## Differential Logic • 7

### Differential Expansions of Propositions

#### Panoptic View • Enlargement Maps

The enlargement or shift operator $\mathrm{E}$ exhibits a wealth of interesting and useful properties in its own right, so it pays to examine a few of the more salient features playing out on the surface of our initial example, $f(p, q) = pq.$

A suitably generic definition of the extended universe of discourse is afforded by the following set-up.

$\begin{array}{cccl} \text{Let} & X & = & X_1 \times \ldots \times X_k. \\[6pt] \text{Let} & \mathrm{d}X & = & \mathrm{d}X_1 \times \ldots \times \mathrm{d}X_k. \\[6pt] \text{Then} & \mathrm{E}X & = & X \times \mathrm{d}X \\[6pt] & & = & X_1 \times \ldots \times X_k ~\times~ \mathrm{d}X_1 \times \ldots \times \mathrm{d}X_k \end{array}$

For a proposition of the form $f : X_1 \times \ldots \times X_k \to \mathbb{B},$ the (first order) enlargement of $f$ is the proposition $\mathrm{E}f : \mathrm{E}X \to \mathbb{B}$ defined by the following equation.

$\mathrm{E}f(x_1, \ldots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k) ~=~ f(x_1 + \mathrm{d}x_1, \ldots, x_k + \mathrm{d}x_k) ~=~ f(\texttt{(} x_1 \texttt{,} \mathrm{d}x_1 \texttt{)}, \ldots, \texttt{(} x_k \texttt{,} \mathrm{d}x_k \texttt{)})$

The differential variables $\mathrm{d}x_j$ are boolean variables of the same type as the ordinary variables $x_j.$  Although it is conventional to distinguish the (first order) differential variables with the operational prefix ${}^{\backprime\backprime} \mathrm{d} {}^{\prime\prime}$ this way of notating differential variables is entirely optional.  It is their existence in particular relations to the initial variables, not their names, which defines them as differential variables.

In the example of logical conjunction, $f(p, q) = pq,$ the enlargement $\mathrm{E}f$ is formulated as follows.

$\begin{matrix} \mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q) & = & (p + \mathrm{d}p)(q + \mathrm{d}q) & = & \texttt{(} p \texttt{,} \mathrm{d}p \texttt{)(} q \texttt{,} \mathrm{d}q \texttt{)} \end{matrix}$

Given that this expression uses nothing more than the boolean ring operations of addition and multiplication, it is permissible to “multiply things out” in the usual manner to arrive at the following result.

$\begin{matrix} \mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q) & = & p~q & + & p~\mathrm{d}q & + & q~\mathrm{d}p & + & \mathrm{d}p~\mathrm{d}q \end{matrix}$

To understand what the enlarged or shifted proposition means in logical terms, it serves to go back and analyze the above expression for $\mathrm{E}f$ in the same way we did for $\mathrm{D}f.$  To that end, the value of $\mathrm{E}f_x$ at each $x \in X$ may be computed in graphical fashion as shown below.

Collating the data of this analysis yields a boolean expansion or disjunctive normal form (DNF) equivalent to the enlarged proposition $\mathrm{E}f.$

$\begin{matrix} \mathrm{E}f & = & pq \cdot \mathrm{E}f_{pq} & + & p(q) \cdot \mathrm{E}f_{p(q)} & + & (p)q \cdot \mathrm{E}f_{(p)q} & + & (p)(q) \cdot \mathrm{E}f_{(p)(q)} \end{matrix}$

Here is a summary of the result, illustrated by means of a digraph picture, where the “no change” element $\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}$ is drawn as a loop at the point $p~q.$

$\begin{array}{rcccccc} f & = & p & \cdot & q \\[4pt] \mathrm{E}f & = & p & \cdot & q & \cdot & \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{(} \mathrm{d}p \texttt{)} \texttt{~} \mathrm{d}q \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \mathrm{d}p \texttt{~} \texttt{(} \mathrm{d}q \texttt{)} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \mathrm{d}p \texttt{~~} \mathrm{d}q \end{array}$

We may understand the enlarged proposition $\mathrm{E}f$ as telling us all the ways of reaching a model of the proposition $f$ from the points of the universe $X.$

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