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| 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. |
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− | 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: |
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| #<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> |
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| 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> |
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− | <pre>
| + | 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: | |
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− | 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
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− | construct a coordinate list ex as a model for eq in EX just
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− | by appending any combination of values for the differential
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− | variables in d!X!.
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− | For example, to focus once again on the center cell c,
| + | <pre> |
− | which happens to be a model of the proposition q in X,
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− | one can extend c in eight different ways into EX, and
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− | thus get eight models of the tacit extension eq in EX.
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| 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. |