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==Place for Discussion==
+
==Work Area • Logical Cacti==
−
 
 
−
…
 
−
 
 
−
==Work Area==
 
−
 
 
−
===Logical Cacti===
 
  
 
; Theme One Program — Logical Cacti
 
; Theme One Program — Logical Cacti
Line 12: Line 6:
 
: http://stderr.org/pipermail/inquiry/2005-February/002361.html
 
: http://stderr.org/pipermail/inquiry/2005-February/002361.html
  
−
<pre>
+
===Original Version===
−
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),
+
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.
−
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 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.
  
−
But the purpose of a logical syntax is to support a logical semantics,
+
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.
−
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
+
<br>
−
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
+
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
−
at least two different interpretations as logical sentences denoting
+
|+ <math>\text{Table A.}~~\text{Existential Interpretation}</math>
−
logical propositions. The two interpretations that I know about are
+
|- style="background:#f0f0ff"
−
descended from the ones that C.S. Peirce called the "entitative" and
+
| <math>\text{Cactus Graph}\!</math>
−
the "existential" interpretations of his systems of graphical logics.
+
| <math>\text{Cactus Expression}\!</math>
−
For our present aims, I shall briefly introduce the alternatives and
+
| <math>\text{Interpretation}\!</math>
−
then quickly move to the existential interpretation of logical cacti.
+
|-
 +
| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
 +
| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
 +
| <math>\operatorname{true}.</math>
 +
|-
 +
| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
 +
| <math>\texttt{(~)}</math>
 +
| <math>\operatorname{false}.</math>
 +
|-
 +
| height="100px" | [[Image:Cactus A Big.jpg|20px]]
 +
| <math>a\!</math>
 +
| <math>a.\!</math>
 +
|-
 +
| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
 +
| <math>\texttt{(} a \texttt{)}</math>
 +
|
 +
<math>\begin{matrix}
 +
\tilde{a}
 +
\\[2pt]
 +
a^\prime
 +
\\[2pt]
 +
\lnot a
 +
\\[2pt]
 +
\operatorname{not}~ a.
 +
\end{matrix}</math>
 +
|-
 +
| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
 +
| <math>a~b~c</math>
 +
|
 +
<math>\begin{matrix}
 +
a \land b \land c
 +
\\[6pt]
 +
a ~\operatorname{and}~ b ~\operatorname{and}~ 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}
 +
a \lor b \lor c
 +
\\[6pt]
 +
a ~\operatorname{or}~ b ~\operatorname{or}~ c.
 +
\end{matrix}</math>
 +
|-
 +
| height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]]
 +
| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
 +
|
 +
<math>\begin{matrix}
 +
a \Rightarrow b
 +
\\[2pt]
 +
a ~\operatorname{implies}~ b.
 +
\\[2pt]
 +
\operatorname{if}~ a ~\operatorname{then}~ b.
 +
\\[2pt]
 +
\operatorname{not}~ a ~\operatorname{without}~ b.
 +
\end{matrix}</math>
 +
|-
 +
| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
 +
| <math>\texttt{(} a \texttt{,} b \texttt{)}</math>
 +
|
 +
<math>\begin{matrix}
 +
a + b
 +
\\[2pt]
 +
a \neq b
 +
\\[2pt]
 +
a ~\operatorname{exclusive-or}~ b.
 +
\\[2pt]
 +
a ~\operatorname{not~equal~to}~ b.
 +
\end{matrix}</math>
 +
|-
 +
| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
 +
| <math>\texttt{((} a \texttt{,} b \texttt{))}</math>
 +
|
 +
<math>\begin{matrix}
 +
a = b
 +
\\[2pt]
 +
a \iff b
 +
\\[2pt]
 +
a ~\operatorname{equals}~ b.
 +
\\[2pt]
 +
a ~\operatorname{if~and~only~if}~ b.
 +
\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}
 +
\operatorname{just~one~of}
 +
\\
 +
a, b, c
 +
\\
 +
\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}
 +
\operatorname{just~one~of}
 +
\\
 +
a, b, c
 +
\\
 +
\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}
 +
\operatorname{genus}~ a ~\operatorname{of~species}~ b, c.
 +
\\[6pt]
 +
\operatorname{partition}~ a ~\operatorname{into}~ b, c.
 +
\\[6pt]
 +
\operatorname{pie}~ a ~\operatorname{of~slices}~ b, c.
 +
\end{matrix}</math>
 +
|}
  
−
Table 13 illustrates the "existential interpretation"
+
<br>
−
of cactus graphs and cactus expressions by providing
 
−
English translations for a few of the most basic and
 
−
commonly occurring forms.
 
