## Animated Logical Graphs • 22

The step of controlled reflection we just took can be repeated at will, as suggested by the following series of forms: Written inline, we have the series ${}^{\backprime\backprime} \texttt{(} a \texttt{)} {}^{\prime\prime},$ ${}^{\backprime\backprime} \texttt{(} a \texttt{,} b \texttt{)} {}^{\prime\prime},$ ${}^{\backprime\backprime} \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)} {}^{\prime\prime},$ ${}^{\backprime\backprime} \texttt{(} a \texttt{,} b \texttt{,} c \texttt{,} d \texttt{)} {}^{\prime\prime},$ and so on, whose general form is ${}^{\backprime\backprime} \texttt{(} x_1 \texttt{,} x_2 \texttt{,} \ldots \texttt{,} x_k \texttt{)} {}^{\prime\prime}.$  With this move we have passed beyond the graph-theoretical form of rooted trees to what graph theorists know as rooted cacti.

I will discuss this cactus language and its logical interpretations next.

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