## Frankl, My Dear : 6

Figure 3 shows the eight terms of the enlarged proposition $\mathrm{E}f$ as arcs, arrows, or directed edges in the venn diagram of the original proposition $f(p, q, r) = pqr.$ Each term of the enlargement $\mathrm{E}f$ corresponds to an arc into the cell where $f$ is true from one of the eight cells of the venn diagram.

For ease of reference, here is the expansion of $\mathrm{E}f$ from the previous post: $\mathrm{E}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =$ $\begin{smallmatrix} & p q r \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q \texttt{(} r \texttt{)} \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r & + & p \texttt{(} q \texttt{)} r \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} & + & p \texttt{(} q \texttt{)(} r \texttt{)} \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \, \mathrm{d}r \\[4pt] + & \texttt{(} p \texttt{)} q r \,\cdot\, \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)} q \texttt{(} r \texttt{)} \,\cdot\, \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r & + & \texttt{(} p \texttt{)(} q \texttt{)} r \,\cdot\, \mathrm{d}p \, \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \,\cdot\, \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r \end{smallmatrix}$

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

• The term $p q r \cdot \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}$ is shown as a looped arc starting in the cell where $p q r$ 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 $\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \cdot \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r$ is shown as an arc from the cell where $\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)}$ is true to the cell where $p q r$ 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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