  
−
Table 13.  The Existential Interpretation
+
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.
−
o----o-------------------o-------------------o-------------------o
 
−
| Ex |  Cactus Graph    | Cactus Expression |    Existential    |
 
−
|    |                  |                  |  Interpretation  |
 
−
o----o-------------------o-------------------o-------------------o
 
−
|    |                  |                  |                  |
 
−
|  1 |        @        |        " "        |      true.      |
 
−
|    |                  |                  |                  |
 
−
o----o-------------------o-------------------o-------------------o
 
−
|    |                  |                  |                  |
 
−
|    |        o        |                  |                  |
 
−
|    |        |        |                  |                  |
 
−
|  2 |        @        |        ( )        |      untrue.      |
 
−
|    |                  |                  |                  |
 
−
o----o-------------------o-------------------o-------------------o
 
−
|    |                  |                  |                  |
 
−
|    |        a        |                  |                  |
 
−
|  3 |        @        |        a        |        a.        |
 
−
|    |                  |                  |                  |
 
−
o----o-------------------o-------------------o-------------------o
 
−
|    |                  |                  |                  |
 
−
|    |        a        |                  |                  |
 
−
|    |        o        |                  |                  |
 
−
|    |        |        |                  |                  |
 
−
|  4 |        @        |        (a)        |      not a.      |
 
−
|    |                  |                  |                  |
 
−
o----o-------------------o-------------------o-------------------o
 
−
|    |                  |                  |                  |
 
−
|    |      a b c      |                  |                  |
 
−
|  5 |        @        |      a b c      |  a and b and c.  |
 
−
|    |                  |                  |                  |
 
−
o----o-------------------o-------------------o-------------------o
 
−
|    |                  |                  |                  |
 
−
|    |      a b c      |                  |                  |
 
−
|    |      o o o      |                  |                  |
 
−
|    |        \|/        |                  |                  |
 
−
|    |        o        |                  |                  |
 
−
|    |        |        |                  |                  |
 
−
|  6 |        @        |    ((a)(b)(c))    |    a or b or c.  |
 
−
|    |                  |                  |                  |
 
−
o----o-------------------o-------------------o-------------------o
 
−
|    |                  |                  |                  |
 
−
|    |                  |                  |    a implies b.  |
 
−
|    |        a  b    |                  |                  |
 
−
|    |        o---o    |                  |    if a then b.  |
 
−
|    |        |        |                  |                  |
 
−
|  7 |        @        |    ( a (b))      |    no a sans b.  |
 
−
|    |                  |                  |                  |
 
−
o----o-------------------o-------------------o-------------------o
 
−
|    |                  |                  |                  |
 
−
|    |      a  b      |                  |                  |
 
−
|    |      o---o      |                  | a exclusive-or b. |
 
−
|    |        \ /        |                  |                  |
 
−
|  8 |        @        |    ( a , b )    | a not equal to b. |
 
−
|    |                  |                  |                  |
 
−
o----o-------------------o-------------------o-------------------o
 
−
|    |                  |                  |                  |
 
−
|    |      a  b      |                  |                  |
 
−
|    |      o---o      |                  |                  |
 
−
|    |        \ /        |                  |                  |
 
−
|    |        o        |                  | a if & only if b. |
 
−
|    |        |        |                  |                  |
 
−
|  9 |        @        |    (( a , b ))    | a equates with b. |
 
−
|    |                  |                  |                  |
 
−
o----o-------------------o-------------------o-------------------o
 
−
|    |                  |                  |                  |
 
−
|    |      a  b  c      |                  |                  |
 
−
|    |      o--o--o      |                  |                  |
 
−
|    |      \  /      |                  |                  |
 
−
|    |        \ /        |                  |  just one false  |
 
−
| 10 |        @        |  ( a , b , c )  |  out of a, b, c.  |
 
−
|    |                  |                  |                  |
 
−
o----o-------------------o-------------------o-------------------o
 
−
|    |                  |                  |                  |
 
−
|    |      a  b  c      |                  |                  |
 
−
|    |      o  o  o      |                  |                  |
 
−
|    |      |  |  |      |                  |                  |
 
−
|    |      o--o--o      |                  |                  |
 
−
|    |      \  /      |                  |                  |
 
−
|    |        \ /        |                  |  just one true  |
 
−
| 11 |        @        |  ((a),(b),(c))  |  among a, b, c.  |
 
−
|    |                  |                  |                  |
 
−
o----o-------------------o-------------------o-------------------o
 
−
|    |                  |                  |                  |
 
−
|    |                  |                  |  genus a over    |
 
−
|    |        b  c      |                  |  species b, c.  |
 
−
|    |        o  o      |                  |                  |
 
−
|    |      a  |  |      |                  |  partition a    |
 
−
|    |      o--o--o      |                  |  among b & c.    |
 
−
|    |      \  /      |                  |                  |
 
−
|    |        \ /        |                  |  whole pie a:    |
 
−
| 12 |        @        |  ( a ,(b),(c))  |  slices b, c.   |
 
−
|    |                  |                  |                  |
 
−
o----o-------------------o-------------------o-------------------o
 
  
−
Table 14 illustrates the "entitative interpretation"
+
<br>
−
of cactus graphs and cactus expressions by providing
 
−
English translations for a few of the most basic and
 
−
commonly occurring forms.
 
