Changes

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A function like this has an abstract type and a concrete type.  The abstract type is what we invoke when we write things like <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}</math> or <math>f : \mathbb{B}^2 \to \mathbb{B}.</math>  The concrete type takes into account the qualitative dimensions or the "units" of the case, which can be explained as follows.

A function like this has an abstract type and a concrete type.  The abstract type is what we invoke when we write things like <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}</math> or <math>f : \mathbb{B}^2 \to \mathbb{B}.</math>  The concrete type takes into account the qualitative dimensions or the &ldquo;units&rdquo; of the case, which can be explained as follows.

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The interpretations of these new symbols can be diverse, but the easiest

The interpretations of these new symbols can be diverse, but the easiest

option for now is just to say that <math>\operatorname{d}p</math> means "change <math>p\!</math>" and <math>\operatorname{d}q</math> means "change <math>q\!</math>".

option for now is just to say that <math>\operatorname{d}p</math> means &ldquo;change <math>p\!</math>&rdquo; and <math>\operatorname{d}q</math> means &ldquo;change <math>q\!</math>&rdquo;.

Drawing a venn diagram for the differential extension <math>\operatorname{E}X = X \times \operatorname{d}X</math> requires four logical dimensions, <math>P, Q, \operatorname{d}P, \operatorname{d}Q,</math> but it is possible to project a suggestion of what the differential features <math>\operatorname{d}p</math> and <math>\operatorname{d}q</math> are about on the 2-dimensional base space <math>X = P \times Q</math> by drawing arrows that cross the boundaries of the basic circles in the venn diagram for <math>X\!,</math> reading an arrow as <math>\operatorname{d}p</math> if it crosses the boundary between <math>p\!</math> and <math>\texttt{(} p \texttt{)}</math> in either direction and reading an arrow as <math>\operatorname{d}q</math> if it crosses the boundary between <math>q\!</math> and <math>\texttt{(} q \texttt{)}</math> in either direction.

Drawing a venn diagram for the differential extension <math>\operatorname{E}X = X \times \operatorname{d}X</math> requires four logical dimensions, <math>P, Q, \operatorname{d}P, \operatorname{d}Q,</math> but it is possible to project a suggestion of what the differential features <math>\operatorname{d}p</math> and <math>\operatorname{d}q</math> are about on the 2-dimensional base space <math>X = P \times Q</math> by drawing arrows that cross the boundaries of the basic circles in the venn diagram for <math>X\!,</math> reading an arrow as <math>\operatorname{d}p</math> if it crosses the boundary between <math>p\!</math> and <math>\texttt{(} p \texttt{)}</math> in either direction and reading an arrow as <math>\operatorname{d}q</math> if it crosses the boundary between <math>q\!</math> and <math>\texttt{(} q \texttt{)}</math> in either direction.

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The differential proposition that results may be interpreted to say "change <math>p\!</math> or change <math>q\!</math> or both".  And this can be recognized as just what you need to do if you happen to find yourself in the center cell and require a complete and detailed description of ways to escape it.

The differential proposition that results may be interpreted to say &ldquo;change <math>p\!</math> or change <math>q\!</math> or both&rdquo;.  And this can be recognized as just what you need to do if you happen to find yourself in the center cell and require a complete and detailed description of ways to escape it.

Last time we computed what is variously called the ''difference map'', the ''difference proposition'', or the ''local proposition'' <math>\operatorname{D}f_x</math> of the proposition <math>f(p, q) = pq\!</math> at the point <math>x\!</math> where <math>p = 1\!</math> and <math>q = 1.\!</math>

Last time we computed what is variously called the ''difference map'', the ''difference proposition'', or the ''local proposition'' <math>\operatorname{D}f_x</math> of the proposition <math>f(p, q) = pq\!</math> at the point <math>x\!</math> where <math>p = 1\!</math> and <math>q = 1.\!</math>

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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>" 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.

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 &ldquo;<math>\operatorname{d}</math>&rdquo; 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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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 &ldquo;multiply things out&rdquo; in the usual manner to arrive at the following result:

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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>

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

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