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 and act on that set.  There being some advantage to singling out the enlargement or shift operator in its own right, we’ll begin by computing for each of the functions

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 and contemplated the value of that proposition in various other states, as determined by the differential propositions and describing the steps we might take to change our state.

Those thoughts led us from the boolean function of two variables to the boolean function of four variables as shown in the entry for in the first three columns of Table A3.

Let’s catch a breath here and discuss the full Table next time.

Resources

• Logic Syllabus
• Survey of Differential Logic

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