## Animated Logical Graphs • 18

We had been contemplating the penultimately simple algebraic expression ${}^{\backprime\backprime} \texttt{(} a \texttt{)} {}^{\prime\prime}$ as a name for a set of arithmetic expressions, namely, $\texttt{(} a \texttt{)} = \{ \,\texttt{()}\, , \,\texttt{(())}\, \},$ taking the equality sign in the appropriate sense.

Then we asked the corresponding question about the operator ${}^{\backprime\backprime} \texttt{(} ~ \texttt{)} {}^{\prime\prime}.$  The above selection of arithmetic expressions is what it means to contemplate the absence or presence of the operand ${}^{\backprime\backprime} a {}^{\prime\prime}$ in the algebraic expression ${}^{\backprime\backprime} \texttt{(} a \texttt{)} {}^{\prime\prime}.$  But what would it mean to contemplate the absence or presence of the operator ${}^{\backprime\backprime} \texttt{(} ~ \texttt{)} {}^{\prime\prime}$ in the algebraic expression ${}^{\backprime\backprime} \texttt{(} a \texttt{)} {}^{\prime\prime}?$

Clearly, a variation between the absence and the presence of the operator ${}^{\backprime\backprime} \texttt{(} ~ \texttt{)} {}^{\prime\prime}$ in the algebraic expression ${}^{\backprime\backprime} \texttt{(} a \texttt{)} {}^{\prime\prime}$ refers to a variation between the algebraic expressions ${}^{\backprime\backprime} a {}^{\prime\prime}$ and ${}^{\backprime\backprime} \texttt{(} a \texttt{)} {}^{\prime\prime},$ respectively, somewhat as pictured below:

But how shall we signify such variations in a coherent calculus?

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