Work J

Syntactic Correspondences

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

Operation Tables

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

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

Interpretation Rules

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