Changes

We may understand the enlarged proposition <math>\operatorname{E}f</math> as telling us all the different ways to reach a model of the proposition <math>f\!</math> from each point of the universe <math>X.\!</math>

We may understand the enlarged proposition <math>\operatorname{E}f</math> as telling us all the different ways to reach a model of the proposition <math>f\!</math> from each point of the universe <math>X.\!</math>

==Note 7==

==Propositional Forms on Two Variables==

To broaden our experience with simple examples, let us examine the sixteen functions of concrete type <math>P \times Q \to \mathbb{B}</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math>  A few Tables are set here that detail the actions of <math>\operatorname{E}</math> and <math>\operatorname{D}</math> on each of these functions, allowing us to view the results in several different ways.

To broaden our experience with simple examples, let us examine the sixteen functions of concrete type <math>P \times Q \to \mathbb{B}</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math>  A few Tables are set here that detail the actions of <math>\operatorname{E}</math> and <math>\operatorname{D}</math> on each of these functions, allowing us to view the results in several different ways.

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==Note 8==

===Enlargement and Difference Expanded over Differential Features===

The next four Tables expand the expressions of <math>\operatorname{E}f</math> and <math>\operatorname{D}f</math> in two different ways, for each of the sixteen functions.  Notice that the functions are given in a different order, partitioned into seven natural classes by a group action.

The next four Tables expand the expressions of <math>\operatorname{E}f</math> and <math>\operatorname{D}f</math> in two different ways, for each of the sixteen functions.  Notice that the functions are given in a different order, partitioned into seven natural classes by a group action.

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==Note 9==

===Enlargement and Difference Expanded over Ordinary Features===

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==Note 10==

==Transformation Groups==

If you think that I linger in the realm of logical difference calculus out of sheer vacillation about getting down to the differential proper, it is probably out of a prior expectation that you derive from the art or the long-engrained practice of real analysis.  But the fact is that ordinary calculus only rushes on to the sundry orders of approximation because the strain of comprehending the full import of <math>\operatorname{E}</math> and <math>\operatorname{D}</math> at once whelm over its discrete and finite powers to grasp them.  But here, in the fully serene idylls of ZOL, we find ourselves fit with the compass of a wit that is all we'd ever need to explore their effects with care.

If you think that I linger in the realm of logical difference calculus out of sheer vacillation about getting down to the differential proper, it is probably out of a prior expectation that you derive from the art or the long-engrained practice of real analysis.  But the fact is that ordinary calculus only rushes on to the sundry orders of approximation because the strain of comprehending the full import of <math>\operatorname{E}</math> and <math>\operatorname{D}</math> at once whelm over its discrete and finite powers to grasp them.  But here, in the fully serene idylls of ZOL, we find ourselves fit with the compass of a wit that is all we'd ever need to explore their effects with care.