## Differential Propositional Calculus • Discussion 5

HR:
1. I think I like very much your Cactus Graphs.  Meaning that I am in the process of understanding them, and finding it much better not to have to draw circles, but lines.
2. Less easy for me is the differential calculus.  Where is the consistency between $\texttt{(} x \texttt{,} y \texttt{)}$ and $\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}?$  $\texttt{(} x \texttt{,} y \texttt{)}$ means that $x$ and $y$ are not equal and $\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}$ means that one of them is false.  Unequality and truth/falsity for me are two concepts so different I cannot think them together or see a consistency between them.
3. What about $\texttt{(} w \texttt{,} x \texttt{,} y \texttt{,} z \texttt{)}?$
MB:
So, if I want to transform a circle into a line I have to use a function $f : \mathbb{B}^n \to \mathbb{B}?$
This is the base of temporal logic?  I’m using $f : \mathbb{N}^n \to \mathbb{N}.$

Dear Mauro,

If I understand what Helmut is saying about “circles” and “lines”, he is talking about the passage from forms of enclosure on plane sheets of paper — such as those used by Peirce and Spencer Brown — to their topological duals in the form of rooted trees.  There is more discussion of this transformation at the following sites.

This is the first step in the process of converting planar maps to graph-theoretic data structures.  Further transformations take us from trees to the more general class of cactus graphs, which implement a highly efficient family of logical primitives called minimal negation operators.  These are described in the following article.

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