  
−
Table 14. The Entitative Interpretation
+
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
−
o----o-------------------o-------------------o-------------------o
+
|+ <math>\text{Table B.}~~\text{Entitative Interpretation}</math>
−
| En |  Cactus Graph   | Cactus Expression |    Entitative    |
+
|- style="background:#f0f0ff"
−
|   |                  |                  |  Interpretation   |
+
| <math>\text{Cactus Graph}\!</math>
−
o----o-------------------o-------------------o-------------------o
+
| <math>\text{Cactus Expression}\!</math>
−
|    |                  |                  |                  |
+
| <math>\text{Interpretation}\!</math>
−
| 1 |        @        |        " "       |     untrue.     |
+
|-
−
|   |                  |                  |                  |
+
| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
−
o----o-------------------o-------------------o-------------------o
+
| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
−
|   |                  |                  |                  |
+
| <math>\operatorname{false}.</math>
−
|   |        o        |                  |                  |
+
|-
−
|   |         |        |                  |                  |
+
| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
−
| 2 |        @        |        ( )       |       true.       |
+
| <math>\texttt{(~)}</math>
−
|   |                  |                  |                  |
+
| <math>\operatorname{true}.</math>
−
o----o-------------------o-------------------o-------------------o
+
|-
−
|   |                   |                  |                  |
+
| height="100px" | [[Image:Cactus A Big.jpg|20px]]
−
|   |        a         |                  |                  |
+
| <math>a\!</math>
−
| 3 |        @        |        a        |        a.       |
+
| <math>a.\!</math>
−
|   |                  |                  |                  |
+
|-
−
o----o-------------------o-------------------o-------------------o
+
| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
−
|   |                   |                   |                   |
+
| <math>\texttt{(} a \texttt{)}</math>
−
|    |        a         |                  |                  |
+
|
−
|    |        o        |                  |                  |
+
<math>\begin{matrix}
−
|    |        |        |                  |                  |
+
\tilde{a}
−
|  4 |        @        |        (a)        |      not a.     |
+
\\[2pt]
−
|   |                  |                  |                  |
+
a^\prime
−
o----o-------------------o-------------------o-------------------o
+
\\[2pt]
−
|   |                   |                  |                  |
+
\lnot a
−
|   |      a b c       |                  |                  |
+
\\[2pt]
−
|  5 |        @        |      a b c       |    a or b or c.   |
+
\operatorname{not}~ a.
−
|    |                  |                  |                  |
+
\end{matrix}</math>
−
o----o-------------------o-------------------o-------------------o
+
|-
−
|    |                  |                  |                  |
+
| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
−
|    |      a b c      |                  |                  |
+
| <math>a~b~c</math>
−
|    |      o o o      |                  |                  |
+
|
−
|    |        \|/       |                  |                  |
+
<math>\begin{matrix}
−
|   |        o        |                  |                  |
+
a \lor b \lor c
−
|   |         |        |                  |                  |
+
\\[6pt]
−
| 6 |        @        |    ((a)(b)(c))   |   a and b and c. |
+
a ~\operatorname{or}~ b ~\operatorname{or}~ c.
−
|   |                   |                   |                   |
+
\end{matrix}</math>
−
o----o-------------------o-------------------o-------------------o
+
|-
−
|    |                  |                  |                  |
+
| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
−
|    |                  |                  |    a implies b.   |
+
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
−
|    |                  |                  |                  |
+
|
−
|    |        o a      |                  |    if a then b.   |
+
<math>\begin{matrix}
−
|    |        |        |                  |                  |
+
a \land b \land c
−
|  7 |        @ b      |      (a) b        |    not a, or b.   |
+
\\[6pt]
−
|   |                   |                   |                   |
+
a ~\operatorname{and}~ b ~\operatorname{and}~ c.
−
o----o-------------------o-------------------o-------------------o
+
\end{matrix}</math>
−
|    |                  |                  |                  |
+
|-
−
|    |      a   b       |                  |                  |
+
| height="120px" | [[Image:Cactus (A)B Big.jpg|35px]]
−
|    |      o---o      |                  | a if & only if b. |
+
| <math>\texttt{(} a \texttt{)} b</math>
−
|    |        \ /       |                   |                   |
+
|
−
| 8 |        @        |    ( a , b )     | a equates with b. |
+
<math>\begin{matrix}
−
|    |                  |                  |                  |
+
a \Rightarrow b
−
o----o-------------------o-------------------o-------------------o
+
\\[2pt]
−
|    |                  |                  |                  |
+
a ~\operatorname{implies}~ b.
−
|    |      a   b       |                  |                  |
+
\\[2pt]
−
|    |      o---o      |                  |                  |
+
\operatorname{if}~ a ~\operatorname{then}~ b.
−
|    |        \ /        |                  |                  |
+
\\[2pt]
−
|    |        o        |                  | a exclusive-or b. |
+
\operatorname{not}~ a, ~\operatorname{or}~ b.
−
|    |        |        |                  |                  |
+
\end{matrix}</math>
−
|  9 |        @        |    (( a , b ))    | a not equal to b. |
+
|-
−
|   |                  |                  |                  |
+
| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
−
o----o-------------------o-------------------o-------------------o
+
| <math>\texttt{(} a \texttt{,} b \texttt{)}</math>
−
|   |                   |                  |                  |
+
|
−
|   |      a b c     |                  |                  |
+
<math>\begin{matrix}
−
|    |      o--o--o      |                  |                  |
+
a = b
−
|    |      \   /      |                  |                  |
+
\\[2pt]
−
|    |        \ /        |                  | not just one true |
+
a \iff b
−
| 10 |         @        |  ( a , b , c )   | out of a, b, c.  |
+
\\[2pt]
−
|    |                  |                  |                  |
+
a ~\operatorname{equals}~ b.
−
o----o-------------------o-------------------o-------------------o
+
\\[2pt]
−
|    |                  |                  |                  |
+
a ~\operatorname{if~and~only~if}~ b.
−
|    |      a b c     |                  |                  |
+
\end{matrix}</math>
−
|    |      o--o--o      |                  |                  |
+
|-
−
|    |      \   /      |                  |                  |
+
| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
−
|    |        \ /       |                  |                  |
+
| <math>\texttt{((} a \texttt{,} b \texttt{))}</math>
−
|   |        o        |                  |                  |
+
|
−
|   |         |        |                  |  just one true  |
+
<math>\begin{matrix}
−
| 11 |        @        |  (( a , b , c )) |  among a, b, c.  |
+
a + b
−
|    |                  |                  |                  |
+
\\[2pt]
−
o----o-------------------o-------------------o-------------------o
+
a \neq b
−
|    |                  |                  |                  |
+
\\[2pt]
−
|    |      a            |                  |                  |
+
a ~\operatorname{exclusive-or}~ b.
−
|    |      o            |                  |  genus a over    |
+
\\[2pt]
−
|    |      |  b  c      |                  |  species b, c.   |
+
a ~\operatorname{not~equal~to}~ b.
−
|    |      o--o--o      |                  |                  |
+
\end{matrix}</math>
−
|    |      \   /      |                  |  partition a     |
+
|-
−
|    |        \ /        |                  |  among b & c.   |
+
| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
−
|    |        o        |                  |                  |
+
| <math>\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}</math>
−
|    |        |        |                  |  whole pie a:    |
+
|
−
| 12 |        @        |  (((a), b , c ))  |  slices b, c.   |
+
<math>\begin{matrix}
−
|   |                  |                  |                  |
+
\operatorname{not~just~one~of}
−
o----o-------------------o-------------------o-------------------o
+
\\
 +
a, b, c
 +
\\
 +
\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}
 +
\operatorname{just~one~of}
 +
\\
 +
a, b, c
 +
\\
 +
\operatorname{is~true}.
 +
\end{matrix}</math>
 +
|-
 +
| height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|65px]]
 +
| <math>\texttt{(((} a \texttt{),} b \texttt{,} c \texttt{))}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{genus}~ a ~\operatorname{of~species}~ b, c.
 +
\\[6pt]
 +
\operatorname{partition}~ a ~\operatorname{into}~ b, c.
 +
\\[6pt]
 +
\operatorname{pie}~ a ~\operatorname{of~slices}~ b, c.
 +
\end{matrix}</math>
 +
|}
  
