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Justifying a notion of approximation is a little more involved in general, and especially in these discrete logical spaces, than it would be expedient for people in a hurry to tangle with right now.  I will just say that there are ''naive'' or ''obvious'' notions and there are ''sophisticated'' or ''subtle'' notions that we might choose among.  The later would engage us in trying to construct proper logical analogues of Lie derivatives, and so let's save that for when we have become subtle or sophisticated or both.  Against or toward that day, as you wish, let's begin with an option in plain view.

Justifying a notion of approximation is a little more

involved in general, and especially in these discrete

logical spaces, than it would be expedient for people

in a hurry to tangle with right now.  I will just say

that there are "naive" or "obvious" notions and there

are "sophisticated" or "subtle" notions that we might

choose among.  The later would engage us in trying to

construct proper logical analogues of Lie derivatives,

and so let's save that for when we have become subtle

or sophisticated or both.  Against or toward that day,

as you wish, let's begin with an option in plain view.

Figure 1.4 illustrates one way of ranging over the cells of the

Figure&nbsp;1.4 illustrates one way of ranging over the cells of the underlying universe <math>U^\circ = [u, v]\!</math> and selecting at each cell the linear proposition in <math>\operatorname{d}U^\circ = [du, dv]</math> that best approximates the patch of the difference map <math>\operatorname{D}f</math> that is located there, yielding the following formula for the differential <math>\operatorname{d}f.</math>

underlying universe U% = [u, v] and selecting at each cell the

linear proposition in dU% = [du, dv] that best approximates

the patch of the difference map Df that is located there,

yielding the following formula for the differential df.

df  = uv.0 + u(v).du + (u)v.dv + (u)(v).(du, dv)

{| align="center" cellpadding="8" width="90%"

| <math>\operatorname{d}f ~=~ \texttt{uv} \cdot \texttt{0} ~+~ \texttt{u(v)} \cdot \texttt{du} ~+~ \texttt{(u)v} \cdot \texttt{dv} ~+~ \texttt{(u)(v)} \cdot \texttt{(du, dv)}</math>

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