[4pt]
 +
\lnot p \land q
 +
\\[4pt]
 +
p \land \lnot q
 +
\\[4pt]
 +
p \land q
 +
\end{matrix}\!</math>
 
|-
 
|-
−
| height="100px" | [[Image:Cactus A Big.jpg|20px]]
+
|
−
| <math>a\!</math>
+
<math>\begin{matrix}
−
| <math>a.\!</math>
+
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0011}
 +
\\[4pt]
 +
f_{1100}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
0~0~1~1
 +
\\[4pt]
 +
1~1~0~0
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{not}~ p
 +
\\[4pt]
 +
p
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot p
 +
\\[4pt]
 +
p
 +
\end{matrix}\!</math>
 
|-
 
|-
−
| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
  −
| <math>\texttt{(} a \texttt{)}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\tilde{a}
+
f_6
−
\\[2pt]
+
\\[4pt]
−
a^\prime
+
f_9
−
\\[2pt]
+
\end{matrix}\!</math>
−
\lnot a
+
|
−
\\[2pt]
+
<math>\begin{matrix}
−
\operatorname{not}~ a.
+
f_{0110}
−
\end{matrix}</math>
+
\\[4pt]
 +
f_{1001}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
0~1~1~0
 +
\\[4pt]
 +
1~0~0~1
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
p ~\text{not equal to}~ q
 +
\\[4pt]
 +
p ~\text{equal to}~ q
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
p \ne q
 +
\\[4pt]
 +
p = q
 +
\end{matrix}\!</math>
 
|-
 
|-
−
| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
  −
| <math>a~b~c</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
a \land b \land c
+
f_5
−
\\[6pt]
+
\\[4pt]
−
a ~\operatorname{and}~ b ~\operatorname{and}~ c.
+
f_{10}
−
\end{matrix}</math>
+
\end{matrix}\!</math>
−
|-
+
|
−
| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
+
<math>\begin{matrix}
−
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
+
f_{0101}
 +
\\[4pt]
 +
f_{1010}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
0~1~0~1
 +
\\[4pt]
 +
1~0~1~0
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{not}~ q
 +
\\[4pt]
 +
q
 +
\end{matrix}\!</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
a \lor b \lor c
+
\lnot q
−
\\[6pt]
+
\\[4pt]
−
a ~\operatorname{or}~ b ~\operatorname{or}~ c.
+
q
−
\end{matrix}</math>
+
\end{matrix}\!</math>
 
|-
 
|-
−
| height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]]
  −
| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
a \Rightarrow b
+
f_7
−
\\[2pt]
+
\\[4pt]
−
a ~\operatorname{implies}~ b.
+
f_{11}
−
\\[2pt]
+
\\[4pt]
−
\operatorname{if}~ a ~\operatorname{then}~ b.
+
f_{13}
−
\\[2pt]
+
\\[4pt]
−
\operatorname{not}~ a ~\operatorname{without}~ b.
+
f_{14}
−
\end{matrix}</math>
+
\end{matrix}\!</math>
−
|-
  −
| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
  −
| <math>\texttt{(} a \texttt{,} b \texttt{)}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
a + b
+
f_{0111}
−
\\[2pt]
+
\\[4pt]
−
a \neq b
+
f_{1011}
−
\\[2pt]
+
\\[4pt]
−
a ~\operatorname{exclusive-or}~ b.
+
f_{1101}
−
\\[2pt]
+
\\[4pt]
−
a ~\operatorname{not~equal~to}~ b.
+
f_{1110}
−
\end{matrix}</math>
+
\end{matrix}\!</math>
−
|-
  −
| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
  −
| <math>\texttt{((} a \texttt{,} b \texttt{))}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
a = b
+
0~1~1~1
−
\\[2pt]
+
\\[4pt]
−
a \iff b
+
1~0~1~1
−
\\[2pt]
+
\\[4pt]
−
a ~\operatorname{equals}~ b.
+
1~1~0~1
−
\\[2pt]
+
\\[4pt]
−
a ~\operatorname{if~and~only~if}~ b.
+
1~1~1~0
−
\end{matrix}</math>
+
\end{matrix}\!</math>
−
|-
  −
| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
  −
| <math>\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\operatorname{just~one~of}
+
~(p~~q)~
−
\\
+
\\[4pt]
−
a, b, c
+
~(p~(q))
−
\\
+
\\[4pt]
−
\operatorname{is~false}.
+
((p)~q)~
−
\end{matrix}</math>
+
\\[4pt]
−
|-
+
((p)(q))
−
| height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]]
+
\end{matrix}\!</math>
−
| <math>\texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\operatorname{just~one~of}
+
\text{not both}~ p ~\text{and}~ q
−
\\
+
\\[4pt]
−
a, b, c
+
\text{not}~ p ~\text{without}~ q
−
\\
+
\\[4pt]
−
\operatorname{is~true}.
+
\text{not}~ q ~\text{without}~ p
−
\end{matrix}</math>
+
\\[4pt]
−
|-
+
p ~\text{or}~ q
−
| height="160px" | [[Image:Cactus (A,(B),(C)) Big.jpg|65px]]
+
\end{matrix}\!</math>
−
| <math>\texttt{(} a \texttt{,(} b \texttt{),(} c \texttt{))}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\operatorname{genus}~ a ~\operatorname{of~species}~ b, c.
+
\lnot p \lor \lnot q
−
\\[6pt]
+
\\[4pt]
−
\operatorname{partition}~ a ~\operatorname{into}~ b, c.
+
p \Rightarrow q
−
\\[6pt]
+
\\[4pt]
−
\operatorname{pie}~ a ~\operatorname{of~slices}~ b, c.
+
p \Leftarrow q
−
\end{matrix}</math>
+
\\[4pt]
 +
p \lor q
 +
\end{matrix}\!</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>f_{1111}\!</math>
 +
| <math>1~1~1~1\!</math>
 +
| <math>((~))\!</math>
 +
| <math>\text{true}\!</math>
 +
| <math>1\!</math>
 
|}
 
|}
    
<br>
 
<br>
   −
Table&nbsp;B illustrates the entitative interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.
+
===Transforms Expanded over Differential Features===
 +
 
 +
The next four Tables expand the expressions of <math>\mathrm{E}f\!</math> and <math>\mathrm{D}f~\!</math> in two different ways, for each of the sixteen functions.  Notice that the functions are given in a different order, partitioned into seven natural classes by a group action.
    