−
For the time being, the main things to take away from Tables 13 and 14 are
+
<br>
−
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:
 
  
−
1.  The "node connective" joins a number of
+
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:
−
    component cacti C_1, ..., C_k at a node:
 
  
 +
{| align="center" cellpadding="6" width="90%"
 +
| valign="top" | 1.
 +
| The ''node connective'' joins a number of component cacti <math>C_1, \ldots, C_k</math> at a node:
 +
|-
 +
| &nbsp;
 +
|
 +
<pre>
 
     C_1 ... C_k
 
     C_1 ... C_k
 
         @
 
         @
−
 
+
</pre>
−
2. The "lobe connective" joins a number of
+
|-
−
    component cacti C_1, ..., C_k to a lobe:
+
| valign="top" | 2.
−
 
+
| The ''lobe connective'' joins a number of component cacti <math>C_1, \ldots, C_k</math> to a lobe:
 +
|-
 +
| &nbsp;
 +
|
 +
<pre>
 
     C_1 C_2  C_k
 
     C_1 C_2  C_k
 
     o---o-...-o
 
     o---o-...-o
Line 261: Line 300:
 
         \ /
 
         \ /
 
           @
 
           @
 +
</pre>
 +
|}
  
−
Table 15 summarizes the existential and entitative
+
Table&nbsp;15 summarizes the existential and entitative interpretations of the primitive cactus structures, in effect, the graphical constants and connectives.
−
interpretations of the primitive cactus structures,
 
−
in effect, the graphical constants and connectives.
 
  
 +
{| align="center" cellpadding="6" style="text-align:center; width:90%"
 +
|
 +
<pre>
 
Table 15.  Existential & Entitative Interpretations of Cactus Structures
 
Table 15.  Existential & Entitative Interpretations of Cactus Structures
 
o-----------------o-----------------o-----------------o-----------------o
 
o-----------------o-----------------o-----------------o-----------------o
Line 296: Line 338:
 
|                |                |                |                |
 
|                |                |                |                |
 
o-----------------o-----------------o-----------------o-----------------o
 
o-----------------o-----------------o-----------------o-----------------o
 +
</pre>
 +
|}
  
−
It is possible to specify "abstract rules of equivalence" (AROE's)
+
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.
−
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
+
A ''reduction'' is an equivalence transformation that is applied in the direction of decreasing graphical complexity.
−
that is applied in the direction of decreasing
 
−
graphical complexity.
 
  
−
A "basic reduction" is a reduction that applies
+
A ''basic reduction'' is a reduction that applies to one of the two families of basic connectives.
−
to one of the two families of basic connectives.
 
  
−
Table 16 schematizes the two types of basic reductions
+
Table&nbsp;16 schematizes the two types of basic reductions in a purely formal, interpretation-independent fashion.
−
in a purely formal, interpretation-independent fashion.
 
