## Animated Logical Graphs • 25

Let’s examine the formal operation table for the third in our series of reflective forms to see if we can elicit the general pattern:

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

Or, thinking in terms of the corresponding cactus graphs, writing ${}^{\backprime\backprime} \texttt{o} {}^{\prime\prime}$ for a blank node and ${}^{\backprime\backprime} \texttt{|} {}^{\prime\prime}$ for a terminal edge, we get the following Table:

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

Evidently, the rule is that ${}^{\backprime\backprime} \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)} {}^{\prime\prime}$ denotes the value denoted by ${}^{\backprime\backprime} \texttt{o} {}^{\prime\prime}$ if and only if exactly one of the variables $a, b, c$ has the value denoted by ${}^{\backprime\backprime} \texttt{|} {}^{\prime\prime},$ otherwise ${}^{\backprime\backprime} \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)} {}^{\prime\prime}$ denotes the value denoted by ${}^{\backprime\backprime} \texttt{|} {}^{\prime\prime}.$  Examining the whole series of reflective forms shows this is the general rule.

• In the Entitative Interpretation $(\mathrm{En}),$ where $\texttt{o}$ = false and $\texttt{|}$ = true, ${}^{\backprime\backprime} \texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)} {}^{\prime\prime}$ translates as “not just one of the $x_j$ is true”.
• In the Existential Interpretation $(\mathrm{Ex}),$ where $\texttt{o}$ = true and $\texttt{|}$ = false, ${}^{\backprime\backprime} \texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)} {}^{\prime\prime}$ translates as “just one of the $x_j$ is not true”.

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