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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>  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>", 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.

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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Given that this expression uses nothing more than the boolean ring operations of addition and multiplication, it is permissible to "multiply things out" in the usual manner to arrive at the following result:

Given that this expression uses nothing more than the "boolean ring"

operations of addition (+) and multiplication (*), it is permissible

to "multiply things out" in the usual manner to arrive at the result:

  Ef<p, q, dp, dq>

 

  =  p q + p dq  + q dp  + dp dq

<math>\begin{matrix}

\operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q)

p~q

& + &

p~\operatorname{d}q

& + &

q~\operatorname{d}p

& + &

To understand what this means in logical terms,

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>

or a "disjunctive normal form" (DNF), it is perhaps

a little better to go back and analyze the expression

the same way that we did for Df.  Thus, let us compute

the value of the enlarged proposition Ef at each of the

points in the initial domain of discourse X = !P! x !Q!.

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