  
 +
{| align="center" cellpadding="6" style="text-align:center; width:90%"
 +
|
 +
<pre>
 
Table 16.  Basic Reductions
 
Table 16.  Basic Reductions
 
o---------------------------------------o
 
o---------------------------------------o
Line 340: Line 379:
 
|                                      |
 
|                                      |
 
o---------------------------------------o
 
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>
 +
o---------------------------------------o
 +
|                                      |
 +
|                  o                  |
 +
|                  |                  |
 +
|                  @                  |
 +
|                                      |
 +
o---------------------------------------o
 +
|                  ( )                  |
 +
o---------------------------------------o
 +
</pre>
 +
|}
 +
 +
The rule of reduction for a lobe is:
  
−
The careful reader will have noticed that we have begun to use
+
{| align="center" cellpadding="6" style="text-align:center; width:90%"
−
graphical paints like "a", "b", "c" and schematic proxies like
+
|
−
"C_1", "C_j", "C_k" in a variety of novel and unjustified ways.
+
<pre>
 +
o---------------------------------------o
 +
|                                      |
 +
|  x_1  x_2  ...  x_k                |
 +
|  o-----o--- ... ---o                |
 +
|    \              /                  |
 +
|    \            /                  |
 +
|      \          /                    |
 +
|      \        /                    |
 +
|        \      /                      |
 +
|        \    /                      |
 +
|          \  /                        |
 +
|          \ /                        |
 +
|            @      =      @            |
 +
|                                      |
 +
o---------------------------------------o
 +
</pre>
 +
|}
  
−
The careful writer would have already introduced a whole bevy of
+
if and only if exactly one of the <math>x_j\!</math> is a spike.
−
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".
+
In Ref Log, an expression of the form <math>\texttt{((}~ e_1 ~\texttt{),(}~ e_2 ~\texttt{),(}~ \ldots ~\texttt{),(}~ e_k ~\texttt{))}</math>
−
So let me assist the prospective exercise
+
expresses the fact that ''exactly one of the <math>e_j\!</math> is true''.  Expressions of this form are called ''universal partition'' expressions, and
−
with a few hints of what it would take to
+
they parse into a type of graph called a ''painted and rooted cactus'' (PARC):
−
guarantee that these practices make sense.
 
  
−
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
+
{| align="center" cellpadding="6" style="text-align:center; width:90%"
 +
|
 +
<pre>
 +
o---------------------------------------o
 +
|                                      |
 +
|  e_1  e_2  ...  e_k                |
 +
|  o     o           o                 |
 +
|  |    |          |                |
 +
|  o-----o--- ... ---o                |
 +
|    \              /                  |
 +
|    \            /                  |
 +
|      \          /                    |
 +
|      \        /                    |
 +
|        \      /                      |
 +
|        \    /                      |
 +
|          \  /                        |
 +
|          \ /                        |
 +
|            @                          |
 +
|                                      |
 +
o---------------------------------------o
 
