Changes

One more piece of notation will save us a few bytes in the length of many of our schematic formulations.

One more piece of notation will save us a few bytes in the length of many of our schematic formulations.

Let <math>\mathcal{X} = \{ x_1, \ldots, x_k \}</math> be a finite class of variables, regarded as a formal alphabet of formal symbols but listed here without quotation marks.  Starting from this initial alphabet, the following items may then be defined:

Let <math>\mathcal{X} = \{ x_1, \ldots, x_k \}</math> be a finite set of variables, regarded as a formal alphabet of formal symbols but listed here without quotation marks.  Starting from this initial alphabet, the following items may then be defined:

#<p>The "(first order) differential alphabet",</p><p><math>\operatorname{d}\mathcal{X} = \{ \operatorname{d}x_1, \ldots, \operatorname{d}x_k \}.</math></p>

#<p>The "(first order) differential alphabet",</p><p><math>\operatorname{d}\mathcal{X} = \{ \operatorname{d}x_1, \ldots, \operatorname{d}x_k \}.</math></p>

Before we continue with the differential analysis of the source proposition <math>q\!</math>, we need to pause and take another look at just how it shapes up in the light of the extended universe <math>\operatorname{E}X,</math> in other words, to examine in detail its tacit extension <math>\operatorname{e}q.\!</math>

Before we continue with the differential analysis of the source proposition <math>q\!</math>, we need to pause and take another look at just how it shapes up in the light of the extended universe <math>\operatorname{E}X,</math> in other words, to examine in detail its tacit extension <math>\operatorname{e}q.\!</math>

The models of <math>\operatorname{e}q\!</math> in <math>\operatorname{E}X\!</math> can be comprehended as follows:

The models of eq in EX can be comprehended as follows:

1.  Working in the "summary coefficient" form of representation,

*<p>Working in the ''summary coefficient'' form of representation, if the coordinate list <math>\mathbf{x}\!</math> is a model of <math>q\!</math> in <math>X,\!</math> then one can construct a coordinate list <math>\operatorname{e}\mathbf{x}\!</math> as a model for <math>\operatorname{e}q\!</math> in <math>\operatorname{E}X\!</math> just by appending any combination of values for the differential variables in <math>\operatorname{d}\mathcal{X}.</math></p><p>For example, to focus once again on the center cell <math>c,\!</math> which happens to be a model of the proposition <math>q\!</math> in <math>X,\!</math> one can extend <math>c\!</math> in eight different ways into <math>\operatorname{E}X,\!</math> and thus get eight models of the tacit extension <math>\operatorname{e}q\!</math> in <math>\operatorname{E}X.\!</math></p>

    if the coordinate list x is a model of q in X, then one can

    construct a coordinate list ex as a model for eq in EX just

    by appending any combination of values for the differential

    variables in d!X!.

     Though it may seem an utter triviality to write these

     Though it may seem an utter triviality to write these

     out, I will do it for the sake of seeing the patterns.

     out, I will do it for the sake of seeing the patterns.