<br>
 
<br>
   −
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
−
|+ <math>\text{Table B.}~~\text{Entitative Interpretation}</math>
+
|+ <math>\text{Table A3.}~~\mathrm{E}f ~\text{Expanded over Differential Features}~ \{ \mathrm{d}p, \mathrm{d}q \}\!</math>
 
|- style="background:#f0f0ff"
 
|- style="background:#f0f0ff"
−
| <math>\text{Cactus Graph}\!</math>
+
| width="10%" | &nbsp;
−
| <math>\text{Cactus Expression}\!</math>
+
| width="18%" | <math>f\!</math>
−
| <math>\text{Interpretation}\!</math>
+
| width="18%" |
 +
<p><math>\mathrm{T}_{11} f\!</math></p>
 +
<p><math>\mathrm{E}f|_{\mathrm{d}p~\mathrm{d}q}\!</math></p>
 +
| width="18%" |
 +
<p><math>\mathrm{T}_{10} f\!</math></p>
 +
<p><math>\mathrm{E}f|_{\mathrm{d}p(\mathrm{d}q)}\!</math></p>
 +
| width="18%" |
 +
<p><math>\mathrm{T}_{01} f\!</math></p>
 +
<p><math>\mathrm{E}f|_{(\mathrm{d}p)\mathrm{d}q}\!</math></p>
 +
| width="18%" |
 +
<p><math>\mathrm{T}_{00} f\!</math></p>
 +
<p><math>\mathrm{E}f|_{(\mathrm{d}p)(\mathrm{d}q)}\!</math></p>
 
|-
 
|-
−
| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
+
| <math>f_0\!</math>
−
| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
+
| <math>(~)\!</math>
−
| <math>\operatorname{false}.</math>
+
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 
|-
 
|-
−
| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
+
|
−
| <math>\texttt{(~)}</math>
+
<math>\begin{matrix}
−
| <math>\operatorname{true}.</math>
+
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
(p)(q)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\\[4pt]
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)~q~
 +
\\[4pt]
 +
(p)(q)
 +
\\[4pt]
 +
~p~~q~
 +
\\[4pt]
 +
~p~(q)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}\!</math>
 
|-
 
|-
−
| height="100px" | [[Image:Cactus A Big.jpg|20px]]
+
|
−
| <math>a\!</math>
+
<math>\begin{matrix}
−
| <math>a.\!</math>
+
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~
 +
\\[4pt]
 +
(p)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~
 +
\\[4pt]
 +
(p)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}\!</math>
 
|-
 
|-
−
| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
  −
| <math>\texttt{(} a \texttt{)}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\tilde{a}
+
f_6
−
\\[2pt]
+
\\[4pt]
−
a^\prime
+
f_9
−
\\[2pt]
+
\end{matrix}\!</math>
−
\lnot a
+
|
−
\\[2pt]
+
<math>\begin{matrix}
−
\operatorname{not}~ a.
+
~(p,~q)~
−
\end{matrix}</math>
+
\\[4pt]
 +
((p,~q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((p,~q))
 +
\\[4pt]
 +
~(p,~q)~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((p,~q))
 +
\\[4pt]
 +
~(p,~q)~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}\!</math>
 
|-
 
|-
−
| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
  −
| <math>a~b~c</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
a \lor b \lor c
+
f_5
−
\\[6pt]
+
\\[4pt]
−
a ~\operatorname{or}~ b ~\operatorname{or}~ c.
+
f_{10}
−
\end{matrix}</math>
+
\end{matrix}\!</math>
−
|-
+
|
−
| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
+
<math>\begin{matrix}
−
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
+
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~q~
 +
\\[4pt]
 +
(q)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~q~
 +
\\[4pt]
 +
(q)
 +
\end{matrix}\!</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
a \land b \land c
+
(q)
−
\\[6pt]
+
\\[4pt]
−
a ~\operatorname{and}~ b ~\operatorname{and}~ c.
+
~q~
−
\end{matrix}</math>
+
\end{matrix}\!</math>
 
|-
 
|-
−
| height="120px" | [[Image:Cactus (A)B Big.jpg|35px]]
  −
| <math>\texttt{(} a \texttt{)} b</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
a \Rightarrow b
+
f_7
−
\\[2pt]
+
\\[4pt]
−
a ~\operatorname{implies}~ b.
+
f_{11}
−
\\[2pt]
+
\\[4pt]
−
\operatorname{if}~ a ~\operatorname{then}~ b.
+
f_{13}
−
\\[2pt]
+
\\[4pt]
−
\operatorname{not}~ a, ~\operatorname{or}~ b.
+
f_{14}
−
\end{matrix}</math>
+
\end{matrix}\!</math>
−
|-
  −
| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
  −
| <math>\texttt{(} a \texttt{,} b \texttt{)}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
a = b
+
(~p~~q~)
−
\\[2pt]
+
\\[4pt]
−
a \iff b
+
(~p~(q))
−
\\[2pt]
+
\\[4pt]
−
a ~\operatorname{equals}~ b.
+
((p)~q~)
−
\\[2pt]
+
\\[4pt]
−
a ~\operatorname{if~and~only~if}~ b.
+
((p)(q))
−
\end{matrix}</math>
+
\end{matrix}\!</math>
−
|-
  −
| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
  −
| <math>\texttt{((} a \texttt{,} b \texttt{))}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
a + b
+
((p)(q))
−
\\[2pt]
+
\\[4pt]
−
a \neq b
+
((p)~q~)
−
\\[2pt]
+
\\[4pt]
−
a ~\operatorname{exclusive-or}~ b.
+
(~p~(q))
−
\\[2pt]
+
\\[4pt]
−
a ~\operatorname{not~equal~to}~ b.
+
(~p~~q~)
−
\end{matrix}</math>
+
\end{matrix}\!</math>
−
|-
  −
| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
  −
| <math>\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\operatorname{not~just~one~of}
+
((p)~q~)
−
\\
+
\\[4pt]
−
a, b, c
+
((p)(q))
−
\\
+
\\[4pt]
−
\operatorname{is~true}.
+
(~p~~q~)
−
\end{matrix}</math>
+
\\[4pt]
−
|-
+
(~p~(q))
−
| height="160px" | [[Image:Cactus ((A,B,C)) Big.jpg|65px]]
+
\end{matrix}\!</math>
−
| <math>\texttt{((} a \texttt{,} b \texttt{,} c \texttt{))}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\operatorname{just~one~of}
+
(~p~(q))
−
\\
+
\\[4pt]
−
a, b, c
+
(~p~~q~)
−
\\
+
\\[4pt]
−
\operatorname{is~true}.
+
((p)(q))
−
\end{matrix}</math>
+
\\[4pt]
−
|-
+
((p)~q~)
−
| height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|65px]]
+
\end{matrix}\!</math>
−
| <math>\texttt{(((} a \texttt{),} b \texttt{,} c \texttt{))}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\operatorname{genus}~ a ~\operatorname{of~species}~ b, c.
+
(~p~~q~)
−
\\[6pt]
+
\\[4pt]
−
\operatorname{partition}~ a ~\operatorname{into}~ b, c.
+
(~p~(q))
−
\\[6pt]
+
\\[4pt]
−
\operatorname{pie}~ a ~\operatorname{of~slices}~ b, c.
+
((p)~q~)
−
\end{matrix}</math>
+
\\[4pt]
 +
((p)(q))
 +
\end{matrix}\!</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 +
|- style="background:#f0f0ff"
 +
| colspan="2" | <math>\text{Fixed Point Total}\!</math>
 +
| <math>4\!</math>
 +
| <math>4\!</math>
 +
| <math>4\!</math>
 +
| <math>16\!</math>
 