</pre>
 
</pre>
 +
|}
 +
 +
{| align="center" cellpadding="6" style="text-align:center; width:90%"
 +
|
 +
<pre>
 +
o---------------------------------------o
 +
|                                      |
 +
| ( x1, x2, ..., xk )  =  [blank]      |
 +
|                                      |
 +
| iff                                  |
 +
|                                      |
 +
| Just one of the arguments            |
 +
| x1, x2, ..., xk  =  ()                |
 +
|                                      |
 +
o---------------------------------------o
 +
</pre>
 +
|}
 +
 +
The interpretation of these operators, read as assertions about the values of their listed arguments, is as follows:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| Existential Interpretation:
 +
| Just one of the k argument is false.
 +
|-
 +
| Entitative  Interpretation:
 +
| Not just one of the k arguments is true.
 +
|}
 +
 +
===Tables===
 +
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
 +
|+ <math>\text{Table 1.}~~\text{Syntax and Semantics of a Calculus for Propositional Logic}</math>
 +
|- style="background:#f0f0ff"
 +
| <math>\text{Graph}\!</math>
 +
| <math>\text{Expression}\!</math>
 +
| <math>\text{Interpretation}\!</math>
 +
| <math>\text{Other Notations}\!</math>
 +
|-
 +
| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
 +
| <math>~</math>
 +
| <math>\operatorname{true}</math>
 +
| <math>1\!</math>
 +
|-
 +
| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
 +
| <math>\texttt{(~)}</math>
 +
| <math>\operatorname{false}</math>
 +
| <math>0\!</math>
 +
|-
 +
| height="100px" | [[Image:Cactus A Big.jpg|20px]]
 +
| <math>a\!</math>
 +
| <math>a\!</math>
 +
| <math>a\!</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}
 +
a ~\operatorname{implies}~ b
 +
\\[6pt]
 +
\operatorname{if}~ a ~\operatorname{then}~ b
 +
\end{matrix}</math>
 +
| <math>a \Rightarrow b</math>
 +
|-
 +
| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
 +
| <math>\texttt{(} a \texttt{,} b \texttt{)}</math>
 +
|
 +
<math>\begin{matrix}
 +
a ~\operatorname{not~equal~to}~ b
 +
\\[6pt]
 +
a ~\operatorname{exclusive~or}~ b
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
a \neq b
 +
\\[6pt]
 +
a + b
 +
\end{matrix}</math>
 +
|-
 +
| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
 +
| <math>\texttt{((} a \texttt{,} b \texttt{))}</math>
 +
|
 +
<math>\begin{matrix}
 +
a ~\operatorname{is~equal~to}~ b
 +
\\[6pt]
 +
a ~\operatorname{if~and~only~if}~ b
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
a = b
 +
\\[6pt]
 +
a \Leftrightarrow b
 +
\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}
 +
\operatorname{just~one~of}
 +
\\
 +
a, b, c
 +
\\
 +
\operatorname{is~false}.
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
& \bar{a} ~ b ~ c
 +
\\
 +
\lor & a ~ \bar{b} ~ c
 +
\\
 +
\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}
 +
\operatorname{just~one~of}
 +
\\
 +
a, b, c
 +
\\
 +
\operatorname{is~true}.
 +
\\[6pt]
 +
\operatorname{partition~all}
 +
\\
 +
\operatorname{into}~ a, b, c.
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
& a ~ \bar{b} ~ \bar{c}
 +
\\
 +
\lor & \bar{a} ~ b ~ \bar{c}
 +
\\
 +
\lor & \bar{a} ~ \bar{b} ~ c
 +
\end{matrix}</math>
 +
|-
 +
| height="160px" | [[Image:Cactus (A,(B,C)) Big.jpg|90px]]
 +
| <math>\texttt{(} a \texttt{,(} b \texttt{,} c \texttt{))}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{oddly~many~of}
 +
\\
 +
a, b, c
 +
\\
 +
\operatorname{are~true}.
 +
\end{matrix}</math>
 +
|
 +
<p><math>a + b + c\!</math></p>
 +
<br>
 +
<p><math>\begin{matrix}
 +
& a ~ b ~ c
 +
\\
 +
\lor & a ~ \bar{b} ~ \bar{c}
 +
\\
 +
\lor & \bar{a} ~ b ~ \bar{c}
 +
\\
 +
\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}
 +
\operatorname{partition}~ x
 +
\\
 +
\operatorname{into}~ a, b, c.
 +
\\[6pt]
 +
\operatorname{genus}~ x ~\operatorname{comprises}
 +
\\
 +
\operatorname{species}~ a, b, c.
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
& \bar{x} ~ \bar{a} ~ \bar{b} ~ \bar{c}
 +
\\
 +
\lor & x ~ a ~ \bar{b} ~ \bar{c}
 +
\\
 +
\lor & x ~ \bar{a} ~ b ~ \bar{c}
 +
\\
 +
\lor & x ~ \bar{a} ~ \bar{b} ~ c
 +
\end{matrix}</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
 +
|+ <math>\text{Table C.}~~\text{Dualing Interpretations}</math>
 +
|- style="background:#f0f0ff"
 +
| <math>\text{Graph}\!</math>
 +
| <math>\text{String}\!</math>
 +
| <math>\text{Existential}\!</math>
 +
| <math>\text{Entitative}\!</math>
 +
|-
 +
| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
 +
| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
 +
| <math>\operatorname{true}.</math>
 +
| <math>\operatorname{false}.</math>
 +
|-
 +
| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
 +
| <math>\texttt{(~)}</math>
 +
| <math>\operatorname{false}.</math>
 +
| <math>\operatorname{true}.</math>
 +
|-
 +
| height="100px" | [[Image:Cactus A Big.jpg|20px]]
 +
| <math>a\!</math>
 +
| <math>a.\!</math>
 +
| <math>a.\!</math>
 +
|-
 +
| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
 +
| <math>\texttt{(} a \texttt{)}</math>
 +
| <math>\lnot a</math>
 +
| <math>\lnot a</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}
 +
\operatorname{just~one}
 +
\\
 +
\operatorname{of}~ a, b, c
 +
\\
 +
\operatorname{is~false}.
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{not~just~one}
 +
\\
 +
\operatorname{of}~ a, b, c
 +
\\
 +
\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}
 +
\operatorname{just~one}
 +
\\
 +
\operatorname{of}~ a, b, c
 +
\\
 +
\operatorname{is~true}.
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{not~just~one}
 +
\\
 +
\operatorname{of}~ a, b, c
 +
\\
 +
\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}
 +
\operatorname{not~just~one}
 +
\\
 +
\operatorname{of}~ a, b, c
 +
\\
 +
\operatorname{is~false}.
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{just~one}
 +
\\
 +
\operatorname{of}~ a, b, c
 +
\\
 +
\operatorname{is~true}.
 +
\end{matrix}</math>
 +
|-
 +
| height="200px" | [[Image:Cactus (((A),(B),(C))) Big.jpg|65px]]
 +
| <math>\texttt{(((} a \texttt{),(} b \texttt{),(} c \texttt{)))}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{not~just~one}
 +
\\
 +
\operatorname{of}~ a, b, c
 +
\\
 +
\operatorname{is~true}.
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{just~one}
 +
\\
 +
\operatorname{of}~ a, b, c
 +
\\
 +
\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}
 +
\operatorname{partition}~ a
 +
\\
 +
\operatorname{into}~ b, c.
 +
\end{matrix}</math>
 +
| &nbsp;
 +
|-
 +
| height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|65px]]
 +
| <math>\texttt{(((} a \texttt{),} b \texttt{,} c \texttt{))}</math>
 +
| &nbsp;
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{partition}~ a
 +
\\
 +
\operatorname{into}~ b, c.
 +
\end{matrix}</math>
 +
|}
 +
 +
<br>

Latest revision as of 18:38, 2 December 2015

Work Area • Logical Cacti

Theme One Program — Logical Cacti
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

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 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.

