Differential Logic • 10

It’s been a while, so let’s review …

Tables A1 and A2 showed two ways of organizing the sixteen boolean functions or propositional forms on two variables, as expressed in several notations.  For ease of reference, here are fresh copies of those Tables.

Table A1.  Propositional Forms on Two Variables

Table A2.  Propositional Forms on Two Variables

We took as our first example the boolean function $f_{8}(p, q) = pq$ corresponding to the logical conjunction $p \land q$ and examined how the differential operators $\mathrm{E}$ and $\mathrm{D}$ act on $f_{8}.$  Each differential operator takes a boolean function of two variables $f_{8}(p, q)$ and gives back a boolean function of four variables, $\mathrm{E}f_{8}(p, q, \mathrm{d}p, \mathrm{d}q)$ and $\mathrm{D}f_{8}(p, q, \mathrm{d}p, \mathrm{d}q),$ respectively.

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 function $f$ in the above tables.

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