|}
 
|}
    
<br>
 
<br>
   −
For the time being, the main things to take away from Tables&nbsp;A and B are the ideas that the compositional structure of cactus graphs and expressions can be articulated in terms of two different kinds of connective operations, and that there are two distinct ways of mapping this compositional structure into the compositional structure of propositional sentences, say, in English:
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
−
 
+
|+ <math>\text{Table A4.}~~\mathrm{D}f ~\text{Expanded over Differential Features}~ \{ \mathrm{d}p, \mathrm{d}q \}\!</math>
−
{| align="center" cellpadding="6" width="90%"
+
|- style="background:#f0f0ff"
−
| valign="top" | 1.
+
| width="10%" | &nbsp;
−
| The ''node connective'' joins a number of component cacti <math>C_1, \ldots, C_k</math> at a node:
+
| width="18%" | <math>f\!</math>
 +
| width="18%" |
 +
<math>\mathrm{D}f|_{\mathrm{d}p~\mathrm{d}q}\!</math>
 +
| width="18%" |
 +
<math>\mathrm{D}f|_{\mathrm{d}p(\mathrm{d}q)}\!</math>
 +
| width="18%" |
 +
<math>\mathrm{D}f|_{(\mathrm{d}p)\mathrm{d}q}\!</math>
 +
| width="18%" |
 +
<math>\mathrm{D}f|_{(\mathrm{d}p)(\mathrm{d}q)}\!</math>
 +
|-
 +
| <math>f_0\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((p,~q))
 +
\\[4pt]
 +
~(p,~q)~
 +
\\[4pt]
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\\[4pt]
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\\[4pt]
 +
~p~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}\!</math>
 
|-
 
|-
−
| &nbsp;
   
|
 
|
−
<pre>
+
<math>\begin{matrix}
−
    C_1 ... C_k
+
f_3
−
        @
+
\\[4pt]
−
</pre>
+
f_{12}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}~\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}~\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}\!</math>
 
|-
 
|-
−
| valign="top" | 2.
+
|
−
| The ''lobe connective'' joins a number of component cacti <math>C_1, \ldots, C_k</math> to a lobe:
+
<math>\begin{matrix}
 +
f_6
 +
\\[4pt]
 +
f_9
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}~\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}~\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}\!</math>
 
|-
 
|-
−
| &nbsp;
   
|
 
|
−
<pre>
+
<math>\begin{matrix}
−
    C_1 C_2  C_k
+
f_5
−
    o---o-...-o
+
\\[4pt]
−
      \       /
+
f_{10}
−
      \     /
+
\end{matrix}\!</math>
−
        \   /
+
|
−
        \ /
+
<math>\begin{matrix}
−
          @
+
(q)
−
</pre>
+
\\[4pt]
−
|}
+
~q~
−
 
+
\end{matrix}\!</math>
−
Table&nbsp;15 summarizes the existential and entitative interpretations of the primitive cactus structures, in effect, the graphical constants and connectives.
+
|
−
 
+
<math>\begin{matrix}
−
{| align="center" cellpadding="6" style="text-align:center; width:90%"
+
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}~\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}~\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}\!</math>
 +
|-
 
|
 
|
−
<pre>
+
<math>\begin{matrix}
−
Table 15.  Existential & Entitative Interpretations of Cactus Structures
+
f_7
−
o-----------------o-----------------o-----------------o-----------------o
+
\\[4pt]
−
|  Cactus Graph  |  Cactus String  |  Existential    |  Entitative    |
+
f_{11}
−
|                |                | Interpretation  | Interpretation  |
+
\\[4pt]
−
o-----------------o-----------------o-----------------o-----------------o
+
f_{13}
−
|                |                |                |                |
+
\\[4pt]
−
|        @        |      " "      |      true      |      false      |
+
f_{14}
−
|                |                |                |                |
+
\end{matrix}\!</math>
−
o-----------------o-----------------o-----------------o-----------------o
  −
|                |                |                |                |
  −
|        o        |                |                |                |
  −
|        |        |                |                |                |
  −
|        @        |      ( )      |      false      |      true      |
  −
|                |                |                |                |
  −
o-----------------o-----------------o-----------------o-----------------o
  −
|                |                |                |                |
  −
|  C_1 ... C_k  |                |                |                |
  −
|        @        |  C_1 ... C_k  | C_1 & ... & C_k | C_1 v ... v C_k |
  −
|                |                |                |                |
  −
o-----------------o-----------------o-----------------o-----------------o
  −
|                |                |                |                |
  −
|  C_1 C_2  C_k  |                |  Just one      |  Not just one  |
  −
|  o---o-...-o  |                |                |                |
  −
|    \       /    |                |  of the C_j,    |  of the C_j,    |
  −
|    \    /    |                |                |                |
  −
|      \   /      |                |  j = 1 to k,    |  j = 1 to k,    |
  −
|      \ /      |                |                |                |
  −
|        @        | (C_1, ..., C_k) |  is not true.  |  is true.      |
  −
|                |                |                |                |
  −
o-----------------o-----------------o-----------------o-----------------o
  −
</pre>
  −
|}
  −
 