Table 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.


\(\text{Table A.}~~\text{Existential Interpretation}\)
\(\text{Cactus Graph}\!\) \(\text{Cactus Expression}\!\) \(\text{Interpretation}\!\)
Cactus Node Big Fat.jpg \({}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}\) \(\operatorname{true}.\)
Cactus Spike Big Fat.jpg \(\texttt{(~)}\) \(\operatorname{false}.\)
Cactus A Big.jpg \(a\!\) \(a.\!\)
Cactus (A) Big.jpg \(\texttt{(} a \texttt{)}\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


Table 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.


\(\text{Table B.}~~\text{Entitative Interpretation}\)
\(\text{Cactus Graph}\!\) \(\text{Cactus Expression}\!\) \(\text{Interpretation}\!\)
Cactus Node Big Fat.jpg \({}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}\) \(\operatorname{false}.\)
Cactus Spike Big Fat.jpg \(\texttt{(~)}\) \(\operatorname{true}.\)
Cactus A Big.jpg \(a\!\) \(a.\!\)
Cactus (A) Big.jpg \(\texttt{(} a \texttt{)}\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


For the time being, the main things to take away from Tables 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:

1. The node connective joins a number of component cacti \(C_1, \ldots, C_k\) at a node:
  
    C_1 ... C_k
         @
2. The lobe connective joins a number of component cacti \(C_1, \ldots, C_k\) to a lobe:
  
    C_1 C_2   C_k
     o---o-...-o
      \       /
       \     /
        \   /
         \ /
          @

Table 15 summarizes the existential and entitative interpretations of the primitive cactus structures, in effect, the graphical constants and connectives. 

Table 15.  Existential & Entitative Interpretations of Cactus Structures
o-----------------o-----------------o-----------------o-----------------o
|  Cactus Graph   |  Cactus String  |  Existential    |   Entitative    |
|                 |                 | Interpretation  | Interpretation  |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|        @        |       " "       |      true       |      false      |
|                 |                 |                 |                 |
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

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 16 schematizes the two types of basic reductions in a purely formal, interpretation-independent fashion. 

Table 16.  Basic Reductions
o---------------------------------------o
|                                       |
|    C_1 ... C_k                        |
|         @         =         @         |
|                                       |
|    if and only if                     |
|                                       |
|    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

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 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 "\(\texttt{(~)}\)" in a certain way, calling "\(\texttt{(~)}\)" the controlled, moderated, or reflective negation operator of order 1, then adding another such operator for each finite \(k = 2, 3, \ldots .\)

In sum, these operators are symbolized by bracketed argument lists as follows: "\(\texttt{(~)}\)", "\(\texttt{(~,~)}\)", "\(\texttt{(~,~,~)}\)", …, 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. 

o---------------------------------------o
|                                       |
|                   o                   |
|                   |                   |
|                   @                   |
|                                       |
o---------------------------------------o
|                  ( )                  |
o---------------------------------------o

The rule of reduction for a lobe is: 

o---------------------------------------o
|                                       |
|  x_1   x_2   ...   x_k                |
|   o-----o--- ... ---o                 |
|    \               /                  |
|     \             /                   |
|      \           /                    |
|       \         /                     |
|        \       /                      |
|         \     /                       |
|          \   /                        |
|           \ /                         |
|            @      =      @            |
|                                       |
o---------------------------------------o

if and only if exactly one of the \(x_j\!\) is a spike.

In Ref Log, an expression of the form \(\texttt{((}~ e_1 ~\texttt{),(}~ e_2 ~\texttt{),(}~ \ldots ~\texttt{),(}~ e_k ~\texttt{))}\) expresses the fact that exactly one of the \(e_j\!\) 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): 

o---------------------------------------o
|                                       |
|  e_1   e_2   ...   e_k                |
|   o     o           o                 |
|   |     |           |                 |
|   o-----o--- ... ---o                 |
|    \               /                  |
|     \             /                   |
|      \           /                    |
|       \         /                     |
|        \       /                      |
|         \     /                       |
|          \   /                        |
|           \ /                         |
|            @                          |
|                                       |
o---------------------------------------o

o---------------------------------------o
|                                       |
| ( x1, x2, ..., xk )  =  [blank]       |
|                                       |
| iff                                   |
|                                       |
| Just one of the arguments             |
| x1, x2, ..., xk  =  ()                |
|                                       |
o---------------------------------------o

The interpretation of these operators, read as assertions about the values of their listed arguments, is as follows:

Existential Interpretation: Just one of the k argument is false.
Entitative Interpretation: Not just one of the k arguments is true.

