## Differential Propositional Calculus • Discussion 6

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 $\texttt{(} w \texttt{,} x \texttt{,} y \texttt{,} z \texttt{)}?$

Dear Helmut,

Table 1 shows the cactus graphs, the corresponding cactus expressions in “traversal string” or plain text form, their logical meanings under the “existential interpretation”, and their translations into conventional notations for a number of common propositional forms.  I’ll change variables to $\{ x, a, b, c \}$ instead of $\{ w, x, y, z \}$ at this point simply because I’ve already got a Table like that on hand.

As far as the consistency between $\texttt{(} a \texttt{,} b \texttt{)}$ and $\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}$ goes, that’s easy enough to see — if exactly one of two boolean variables is false then the two must have different values.

Out of time for today, so I’ll get to the rest of your questions next time.

Table 1.  Syntax and Semantics of a Calculus for Propositional Logic

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