Differential Logic • 11

Transforms Expanded over Ordinary and Differential Variables

As promised last time, in the next several posts we’ll extend our scope to the full set of boolean functions on two variables and examine how the differential operators $\mathrm{E}$ and $\mathrm{D}$ act on that set. There being some advantage to singling out the enlargement or shift operator $\mathrm{E}$ in its own right, we’ll begin by computing $\mathrm{E}f$ for each of the functions $f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.$

Enlargement Map Expanded over Ordinary Variables

We first encountered the shift operator when we imagined ourselves being in a state described by the truth of a certain proposition and contemplated the value of that proposition in various other states, as determined by a collection of differential propositions describing the steps we might take to change our state.

Restated in terms of our initial example, we imagined ourselves being in a state described by the truth of the proposition $pq$ and contemplated the value of that proposition in various other states, as determined by the differential propositions $\mathrm{d}p$ and $\mathrm{d}q$ describing the steps we might take to change our state.

Those thoughts led us from the boolean function of two variables $f_{8}(p, q) = pq$ to the boolean function of four variables $\mathrm{E}f_{8}(p, q, \mathrm{d}p, \mathrm{d}q) = \texttt{(} p \texttt{,} \mathrm{d}p \texttt{)(} q \texttt{,} \mathrm{d}q \texttt{)},$ as shown in the entry for $f_{8}$ in the first three columns of Table A3.