Tables

\(\text{Table 1.}~~\text{Syntax and Semantics of a Calculus for Propositional Logic}\)
\(\text{Graph}\!\) \(\text{Expression}\!\) \(\text{Interpretation}\!\) \(\text{Other Notations}\!\)
Cactus Node Big Fat.jpg \(~\) \(\operatorname{true}\) \(1\!\)
Cactus Spike Big Fat.jpg \(\texttt{(~)}\) \(\operatorname{false}\) \(0\!\)
Cactus A Big.jpg \(a\!\) \(a\!\) \(a\!\)
Cactus (A) Big.jpg \(\texttt{(} a \texttt{)}\) \(\operatorname{not}~ a\) \(\lnot a \quad \bar{a} \quad \tilde{a} \quad a^\prime\)
Cactus ABC Big.jpg \(a ~ b ~ c\) \(a ~\operatorname{and}~ b ~\operatorname{and}~ c\) \(a \land b \land c\)
Cactus ((A)(B)(C)) Big.jpg \(\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}\) \(a ~\operatorname{or}~ b ~\operatorname{or}~ c\) \(a \lor b \lor c\)
Cactus (A(B)) Big.jpg \(\texttt{(} a \texttt{(} b \texttt{))}\)

\(\begin{matrix} a ~\operatorname{implies}~ b \\[6pt] \operatorname{if}~ a ~\operatorname{then}~ b \end{matrix}\)

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

\(\begin{matrix} a ~\operatorname{not~equal~to}~ b \\[6pt] a ~\operatorname{exclusive~or}~ b \end{matrix}\)

\(\begin{matrix} a \neq b \\[6pt] a + b \end{matrix}\)

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

\(\begin{matrix} a ~\operatorname{is~equal~to}~ b \\[6pt] a ~\operatorname{if~and~only~if}~ b \end{matrix}\)

\(\begin{matrix} a = b \\[6pt] a \Leftrightarrow b \end{matrix}\)

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

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

\(\begin{matrix} & \bar{a} ~ b ~ c \\ \lor & a ~ \bar{b} ~ c \\ \lor & a ~ b ~ \bar{c} \end{matrix}\)

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

\(\begin{matrix} \operatorname{just~one~of} \\ a, b, c \\ \operatorname{is~true}. \\[6pt] \operatorname{partition~all} \\ \operatorname{into}~ a, b, c. \end{matrix}\)

\(\begin{matrix} & a ~ \bar{b} ~ \bar{c} \\ \lor & \bar{a} ~ b ~ \bar{c} \\ \lor & \bar{a} ~ \bar{b} ~ c \end{matrix}\)

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

\(\begin{matrix} \operatorname{oddly~many~of} \\ a, b, c \\ \operatorname{are~true}. \end{matrix}\)

\(a + b + c\!\)


\(\begin{matrix} & a ~ b ~ c \\ \lor & a ~ \bar{b} ~ \bar{c} \\ \lor & \bar{a} ~ b ~ \bar{c} \\ \lor & \bar{a} ~ \bar{b} ~ c \end{matrix}\)

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

\(\begin{matrix} \operatorname{partition}~ x \\ \operatorname{into}~ a, b, c. \\[6pt] \operatorname{genus}~ x ~\operatorname{comprises} \\ \operatorname{species}~ a, b, c. \end{matrix}\)

\(\begin{matrix} & \bar{x} ~ \bar{a} ~ \bar{b} ~ \bar{c} \\ \lor & x ~ a ~ \bar{b} ~ \bar{c} \\ \lor & x ~ \bar{a} ~ b ~ \bar{c} \\ \lor & x ~ \bar{a} ~ \bar{b} ~ c \end{matrix}\)


\(\text{Table C.}~~\text{Dualing Interpretations}\)
\(\text{Graph}\!\) \(\text{String}\!\) \(\text{Existential}\!\) \(\text{Entitative}\!\)
Cactus Node Big Fat.jpg \({}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}\) \(\operatorname{true}.\) \(\operatorname{false}.\)
Cactus Spike Big Fat.jpg \(\texttt{(~)}\) \(\operatorname{false}.\) \(\operatorname{true}.\)
Cactus A Big.jpg \(a\!\) \(a.\!\) \(a.\!\)
Cactus (A) Big.jpg \(\texttt{(} a \texttt{)}\) \(\lnot a\) \(\lnot a\)
Cactus ABC Big.jpg \(a~b~c\) \(a \land b \land c\) \(a \lor b \lor c\)
Cactus ((A)(B)(C)) Big.jpg \(\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}\) \(a \lor b \lor c\) \(a \land b \land c\)
Cactus (A(B)) Big.jpg \(\texttt{(} a \texttt{(} b \texttt{))}\) \(a \Rightarrow b\)  
Cactus (A)B Big.jpg \(\texttt{(} a \texttt{)} b\)   \(a \Rightarrow b\)
Cactus (A,B) Big.jpg \(\texttt{(} a \texttt{,} b \texttt{)}\) \(a \neq b\) \(a = b\!\)
Cactus ((A,B)) Big.jpg \(\texttt{((} a \texttt{,} b \texttt{))}\) \(a = b\!\) \(a \neq b\!\)
Cactus (A,B,C) Big.jpg \(\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}\)

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

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

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

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

\(\begin{matrix} \operatorname{not~just~one} \\ \operatorname{of}~ a, b, c \\ \operatorname{is~false}. \end{matrix}\)

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

\(\begin{matrix} \operatorname{not~just~one} \\ \operatorname{of}~ a, b, c \\ \operatorname{is~false}. \end{matrix}\)

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

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

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

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

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

\(\begin{matrix} \operatorname{partition}~ a \\ \operatorname{into}~ b, c. \end{matrix}\)

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

\(\begin{matrix} \operatorname{partition}~ a \\ \operatorname{into}~ b, c. \end{matrix}\)


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