## Frankl, My Dear : 8

Figure 4 shows the eight terms of the tacit extension $\boldsymbol\varepsilon f$ as arcs, arrows, or directed edges in the venn diagram of the original proposition $f(p, q, r) = pqr.$ Each term of the tacit extension $\boldsymbol\varepsilon f$ corresponds to an arc that starts from the cell where $f$ is true and ends in one of the eight cells of the venn diagram.

For ease of reference, here is the expansion of $\boldsymbol\varepsilon f$ from the previous post: $\boldsymbol\varepsilon f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =$ $\begin{array}{*{8}{l}} & p q r ~ \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \\[4pt] + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \end{array}$

Two examples suffice to convey the general idea of the extended venn diagram:

• The term $pqr \cdot \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}$ is shown as a looped arc starting in the cell where $pqr$ is true and returning back to it. The differential factor $\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}$ corresponds to the fact that the arc crosses no logical feature boundaries from its source to its target.
• The term $pqr \cdot \mathrm{d}p \; \mathrm{d}q \, \mathrm{d}r$ is shown as an arc going from the cell where $pqr$ is true to the cell where $\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)}$ is true. The differential factor $\mathrm{d}p \; \mathrm{d}q \, \mathrm{d}r$ corresponds to the fact that the arc crosses all three logical feature boundaries from its source to its target.

To be continued …

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