# Differential Logic : Sketch 3

**Author: Jon Awbrey**

It would be good to summarize, in rough but intuitive terms, the outlook on differential logic that we have reached so far.

We have been considering a class of operators on universes of discourse, each of which takes us from considering one universe of discourse, to considering a larger universe of discourse,

Each of these operators, in general terms having the form acts on each proposition of the source universe to produce a proposition of the target universe

The two main operators that we have worked with up to this point are the enlargement operator and the difference operator

and take a proposition in that is, a proposition that is said to be *about* the subject matter of and produce the extended propositions which may be interpreted as being about specified collections of changes that might occur in

Here we have need of visual representations, some array of concrete pictures to anchor our more earthy intuitions and to help us keep our wits about us before we try to climb any higher into the ever more rarefied air of abstractions.

One good picture comes to us by way of the “field” concept. Given a space a *field* of a specified type over is formed by assigning to each point of an object of type If that sounds like the same thing as a function from to the space of things of type it is, but it does seems to help to vary the mental pictures and the figures of speech that naturally spring to mind within these fertile fields.

In the field picture a proposition becomes a “scalar” field, that is, a field of values in or a *field of true-false indications*.

Let us take a moment to view an old proposition in this new light, for example, the conjunction that is depicted in Figure 1.

o-------------------------------------------------o | X | | | | o-------------o o-------------o | | / \ / \ | | / o \ | | / /`\ \ | | / /```\ \ | | o o`````o o | | | |`````| | | | | U |`````| V | | | | |`````| | | | o o`````o o | | \ \```/ / | | \ \`/ / | | \ o / | | \ / \ / | | o-------------o o-------------o | | | | | o-------------------------------------------------o | f = u v | o-------------------------------------------------o Figure 1. Conjunction uv : X -> B |

Each of the operators takes us from considering propositions here viewed as “scalar fields” over to considering the corresponding “differential fields” over analogous to what are usually called *vector fields* over

The structure of these differential fields may be described this way. To each point of there is attached an object of the following type, a proposition about changes in that is, a proposition In this setting, if is the universe that is generated by the set of coordinate propositions then is the differential universe that is generated by the set of differential propositions These differential propositions may be interpreted as indicating “change in ” and “change in ”, respectively.

A differential operator of the first order sort that we have been considering, takes a proposition and gives back a differential proposition

In the field view we see the proposition as a scalar field and we see the differential proposition as a vector field, specifically, a field of propositions about contemplated changes in

The field of changes produced by on is shown in Figure 2.

o-------------------------------------------------o | X | | | | o-------------o o-------------o | | / \ / \ | | / U o V \ | | / /`\ \ | | / /```\ \ | | o o.->-.o o | | | u(v)(du)dv |`\`/`| (u)v du(dv) | | | | o---------------|->o<-|---------------o | | | | |``^``| | | | o o``|``o o | | \ \`|`/ / | | \ \|/ / | | \ o / | | \ /|\ / | | o-------------o | o-------------o | | | | | | | | | | | o | | (u)(v) du dv | | | o-------------------------------------------------o | f = u v | o-------------------------------------------------o | | | Ef = u v (du)(dv) | | | | + u (v) (du) dv | | | | + (u) v du (dv) | | | | + (u)(v) du dv | | | o-------------------------------------------------o Figure 2. Enlargement E[uv] : EX -> B |

The differential field specifies the changes that need to be made from each point of in order to reach one of the models of the proposition that is, in order to satisfy the proposition

The field of changes produced by on is shown in Figure 3.

o-------------------------------------------------o | X | | | | o-------------o o-------------o | | / \ / \ | | / U o V \ | | / /`\ \ | | / /```\ \ | | o o`````o o | | | (du)dv |`````| du(dv) | | | | o<--------------|->o<-|-------------->o | | | | |``^``| | | | o o``|``o o | | \ \`|`/ / | | \ \|/ / | | \ o / | | \ /|\ / | | o-------------o | o-------------o | | | | | | | | v | | o | | du dv | | | o-------------------------------------------------o | f = u v | o-------------------------------------------------o | | | Df = u v ((du)(dv)) | | | | + u (v) (du) dv | | | | + (u) v du (dv) | | | | + (u)(v) du dv | | | o-------------------------------------------------o Figure 3. Difference D[uv] : EX -> B |

The differential field specifies the changes that need to be made from each point of in order to change the value of the proposition

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