  −
It is possible to specify ''abstract rules of equivalence'' (AROEs) between cacti, rules for transforming one cactus into another that are ''formal'' in the sense of being indifferent to the above choices for logical or semantic interpretations, and that partition the set of cacti into formal equivalence classes.
  −
 
  −
A ''reduction'' is an equivalence transformation that is applied in the direction of decreasing graphical complexity.
  −
 
  −
A ''basic reduction'' is a reduction that applies to one of the two families of basic connectives.
  −
 
  −
Table&nbsp;16 schematizes the two types of basic reductions in a purely formal, interpretation-independent fashion.
  −
 
  −
{| align="center" cellpadding="6" style="text-align:center; width:90%"
   
|
 
|
−
<pre>
+
<math>\begin{matrix}
−
Table 16.  Basic Reductions
+
~(p~~q)~
−
o---------------------------------------o
+
\\[4pt]
−
|                                      |
+
~(p~(q))
−
|    C_1 ... C_k                        |
+
\\[4pt]
−
|        @        =        @        |
+
((p)~q)~
−
|                                      |
+
\\[4pt]
−
|    if and only if                    |
+
((p)(q))
−
|                                      |
+
\end{matrix}\!</math>
−
|    C_j = @ for all j = 1 to k        |
  −
|                                      |
  −
o---------------------------------------o
  −
|                                      |
  −
|  C_1 C_2  C_k                      |
  −
|    o---o-...-o                        |
  −
|    \       /                        |
  −
|      \    /                          |
  −
|      \   /                          |
  −
|        \ /                            |
  −
|        @        =        @        |
  −
|                                      |
  −
|  if and only if                      |
  −
|                                      |
  −
|        o                            |
  −
|        |                            |
  −
|  C_j = @ for exactly one j in [1, k] |
  −
|                                      |
  −
o---------------------------------------o
  −
</pre>
  −
|}
  −
 
  −
The careful reader will have noticed that we have begun to use graphical paints like "a", "b", "c" and schematic proxies like "C_1", "C_j", "C_k" in a variety of novel and unjustified ways.
  −
 
  −
The careful writer would have already introduced a whole bevy of technical concepts and proved a whole crew of formal theorems to justify their use before contemplating this stage of development, but I have been hurrying to proceed with the informal exposition, and this expedition must leave steps to the reader's imagination.
  −
 
  −
Of course I mean the ''active imagination''.  So let me assist the prospective exercise with a few hints of what it would take to guarantee that these practices make sense.
  −
 
  −
===Partial Rewrites===
  −
 
  −
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.
  −
 
  −
Even though I do most of my thinking in the existential interpretation, I will continue to speak of these forms as ''logical graphs'', because I think it is an important fact about them that the formal validity of the axioms and theorems is not dependent on the choice between the entitative and the existential interpretations.
  −
 
  −
The first extension is the ''reflective extension of logical graphs'' (RefLog).  It is obtained by generalizing the negation operator "<math>\texttt{(~)}</math>" in a certain way, calling "<math>\texttt{(~)}</math>" the ''controlled'', ''moderated'', or ''reflective'' negation operator of order 1, then adding another such operator for each finite <math>k = 2, 3, \ldots .</math>
  −
 
  −
In sum, these operators are symbolized by bracketed argument lists as follows:  "<math>\texttt{(~)}</math>", "<math>\texttt{(~,~)}</math>", "<math>\texttt{(~,~,~)}</math>", &hellip;, where the number of slots is the order of the reflective negation operator in question.
  −
             
  −
The cactus graph and the cactus expression shown here are both described as a ''spike''.
  −
 
  −
{| align="center" cellpadding="6" style="text-align:center; width:90%"
   
|
 
|
−
<pre>
+
<math>\begin{matrix}
−
o---------------------------------------o
+
((p,~q))
−
|                                      |
+
\\[4pt]
−
|                  o                  |
+
~(p,~q)~
−
|                  |                  |
+
\\[4pt]
−
|                  @                  |
+
~(p,~q)~
−
|                                      |
+
\\[4pt]
−
o---------------------------------------o
+
((p,~q))
−
|                  ( )                 |
+
\end{matrix}\!</math>
−
o---------------------------------------o
  −
</pre>
  −
|}
  −
 
  −
The rule of reduction for a lobe is:
  −
 
  −
{| align="center" cellpadding="6" style="text-align:center; width:90%"
   
|
 
|
−
<pre>
+
<math>\begin{matrix}
−
o---------------------------------------o
+
~q~
−
|                                      |
+
\\[4pt]
−
|  x_1  x_2  ...  x_k                |
+
(q)
−
|  o-----o--- ... ---o                |
+
\\[4pt]
−
|    \               /                  |
+
~q~
−
|    \            /                  |
+
\\[4pt]
−
|      \           /                    |
+
(q)
−
|      \         /                    |
+
\end{matrix}\!</math>
−
|        \      /                      |
  −
|        \    /                      |
  −
|          \   /                        |
  −
|          \ /                        |
  −
|            @      =      @            |
  −
|                                      |
  −
o---------------------------------------o
  −
</pre>
  −
|}
  −
 
