Work J

Syntactic Correspondences

\text{Form} \text{String} \text{Graph}
Form (a) \texttt{(} a \texttt{)} Cactus (a)
Form (a,b) \texttt{(} a \texttt{,} b \texttt{)} Cactus (a,b)
Form (a,b,c) \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)} Cactus (a,b,c)

Operation Tables

\begin{array}{|*{3}{c}||c|}  \multicolumn{4}{c}{\text{Formal Operation Table} ~ \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}} \\[4pt]  \hline  a & b & c & \texttt{(}a\texttt{,}b\texttt{,}c\texttt{)} \\  \hline\hline  \texttt{Space}&\texttt{Space}&\texttt{Space}&\texttt{Cross} \\  \texttt{Space}&\texttt{Space}&\texttt{Cross}&\texttt{Space} \\  \texttt{Space}&\texttt{Cross}&\texttt{Space}&\texttt{Space} \\  \texttt{Space}&\texttt{Cross}&\texttt{Cross}&\texttt{Cross} \\  \hline  \texttt{Cross}&\texttt{Space}&\texttt{Space}&\texttt{Space} \\  \texttt{Cross}&\texttt{Space}&\texttt{Cross}&\texttt{Cross} \\  \texttt{Cross}&\texttt{Cross}&\texttt{Space}&\texttt{Cross} \\  \texttt{Cross}&\texttt{Cross}&\texttt{Cross}&\texttt{Cross} \\  \hline  \end{array}

\begin{array}{|*{3}{c}||c|}  \multicolumn{4}{c}{\text{Formal Operation Table} ~ \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}} \\[4pt]  \hline  \quad a \quad & \quad b \quad & \quad c \quad & \texttt{(}a\texttt{,}b\texttt{,}c\texttt{)} \\  \hline\hline  \texttt{o} & \texttt{o} & \texttt{o} & \texttt{|} \\  \texttt{o} & \texttt{o} & \texttt{|} & \texttt{o} \\  \texttt{o} & \texttt{|} & \texttt{o} & \texttt{o} \\  \texttt{o} & \texttt{|} & \texttt{|} & \texttt{|} \\  \hline  \texttt{|} & \texttt{o} & \texttt{o} & \texttt{o} \\  \texttt{|} & \texttt{o} & \texttt{|} & \texttt{|} \\  \texttt{|} & \texttt{|} & \texttt{o} & \texttt{|} \\  \texttt{|} & \texttt{|} & \texttt{|} & \texttt{|} \\  \hline  \end{array}

Evaluation Rules

Node Evaluation Rule

Lobe Evaluation Rule

Interpretation Rules

\text{Logical Interpretations of Cactus Structures}
\text{Graph} \text{Expression} \begin{matrix}  \text{Entitative} \\ \text{Interpretation}  \end{matrix} \begin{matrix}  \text{Existential} \\ \text{Interpretation}  \end{matrix}
“ ” ~ \mathrm{false} \mathrm{true}
( ) \texttt{(} ~ \texttt{)} \mathrm{true} \mathrm{false}
node connective x_1 \ldots x_k x_1 \lor \ldots \lor x_k x_1 \land \ldots \land x_k
lobe connective \texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)} \begin{matrix}  \text{not just one of}  \\[6px]  x_1, \ldots, x_k  \\[6px]  \text{is true}  \end{matrix} \begin{matrix}  \text{just one of}  \\[6px]  x_1, \ldots, x_k  \\[6px]  \text{is false}  \end{matrix}

Logical Interpretations of Cactus Structures