Differential Logic • 16

Propositions and Tacit Extensions

Now that we’ve introduced the field picture as an aid to visualizing propositions and their analytic series, a pleasing way to picture the relationship of a proposition $f : X \to \mathbb{B}$ to its enlargement or shift map $\mathrm{E}f : \mathrm{E}X \to \mathbb{B}$ and its difference map $\mathrm{D}f : \mathrm{E}X \to \mathbb{B}$ can now be drawn.

To illustrate the possibilities, let’s return to the differential analysis of the conjunctive proposition $f(p, q) = pq$ and give its development a slightly different twist at the appropriate point.

The proposition $pq : X \to \mathbb{B}$ is shown again in the venn diagram below. In the field picture it may be seen as a scalar field — analogous to a potential hill in physics but in logic amounting to a potential plateau — where the shaded region indicates an elevation of 1 and the unshaded region indicates an elevation of 0.

$\text{Proposition}~ pq : X \to \mathbb{B}$

Given a proposition $f : X \to \mathbb{B},$ the tacit extension of $f$ to $\mathrm{E}X$ is denoted $\boldsymbol\varepsilon f : \mathrm{E}X \to \mathbb{B}$ and defined by the equation $\boldsymbol\varepsilon f = f,$ so it’s really just the same proposition residing in a bigger universe. Tacit extensions formalize the intuitive idea that a function on a given set of variables can be extended to a function on a superset of those variables in such a way that the new function obeys the same constraints on the old variables, with a “don’t care” condition on the new variables.

The tacit extension of the scalar field $pq : X \to \mathbb{B}$ to the differential field $\boldsymbol\varepsilon (pq) : \mathrm{E}X \to \mathbb{B}$ is shown in the following venn diagram.

$\text{Tacit Extension}~ \boldsymbol\varepsilon (pq) : \mathrm{E}X \to \mathbb{B}$

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