## Differential Logic • 4

### Differential Expansions of Propositions

#### Bird’s Eye View

An efficient calculus for the realm of logic represented by boolean functions and elementary propositions makes it feasible to compute the finite differences and the differentials of those functions and propositions.

For example, consider a proposition of the form ${}^{\backprime\backprime} \, p ~\mathrm{and}~ q \, {}^{\prime\prime}$ graphed as two letters attached to a root node, as shown below. Written as a string, this is just the concatenation $p~q$.

The proposition $pq$ may be taken as a boolean function $f(p, q)$ having the abstract type $f : \mathbb{B} \times \mathbb{B} \to \mathbb{B},$ where $\mathbb{B} = \{ 0, 1 \}$ is read in such a way that $0$ means $\mathrm{false}$ and $1$ means $\mathrm{true}.$

Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition $pq$ is true, as shown in the following Figure. Now ask yourself:  What is the value of the proposition $pq$ at a distance of $\mathrm{d}p$ and $\mathrm{d}q$ from the cell $pq$ where you are standing?

Don’t think about it — just compute: The cactus formula $\texttt{(} p \texttt{,} \mathrm{d}p \texttt{)(} q \texttt{,} \mathrm{d}q \texttt{)}$ and its corresponding graph arise by replacing $p$ with $p + \mathrm{d}p$ and $q$ with $q + \mathrm{d}q$ in the boolean product or logical conjunction $pq$ and writing the result in the two dialects of cactus syntax.  This follows because the boolean sum $p + \mathrm{d}p$ is equivalent to the logical operation of exclusive disjunction, which parses to a cactus graph of the following form. Next question:  What is the difference between the value of the proposition $pq$ over there, at a distance of $\mathrm{d}p$ and $\mathrm{d}q$ from where you are standing, and the value of the proposition $pq$ where you are, all expressed in the form of a general formula, of course?  The answer takes the following form. There is one thing I ought to mention at this point:  Computed over $\mathbb{B},$ plus and minus are identical operations.  This will make the relation between the differential and the integral parts of the appropriate calculus slightly stranger than usual, but we will get into that later.

Last question, for now:  What is the value of this expression from your current standpoint, that is, evaluated at the point where $pq$ is true?  Well, replacing $p$ with $1$ and $q$ with $1$ in the cactus graph amounts to erasing the labels $p$ and $q,$ as shown below. And this is equivalent to the following graph: We have just met with the fact that the differential of the and is the or of the differentials. $\begin{matrix} p ~\mathrm{and}~ q & \quad & \xrightarrow{\quad\mathrm{Diff}\quad} & \quad & \mathrm{d}p ~\mathrm{or}~ \mathrm{d}q \end{matrix}$ It will be necessary to develop a more refined analysis of that statement directly, but that is roughly the nub of it.

If the form of the above statement reminds you of De Morgan’s rule, it is no accident, as differentiation and negation turn out to be closely related operations.  Indeed, one can find discussions of logical difference calculus in the Boole–De Morgan correspondence and Peirce also made use of differential operators in a logical context, but the exploration of these ideas has been hampered by a number of factors, not the least of which has been the lack of a syntax adequate to handle the complexity of expressions evolving in the process.

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