  −
if and only if exactly one of the <math>x_j\!</math> is a spike.
  −
 
  −
In Ref Log, an expression of the form <math>\texttt{((}~ e_1 ~\texttt{),(}~ e_2 ~\texttt{),(}~ \ldots ~\texttt{),(}~ e_k ~\texttt{))}</math>
  −
expresses the fact that ''exactly one of the <math>e_j\!</math> is true''.  Expressions of this form are called ''universal partition'' expressions, and
  −
they parse into a type of graph called a ''painted and rooted cactus'' (PARC):
  −
 
  −
{| align="center" cellpadding="6" style="text-align:center; width:90%"
   
|
 
|
−
<pre>
+
<math>\begin{matrix}
−
o---------------------------------------o
+
~p~
−
|                                      |
+
\\[4pt]
−
|  e_1  e_2  ...  e_k                |
+
~p~
−
|  o    o          o                |
+
\\[4pt]
−
|  |    |          |                |
+
(p)
−
|  o-----o--- ... ---o                |
+
\\[4pt]
−
|    \               /                  |
+
(p)
−
|    \             /                  |
+
\end{matrix}\!</math>
−
|      \          /                    |
  −
|      \         /                    |
  −
|        \       /                      |
  −
|        \    /                      |
  −
|          \   /                        |
  −
|          \ /                        |
  −
|            @                          |
  −
|                                      |
  −
o---------------------------------------o
  −
</pre>
  −
|}
  −
 
  −
{| align="center" cellpadding="6" style="text-align:center; width:90%"
   
|
 
|
−
<pre>
+
<math>\begin{matrix}
−
o---------------------------------------o
+
(~)
−
|                                      |
+
\\[4pt]
−
| ( x1, x2, ..., xk ) =  [blank]       |
+
(~)
−
|                                       |
+
\\[4pt]
−
| iff                                  |
+
(~)
−
|                                       |
+
\\[4pt]
−
| Just one of the arguments            |
+
(~)
−
| x1, x2, ..., xk  =  ()               |
+
\end{matrix}\!</math>
−
|                                       |
+
|-
−
o---------------------------------------o
+
| <math>f_{15}\!</math>
−
</pre>
+
| <math>((~))\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 
|}
 
|}
   −
The interpretation of these operators, read as assertions about the values of their listed arguments, is as follows:
+
<br>
   −
{| align="center" cellpadding="6" width="90%"
+
===Transforms Expanded over Ordinary Features===
−
| Existential Interpretation:
  −
| Just one of the k argument is false.
  −
|-
  −
| Entitative  Interpretation:
  −
| Not just one of the k arguments is true.
  −
|}
     −
===Tables===
+
<br>
   −
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
−
|+ <math>\text{Table 1.}~~\text{Syntax and Semantics of a Calculus for Propositional Logic}</math>
+
|+ <math>\text{Table A5.}~~\mathrm{E}f ~\text{Expanded over Ordinary Features}~ \{ p, q \}\!</math>
 
|- style="background:#f0f0ff"
 
|- style="background:#f0f0ff"
−
| <math>\text{Graph}\!</math>
+
| width="10%" | &nbsp;
−
| <math>\text{Expression}\!</math>
+
| width="18%" | <math>f\!</math>
−
| <math>\text{Interpretation}\!</math>
+
| width="18%" | <math>\mathrm{E}f|_{pq}\!</math>
−
| <math>\text{Other Notations}\!</math>
+
| width="18%" | <math>\mathrm{E}f|_{p(q)}\!</math>
 +
| width="18%" | <math>\mathrm{E}f|_{(p)q}\!</math>
 +
| width="18%" | <math>\mathrm{E}f|_{(p)(q)}\!</math>
 
|-
 
|-
−
| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
+
| <math>f_0\!</math>
−
| <math>~</math>
+
| <math>(~)\!</math>
−
| <math>\operatorname{true}</math>
+
| <math>(~)\!</math>
−
| <math>1\!</math>
+
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 
|-
 
|-
−
| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
+
|
−
| <math>\texttt{(~)}</math>
+
<math>\begin{matrix}
−
| <math>\operatorname{false}</math>
+
f_1
−
| <math>0\!</math>
+
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~\mathrm{d}p~~\mathrm{d}q~
 +
\\[4pt]
 +
~\mathrm{d}p~(\mathrm{d}q)
 +
\\[4pt]
 +
(\mathrm{d}p)~\mathrm{d}q~
 +
\\[4pt]
 +
(\mathrm{d}p)(\mathrm{d}q)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~\mathrm{d}p~(\mathrm{d}q)
 +
\\[4pt]
 +
~\mathrm{d}p~~\mathrm{d}q~
 +
\\[4pt]
 +
(\mathrm{d}p)(\mathrm{d}q)
 +
\\[4pt]
 +
(\mathrm{d}p)~\mathrm{d}q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(\mathrm{d}p)~\mathrm{d}q~
 +
\\[4pt]
 +
(\mathrm{d}p)(\mathrm{d}q)
 +
\\[4pt]
 +
~\mathrm{d}p~~\mathrm{d}q~
 +
\\[4pt]
 +
~\mathrm{d}p~(\mathrm{d}q)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(\mathrm{d}p)(\mathrm{d}q)
 +
\\[4pt]
 +
(\mathrm{d}p)~\mathrm{d}q~
 +
\\[4pt]
 +
~\mathrm{d}p~(\mathrm{d}q)
 +
\\[4pt]
 +
~\mathrm{d}p~~\mathrm{d}q~
 +
\end{matrix}\!</math>
 
|-
 
|-
−
| height="100px" | [[Image:Cactus A Big.jpg|20px]]
+
|
−
| <math>a\!</math>
+
<math>\begin{matrix}
−
| <math>a\!</math>
+
f_3
−
| <math>a\!</math>
+
\\[4pt]
 +
f_{12}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~\mathrm{d}p~
 +
\\[4pt]
 +
(\mathrm{d}p)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~\mathrm{d}p~
 +
\\[4pt]
 +
(\mathrm{d}p)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(\mathrm{d}p)
 +
\\[4pt]
 +
~\mathrm{d}p~
 +
\end{matrix}~\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(\mathrm{d}p)
 +
\\[4pt]
 +
~\mathrm{d}p~
 +
\end{matrix}~\!</math>
 
