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Any proposition worth its salt has many equivalent ways to view it, any one of which may reveal some unsuspected aspect of its meaning.  We will encounter more and more of these variant readings as we go.

Any proposition worth its salt has many equivalent ways to view it, any one of which may reveal some unsuspected aspect of its meaning.  We will encounter more and more of these variant readings as we go.

The enlargement operator ''E'', also known as the ''shift operator'', has many interesting and very useful properties in its own right, so let us not fail to observe a few of the more salient features that play out on the surface of our simple example, ''f''(''x'', ''y'') = ''xy''.

The enlargement operator <math>\operatorname{E},</math> also known as the ''shift operator'', has many interesting and very useful properties in its own right, so let us not fail to observe a few of the more salient features that play out on the surface of our simple example, <math>f(x, y) = xy.\!</math>

Introduce a suitably generic definition of the extended universe of discourse:

Introduce a suitably generic definition of the extended universe of discourse:

: Let ''U'' = ''X''₁&nbsp;&times;&nbsp;&hellip;&nbsp;&times;&nbsp;''X''_''k'' and ''EU'' = ''U''&nbsp;&times;&nbsp;''dU'' = ''X''₁&nbsp;&times;&nbsp;&hellip;&nbsp;&times;&nbsp;''X''_''k''&nbsp;&times;&nbsp;''dX''₁&nbsp;&times;&nbsp;&hellip;&nbsp;&times;&nbsp;''dX''_''k''.

: Let <math>U = X_1 \times \ldots \times X_k</math> and <math>\operatorname{E}U = U \times \operatorname{d}U = X_1 \times \ldots \times X_k \times \operatorname{d}X_1 \times \ldots \times \operatorname{d}X_k.</math>

For a proposition ''f''&nbsp;:&nbsp;''X''₁&nbsp;&times;&nbsp;&hellip;&nbsp;&times;&nbsp;''X''_''k''&nbsp;&rarr;&nbsp;'''B''', the (first order) enlargement of f is the proposition ''Ef''&nbsp;:&nbsp;''EU''&nbsp;&rarr;&nbsp;'''B''' that is defined by:

For a proposition ''f''&nbsp;:&nbsp;''X''₁&nbsp;&times;&nbsp;&hellip;&nbsp;&times;&nbsp;''X''_''k''&nbsp;&rarr;&nbsp;'''B''', the (first order) enlargement of f is the proposition ''Ef''&nbsp;:&nbsp;''EU''&nbsp;&rarr;&nbsp;'''B''' that is defined by: