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=={{anchor|Exemplary Universes}}Back to the Beginning • Exemplary Universes==

==Back to the Beginning • Exemplary Universes==

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To anchor our understanding of differential logic let's look at how the various concepts apply in the simplest possible concrete cases, where the initial dimension is only 1 or 2.  In spite of the simplicity of those cases it is possible to observe how central difficulties of the subject begin to arise already at that stage.

To anchor our understanding of differential logic let's look at how the various concepts apply in the simplest possible concrete cases, where the initial dimension is only 1 or 2.  In spite of the simplicity of those cases it is possible to observe how central difficulties of the subject begin to arise already at that stage.

==={{anchor|One Dim Universe}}A One‑Dimensional Universe===

===A One‑Dimensional Universe===

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Observe how the secular inference rules, used by themselves, involve a loss of information, since nothing in them tells whether the momenta <math>\{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}</math> are changed or unchanged in the next moment.&nbsp; To know that one would have to determine <math>\mathrm{d}^2 A,</math> and so on, pursuing an infinite regress.&nbsp; In order to rest with a finitely determinate system it is necessary to make an infinite assumption, for example, that <math>\mathrm{d}^k A = 0</math> for all <math>k</math> greater than some fixed value <math>M.</math>&nbsp; Another way to escape the regress is through the provision of a dynamic law, in typical form making higher order differentials dependent on lower degrees and estates.

Observe how the secular inference rules, used by themselves, involve a loss of information, since nothing in them tells whether the momenta <math>\{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}</math> are changed or unchanged in the next moment.&nbsp; To know that one would have to determine <math>\mathrm{d}^2 A,</math> and so on, pursuing an infinite regress.&nbsp; In order to rest with a finitely determinate system it is necessary to make an infinite assumption, for example, that <math>\mathrm{d}^k A = 0</math> for all <math>k</math> greater than some fixed value <math>M.</math>&nbsp; Another way to escape the regress is through the provision of a dynamic law, in typical form making higher order differentials dependent on lower degrees and estates.

==={{anchor|Example 1|Rigging}}Example 1. A Square Rigging===

===Example 1. A Square Rigging===

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With growth in the dimensions of our contemplated universes it becomes essential, both for human comprehension and for computer implementation, that dynamic structures of interest be represented not actually, by acquaintance, but virtually, by description.&nbsp; In our present study we are using the language of propositional calculus to express the relevant descriptions, and to grasp the structures embodied in subsets of <i>n</i>&#8209;cubes without being forced to actualize all their points.

With growth in the dimensions of our contemplated universes it becomes essential, both for human comprehension and for computer implementation, that dynamic structures of interest be represented not actually, by acquaintance, but virtually, by description.&nbsp; In our present study we are using the language of propositional calculus to express the relevant descriptions, and to grasp the structures embodied in subsets of <i>n</i>&#8209;cubes without being forced to actualize all their points.

==={{anchor|COSM}}Commentary On Small Models===

===Commentary On Small Models===

One reason for engaging in our present order of extremely reduced but explicitly controlled case study is to throw light on the general study of languages, formal and natural, in their full array of syntactic, semantic, and pragmatic aspects.&nbsp; Propositional calculus is one of the last points of departure where it is possible to see that trio of aspects interacting in a non&#8209;trivial way without being immediately and totally overwhelmed by the complexity they generate.

One reason for engaging in our present order of extremely reduced but explicitly controlled case study is to throw light on the general study of languages, formal and natural, in their full array of syntactic, semantic, and pragmatic aspects.&nbsp; Propositional calculus is one of the last points of departure where it is possible to see that trio of aspects interacting in a non&#8209;trivial way without being immediately and totally overwhelmed by the complexity they generate.

A compression scheme by any other name is a symbolic representation, and this is what the differential extension of propositional calculus, through all of its many universes of discourse, is intended to supply.&nbsp; Why is this particular program of mental calisthenics worth carrying out in general?&nbsp; By providing a uniform logical medium for describing dynamic systems we can make the task of understanding complex systems much easier, both in looking for invariant representations of individual cases and in finding points of comparison among diverse structures that would otherwise appear as isolated systems.&nbsp; All of this goes to facilitate the search for compact knowledge and to adapt what is learned from individual cases to the general realm.

A compression scheme by any other name is a symbolic representation, and this is what the differential extension of propositional calculus, through all of its many universes of discourse, is intended to supply.&nbsp; Why is this particular program of mental calisthenics worth carrying out in general?&nbsp; By providing a uniform logical medium for describing dynamic systems we can make the task of understanding complex systems much easier, both in looking for invariant representations of individual cases and in finding points of comparison among diverse structures that would otherwise appear as isolated systems.&nbsp; All of this goes to facilitate the search for compact knowledge and to adapt what is learned from individual cases to the general realm.

==={{anchor|Feature}}Back to the Feature===

===Back to the Feature===

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In its bearing on the singular propositions over a universe of discourse <math>X</math> the above analysis has an interesting interpretation.&nbsp; The tacit extension takes us from thinking about a particular state, like <math>A</math> or <math>\texttt{(} A \texttt{)},</math> to considering the collection of outcomes, the outgoing changes or the singular dispositions, springing or stemming from that state.

In its bearing on the singular propositions over a universe of discourse <math>X</math> the above analysis has an interesting interpretation.&nbsp; The tacit extension takes us from thinking about a particular state, like <math>A</math> or <math>\texttt{(} A \texttt{)},</math> to considering the collection of outcomes, the outgoing changes or the singular dispositions, springing or stemming from that state.

==={{anchor|Example 2|Drives}}Example 2. Drives and Their Vicissitudes===

===Example 2. Drives and Their Vicissitudes===

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Since <math>\mathrm{d}^4 A</math> and all higher differences <math>\mathrm{d}^k A</math> are fixed, the state vectors vary only with respect to their projections as points of <math>\mathrm{E}^3 X</math> <math>=</math> <math>\langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A \rangle.</math>&nbsp; Thus there is just enough space in a planar venn diagram to plot all the orbits and to show how they partition the points of <math>\mathrm{E}^3 X.</math>&nbsp; It turns out there are exactly two possible orbits, of eight points each, as shown in Figure&nbsp;16.

Since <math>\mathrm{d}^4 A</math> and all higher differences <math>\mathrm{d}^k A</math> are fixed, the state vectors vary only with respect to their projections as points of <math>\mathrm{E}^3 X</math> <math>=</math> <math>\langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A \rangle.</math>&nbsp; Thus there is just enough space in a planar venn diagram to plot all the orbits and to show how they partition the points of <math>\mathrm{E}^3 X.</math>&nbsp; It turns out there are exactly two possible orbits, of eight points each, as shown in Figure&nbsp;16.

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| [[File:Diff Log Dyn Sys &bull; Figure 16 &bull; A Couple of Fourth Gear Orbits.gif]]

| [[File:Diff Log Dyn Sys &bull; Figure 16 &bull; A Couple of Fourth Gear Orbits.gif]]