|-
 
|-
−
| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
  −
| <math>\texttt{(} a \texttt{)}</math>
  −
| <math>\operatorname{not}~ a</math>
  −
| <math>\lnot a \quad \bar{a} \quad \tilde{a} \quad a^\prime</math>
  −
|-
  −
| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
  −
| <math>a ~ b ~ c</math>
  −
| <math>a ~\operatorname{and}~ b ~\operatorname{and}~ c</math>
  −
| <math>a \land b \land c</math>
  −
|-
  −
| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
  −
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
  −
| <math>a ~\operatorname{or}~ b ~\operatorname{or}~ c</math>
  −
| <math>a \lor b \lor c</math>
  −
|-
  −
| height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]]
  −
| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
a ~\operatorname{implies}~ b
+
f_6
−
\\[6pt]
+
\\[4pt]
−
\operatorname{if}~ a ~\operatorname{then}~ b
+
f_9
−
\end{matrix}</math>
+
\end{matrix}\!</math>
−
| <math>a \Rightarrow b</math>
+
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~(\mathrm{d}p,~\mathrm{d}q)~
 +
\\[4pt]
 +
((\mathrm{d}p,~\mathrm{d}q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((\mathrm{d}p,~\mathrm{d}q))
 +
\\[4pt]
 +
~(\mathrm{d}p,~\mathrm{d}q)~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((\mathrm{d}p,~\mathrm{d}q))
 +
\\[4pt]
 +
~(\mathrm{d}p,~\mathrm{d}q)~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~(\mathrm{d}p,~\mathrm{d}q)~
 +
\\[4pt]
 +
((\mathrm{d}p,~\mathrm{d}q))
 +
\end{matrix}\!</math>
 
|-
 
|-
−
| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
  −
| <math>\texttt{(} a \texttt{,} b \texttt{)}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
a ~\operatorname{not~equal~to}~ b
+
f_5
−
\\[6pt]
+
\\[4pt]
−
a ~\operatorname{exclusive~or}~ b
+
f_{10}
−
\end{matrix}</math>
+
\end{matrix}\!</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
a \neq b
+
(q)
−
\\[6pt]
+
\\[4pt]
−
a + b
+
~q~
−
\end{matrix}</math>
+
\end{matrix}\!</math>
−
|-
  −
| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
  −
| <math>\texttt{((} a \texttt{,} b \texttt{))}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
a ~\operatorname{is~equal~to}~ b
+
~\mathrm{d}q~
−
\\[6pt]
+
\\[4pt]
−
a ~\operatorname{if~and~only~if}~ b
+
(\mathrm{d}q)
−
\end{matrix}</math>
+
\end{matrix}\!</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
a = b
+
(\mathrm{d}q)
−
\\[6pt]
+
\\[4pt]
−
a \Leftrightarrow b
+
~\mathrm{d}q~
−
\end{matrix}</math>
+
\end{matrix}\!</math>
−
|-
  −
| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
  −
| <math>\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\operatorname{just~one~of}
+
~\mathrm{d}q~
−
\\
+
\\[4pt]
−
a, b, c
+
(\mathrm{d}q)
−
\\
+
\end{matrix}\!</math>
−
\operatorname{is~false}.
  −
\end{matrix}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
& \bar{a} ~ b ~ c
+
(\mathrm{d}q)
−
\\
+
\\[4pt]
−
\lor & a ~ \bar{b} ~ c
+
~\mathrm{d}q~
−
\\
+
\end{matrix}\!</math>
−
\lor & a ~ b ~ \bar{c}
  −
\end{matrix}</math>
   
|-
 
|-
−
| height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]]
  −
| <math>\texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\operatorname{just~one~of}
+
f_7
−
\\
+
\\[4pt]
−
a, b, c
+
f_{11}
−
\\
+
\\[4pt]
−
\operatorname{is~true}.
+
f_{13}
−
\\[6pt]
+
\\[4pt]
−
\operatorname{partition~all}
+
f_{14}
−
\\
+
\end{matrix}\!</math>
−
\operatorname{into}~ a, b, c.
  −
\end{matrix}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
& a ~ \bar{b} ~ \bar{c}
+
(~p~~q~)
−
\\
+
\\[4pt]
−
\lor & \bar{a} ~ b ~ \bar{c}
+
(~p~(q))
−
\\
+
\\[4pt]
−
\lor & \bar{a} ~ \bar{b} ~ c
+
((p)~q~)
−
\end{matrix}</math>
+
\\[4pt]
−
|-
+
((p)(q))
−
| height="160px" | [[Image:Cactus (A,(B,C)) Big.jpg|90px]]
+
\end{matrix}\!</math>
−
| <math>\texttt{(} a \texttt{,(} b \texttt{,} c \texttt{))}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\operatorname{oddly~many~of}
+
((\mathrm{d}p)(\mathrm{d}q))
−
\\
+
\\[4pt]
−
a, b, c
+
((\mathrm{d}p)~\mathrm{d}q~)
−
\\
+
\\[4pt]
−
\operatorname{are~true}.
+
(~\mathrm{d}p~(\mathrm{d}q))
−
\end{matrix}</math>
+
\\[4pt]
 +
(~\mathrm{d}p~~\mathrm{d}q~)
 +
\end{matrix}\!</math>
 
