Animated Logical Graphs • 18

Posted on July 10, 2019 by Jon Awbrey

Last time we contemplated 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.

Cactus Graph Equation (a) = {(),(())}

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 arithmetic constant {}^{\backprime\backprime} \texttt{(} ~ \texttt{)} {}^{\prime\prime} in the place 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}?

Evidently, 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 expression {}^{\backprime\backprime} a {}^{\prime\prime} and the algebraic expression {}^{\backprime\backprime} \texttt{(} a \texttt{)} {}^{\prime\prime}, somewhat as pictured below.

Cactus Graph Equation ¿a? = {a,(a)}

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

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