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The ''differential variables'' <math>\operatorname{d}x_j</math> are boolean variables of the same basic type as the ordinary variables <math>x_j.\!</math>  Although it is conventional to distinguish the (first order) differential variables with the operative prefix "<math>\operatorname{d}</math>", but this is purely optional.  It is their existence in particular relations to the initial variables, not their names, that defines them as differential variables.

The ''differential variables'' <math>\operatorname{d}x_j</math> are boolean variables of the same basic type as the ordinary variables <math>x_j.\!</math>  Although it is conventional to distinguish the (first order) differential variables with the operative prefix "<math>\operatorname{d}</math>" this way of notating differential variables is entirely optional.  It is their existence in particular relations to the initial variables, not their names, that defines them as differential variables.

In the example of logical conjunction, <math>f(p, q) = pq,\!</math> the enlargement <math>\operatorname{E}f</math> is formulated as follows:

In the example of logical conjunction, <math>f(p, q) = pq,\!</math> the enlargement <math>\operatorname{E}f</math> is formulated as follows:

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To understand what this means in logical terms, it is useful to go back and analyze the above expression for <math>\operatorname{E}f</math> in the same way that we did for <math>\operatorname{D}f.</math>  Toward that end, the next set of Figures represent the computation of the ''enlarged'' or ''shifted'' proposition <math>\operatorname{E}f</math> at each of the 4 points in the universe of discourse <math>X = P \times Q.</math>

To understand what the ''enlarged'' or ''shifted'' proposition means in logical terms, it serves to go back and analyze the above expression for <math>\operatorname{E}f</math> in the same way that we did for <math>\operatorname{D}f.</math>  Toward that end, the value of <math>\operatorname{E}f_x</math> at each <math>x \in X</math> may be computed in graphical fashion as shown below:

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Given the data that develops in this form of analysis, the disjoined ingredients can now be folded back into a boolean expansion or a disjunctive normal form (DNF) that is equivalent to the enlarged proposition <math>\operatorname{E}f.</math>

Given the kind of data that arises from this form of analysis,

we can now fold the disjoined ingredients back into a boolean

expansion or a DNF that is equivalent to the proposition Ef.

  Ef = pq Ef_pq + p(q) Ef_p(q) + (p)q Ef_(p)q + (p)(q) Ef_(p)(q)

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pq \cdot \operatorname{E}f_{pq}

& + &

p(q) \cdot \operatorname{E}f_{p(q)}

& + &

(p)q \cdot \operatorname{E}f_{(p)q}

& + &

(p)(q) \cdot \operatorname{E}f_{(p)(q)}

Here is a summary of the result, illustrated by means of

Here is a summary of the result, illustrated by means of a digraph picture, where the "no change" element <math>(\operatorname{d}p)(\operatorname{d}q)</math> is drawn as a loop at the point <math>p~q.</math>

a digraph picture, where the "no change" element (dp)(dq)

is drawn as a loop at the point p q.

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