|
 
|
−
<p><math>a + b + c\!</math></p>
+
<math>\begin{matrix}
−
<br>
+
((\mathrm{d}p)~\mathrm{d}q~)
−
<p><math>\begin{matrix}
+
\\[4pt]
−
& a ~ b ~ c
+
((\mathrm{d}p)(\mathrm{d}q))
−
\\
+
\\[4pt]
−
\lor & a ~ \bar{b} ~ \bar{c}
+
(~\mathrm{d}p~~\mathrm{d}q~)
−
\\
+
\\[4pt]
−
\lor & \bar{a} ~ b ~ \bar{c}
+
(~\mathrm{d}p~(\mathrm{d}q))
−
\\
+
\end{matrix}\!</math>
−
\lor & \bar{a} ~ \bar{b} ~ c
  −
\end{matrix}</math></p>
  −
|-
  −
| height="160px" | [[Image:Cactus (X,(A),(B),(C)) Big.jpg|90px]]
  −
| <math>\texttt{(} x \texttt{,(} a \texttt{),(} b \texttt{),(} c \texttt{))}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\operatorname{partition}~ x
+
(~\mathrm{d}p~(\mathrm{d}q))
−
\\
+
\\[4pt]
−
\operatorname{into}~ a, b, c.
+
(~\mathrm{d}p~~\mathrm{d}q~)
−
\\[6pt]
+
\\[4pt]
−
\operatorname{genus}~ x ~\operatorname{comprises}
+
((\mathrm{d}p)(\mathrm{d}q))
−
\\
+
\\[4pt]
−
\operatorname{species}~ a, b, c.
+
((\mathrm{d}p)~\mathrm{d}q~)
−
\end{matrix}</math>
+
\end{matrix}\!</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
& \bar{x} ~ \bar{a} ~ \bar{b} ~ \bar{c}
+
(~\mathrm{d}p~~\mathrm{d}q~)
−
\\
+
\\[4pt]
−
\lor & x ~ a ~ \bar{b} ~ \bar{c}
+
(~\mathrm{d}p~(\mathrm{d}q))
−
\\
+
\\[4pt]
−
\lor & x ~ \bar{a} ~ b ~ \bar{c}
+
((\mathrm{d}p)~\mathrm{d}q~)
−
\\
+
\\[4pt]
−
\lor & x ~ \bar{a} ~ \bar{b} ~ c
+
((\mathrm{d}p)(\mathrm{d}q))
−
\end{matrix}</math>
+
\end{matrix}\!</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 
|}
 
|}
    
<br>
 
<br>
   −
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
−
|+ <math>\text{Table C.}~~\text{Dualing Interpretations}</math>
+
|+ <math>\text{Table A6.}~~\mathrm{D}f ~\text{Expanded over Ordinary Features}~ \{ p, q \}\!</math>
 
|- style="background:#f0f0ff"
 
|- style="background:#f0f0ff"
−
| <math>\text{Graph}\!</math>
+
| width="10%" | &nbsp;
−
| <math>\text{String}\!</math>
+
| width="18%" | <math>f\!</math>
−
| <math>\text{Existential}\!</math>
+
| width="18%" | <math>\mathrm{D}f|_{pq}\!</math>
−
| <math>\text{Entitative}\!</math>
+
| width="18%" | <math>\mathrm{D}f|_{p(q)}\!</math>
 +
| width="18%" | <math>\mathrm{D}f|_{(p)q}\!</math>
 +
| width="18%" | <math>\mathrm{D}f|_{(p)(q)}\!</math>
 
|-
 
|-
−
| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
+
| <math>f_0\!</math>
−
| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
+
| <math>(~)\!</math>
−
| <math>\operatorname{true}.</math>
+
| <math>(~)\!</math>
−
| <math>\operatorname{false}.</math>
+
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 
|-
 
|-
−
| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
+
|
−
| <math>\texttt{(~)}</math>
+
<math>\begin{matrix}
−
| <math>\operatorname{false}.</math>
+
f_1
−
| <math>\operatorname{true}.</math>
+
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~~\mathrm{d}p~~\mathrm{d}q~~
 +
\\[4pt]
 +
~~\mathrm{d}p~(\mathrm{d}q)~
 +
\\[4pt]
 +
~(\mathrm{d}p)~\mathrm{d}q~~
 +
\\[4pt]
 +
((\mathrm{d}p)(\mathrm{d}q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~~\mathrm{d}p~(\mathrm{d}q)~
 +
\\[4pt]
 +
~~\mathrm{d}p~~\mathrm{d}q~~
 +
\\[4pt]
 +
((\mathrm{d}p)(\mathrm{d}q))
 +
\\[4pt]
 +
~(\mathrm{d}p)~\mathrm{d}q~~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~(\mathrm{d}p)~\mathrm{d}q~~
 +
\\[4pt]
 +
((\mathrm{d}p)(\mathrm{d}q))
 +
\\[4pt]
 +
~~\mathrm{d}p~~\mathrm{d}q~~
 +
\\[4pt]
 +
~~\mathrm{d}p~(\mathrm{d}q)~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((\mathrm{d}p)(\mathrm{d}q))
 +
\\[4pt]
 +
~(\mathrm{d}p)~\mathrm{d}q~~
 +
\\[4pt]
 +
~~\mathrm{d}p~(\mathrm{d}q)~
 +
\\[4pt]
 +
~~\mathrm{d}p~~\mathrm{d}q~~
 +
\end{matrix}\!</math>
 
|-
 
|-
−
| height="100px" | [[Image:Cactus A Big.jpg|20px]]
+
|
−
| <math>a\!</math>
+
<math>\begin{matrix}
−
| <math>a.\!</math>
+
f_3
−
| <math>a.\!</math>
+
\\[4pt]
 +
f_{12}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}p
 +
\\[4pt]
 +
\mathrm{d}p
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}p
 +
\\[4pt]
 +
\mathrm{d}p
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}p
 +
\\[4pt]
 +
\mathrm{d}p
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}p
 +
\\[4pt]
 +
\mathrm{d}p
 +
\end{matrix}\!</math>
 
|-
 
|-
−
| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
+
|
−
| <math>\texttt{(} a \texttt{)}</math>
+
<math>\begin{matrix}
−
| <math>\lnot a</math>
+
f_6
−
| <math>\lnot a</math>
+
\\[4pt]
 +
f_9
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(\mathrm{d}p,~\mathrm{d}q)
 +
\\[4pt]
 +
(\mathrm{d}p,~\mathrm{d}q)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(\mathrm{d}p,~\mathrm{d}q)
 +
\\[4pt]
 +
(\mathrm{d}p,~\mathrm{d}q)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(\mathrm{d}p,~\mathrm{d}q)
 +
\\[4pt]
 +
(\mathrm{d}p,~\mathrm{d}q)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(\mathrm{d}p,~\mathrm{d}q)
 +
\\[4pt]
 +
(\mathrm{d}p,~\mathrm{d}q)
 +
\end{matrix}\!</math>
 
