## Animated Logical Graphs • 26

This post and the next wrap up the Themes and Variations section of my speculation on Futures Of Logical Graphs.  I made an effort to “show my work”, reviewing the steps I took to arrive at the present perspective on logical graphs, whistling past the least productive of the blind alleys, cul-de-sacs, detours, and forking paths I explored along the way.  It can be useful to tell the story that way, partly because others may find things I missed down those roads, but it does call for a recap of the main ideas I would like readers to take away.

Partly through my reflections on Peirce’s use of operator variables I was led to what I called the reflective extension of logical graphs, or what I now call the “cactus language”, after its principal graph-theoretic data structure.  This graphical formal language arises from generalizing the negation operator ${}^{\backprime\backprime} \texttt{(} ~ \texttt{)} {}^{\prime\prime}$ in a particular direction, treating ${}^{\backprime\backprime} \texttt{(} ~ \texttt{)} {}^{\prime\prime}$ as the controlled, moderated, or reflective negation operator of order 1, and adding another operator for each integer parameter greater than 1.  This family of operators is symbolized by bracketed argument lists of the forms ${}^{\backprime\backprime} \texttt{(} ~ \texttt{)} {}^{\prime\prime},$ ${}^{\backprime\backprime} \texttt{(} ~ \texttt{,} ~ \texttt{)} {}^{\prime\prime},$ ${}^{\backprime\backprime} \texttt{(} ~ \texttt{,} ~ \texttt{,} ~ \texttt{)} {}^{\prime\prime},$ and so on, where the number of places is the order of the reflective negation operator in question.

Two rules suffice for evaluating cactus graphs:

• The rule for evaluating a $k$-node operator, corresponding to an expression of the form ${}^{\backprime\backprime} x_1 x_2 \ldots x_{k-1} x_k {}^{\prime\prime},$ is as follows:

• The rule for evaluating a $k$-lobe operator, corresponding to an expression of the form ${}^{\backprime\backprime} \texttt{(} x_1 \texttt{,} x_2 \texttt{,} \ldots \texttt{,} x_{k-1} \texttt{,} x_k \texttt{)} {}^{\prime\prime},$ is as follows:

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