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====Note 4====
====Note 4====
We have been studying the action of the difference operator <math>\operatorname{D},</math> also known as the ''localization operator'', on the proposition <math>f : X \times Y \to \mathbb{B}</math> that is commonly known as the conjunction <math>x \cdot y.</math> We described <math>\operatorname{D}f</math> as a (first order) differential proposition, that is, a proposition of the type <math>\operatorname{D}f : X \times Y \times \operatorname{d}X \times \operatorname{d}Y \to \mathbb{B}.</math> Abstracting from the augmented venn diagram that illustrates how the ''models'' or ''satisfying interpretations'' of <math>\operatorname{D}f</math> distribute within the extended universe <math>\operatorname{E}U = X \times Y \times \operatorname{d}X \times \operatorname{d}Y,</math> we can depict <math>\operatorname{D}f</math> in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of <math>U = X \times Y</math> and whose arrows are labeled with the elements of <math>\operatorname{d}U = \operatorname{d}X \times \operatorname{d}Y.</math>
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Any proposition worth its salt, as they say,
Any proposition worth its salt, as they say, has many equivalent ways to look at it, any of which may reveal some unsuspected aspect of its meaning. We will encounter more and more of these alternative readings as we go.
has many equivalent ways to look at it, any
of which may reveal some unsuspected aspect
of its meaning. We will encounter more and
more of these alternative readings as we go.
====Note 5====
====Note 5====