|-
 
|-
−
| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
  −
| <math>a~b~c</math>
  −
| <math>a \land b \land c</math>
  −
| <math>a \lor  b \lor  c</math>
  −
|-
  −
| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
  −
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
  −
| <math>a \lor  b \lor  c</math>
  −
| <math>a \land b \land c</math>
  −
|-
  −
| height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]]
  −
| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
  −
| <math>a \Rightarrow b</math>
  −
| &nbsp;
  −
|-
  −
| height="120px" | [[Image:Cactus (A)B Big.jpg|35px]]
  −
| <math>\texttt{(} a \texttt{)} b</math>
  −
| &nbsp;
  −
| <math>a \Rightarrow b</math>
  −
|-
  −
| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
  −
| <math>\texttt{(} a \texttt{,} b \texttt{)}</math>
  −
| <math>a \neq b</math>
  −
| <math>a = b\!</math>
  −
|-
  −
| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
  −
| <math>\texttt{((} a \texttt{,} b \texttt{))}</math>
  −
| <math>a = b\!</math>
  −
| <math>a \neq b\!</math>
  −
|-
  −
| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
  −
| <math>\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\operatorname{just~one}
+
f_5
−
\\
+
\\[4pt]
−
\operatorname{of}~ a, b, c
+
f_{10}
−
\\
+
\end{matrix}\!</math>
−
\operatorname{is~false}.
+
|
−
\end{matrix}</math>
+
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}q
 +
\\[4pt]
 +
\mathrm{d}q
 +
\end{matrix}\!</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\operatorname{not~just~one}
+
\mathrm{d}q
−
\\
+
\\[4pt]
−
\operatorname{of}~ a, b, c
+
\mathrm{d}q
−
\\
+
\end{matrix}\!</math>
−
\operatorname{is~true}.
  −
\end{matrix}</math>
  −
|-
  −
| height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]]
  −
| <math>\texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\operatorname{just~one}
+
\mathrm{d}q
−
\\
+
\\[4pt]
−
\operatorname{of}~ a, b, c
+
\mathrm{d}q
−
\\
+
\end{matrix}\!</math>
−
\operatorname{is~true}.
  −
\end{matrix}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\operatorname{not~just~one}
+
\mathrm{d}q
−
\\
+
\\[4pt]
−
\operatorname{of}~ a, b, c
+
\mathrm{d}q
−
\\
+
\end{matrix}\!</math>
−
\operatorname{is~false}.
  −
\end{matrix}</math>
   
|-
 
|-
−
| height="160px" | [[Image:Cactus ((A,B,C)) Big.jpg|65px]]
  −
| <math>\texttt{((} a \texttt{,} b \texttt{,} c \texttt{))}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\operatorname{not~just~one}
+
f_7
−
\\
+
\\[4pt]
−
\operatorname{of}~ a, b, c
+
f_{11}
−
\\
+
\\[4pt]
−
\operatorname{is~false}.
+
f_{13}
−
\end{matrix}</math>
+
\\[4pt]
 +
f_{14}
 +
\end{matrix}\!</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\operatorname{just~one}
+
(~p~~q~)
−
\\
+
\\[4pt]
−
\operatorname{of}~ a, b, c
+
(~p~(q))
−
\\
+
\\[4pt]
−
\operatorname{is~true}.
+
((p)~q~)
−
\end{matrix}</math>
+
\\[4pt]
−
|-
+
((p)(q))
−
| height="200px" | [[Image:Cactus (((A),(B),(C))) Big.jpg|65px]]
+
\end{matrix}\!</math>
−
| <math>\texttt{(((} a \texttt{),(} b \texttt{),(} c \texttt{)))}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\operatorname{not~just~one}
+
((\mathrm{d}p)(\mathrm{d}q))
−
\\
+
\\[4pt]
−
\operatorname{of}~ a, b, c
+
~(\mathrm{d}p)~\mathrm{d}q~~
−
\\
+
\\[4pt]
−
\operatorname{is~true}.
+
~~\mathrm{d}p~(\mathrm{d}q)~
−
\end{matrix}</math>
+
\\[4pt]
 +
~~\mathrm{d}p~~\mathrm{d}q~~
 +
\end{matrix}\!</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\operatorname{just~one}
+
~(\mathrm{d}p)~\mathrm{d}q~~
−
\\
+
\\[4pt]
−
\operatorname{of}~ a, b, c
+
((\mathrm{d}p)(\mathrm{d}q))
−
\\
+
\\[4pt]
−
\operatorname{is~false}.
+
~~\mathrm{d}p~~\mathrm{d}q~~
−
\end{matrix}</math>
+
\\[4pt]
−
|-
+
~~\mathrm{d}p~(\mathrm{d}q)~
−
| height="160px" | [[Image:Cactus (A,(B),(C)) Big.jpg|65px]]
+
\end{matrix}\!</math>
−
| <math>\texttt{(} a \texttt{,(} b \texttt{),(} c \texttt{))}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\operatorname{partition}~ a
+
~~\mathrm{d}p~(\mathrm{d}q)~
−
\\
+
\\[4pt]
−
\operatorname{into}~ b, c.
+
~~\mathrm{d}p~~\mathrm{d}q~~
−
\end{matrix}</math>
+
\\[4pt]
−
| &nbsp;
+
((\mathrm{d}p)(\mathrm{d}q))
−
|-
+
\\[4pt]
−
| height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|65px]]
+
~(\mathrm{d}p)~\mathrm{d}q~~
−
| <math>\texttt{(((} a \texttt{),} b \texttt{,} c \texttt{))}</math>
+
\end{matrix}\!</math>
−
| &nbsp;
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\operatorname{partition}~ a
+
~~\mathrm{d}p~~\mathrm{d}q~~
−
\\
+
\\[4pt]
−
\operatorname{into}~ b, c.
+
~~\mathrm{d}p~(\mathrm{d}q)~
−
\end{matrix}</math>
+
\\[4pt]
 +
~(\mathrm{d}p)~\mathrm{d}q~~
 +
\\[4pt]
 +
((\mathrm{d}p)(\mathrm{d}q))
 +
\end{matrix}\!</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 
|}
 
|}
    
<br>
 
<br>
Jon Awbrey
12,080

edits

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