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{{DISPLAYTITLE:Differential Logic and Dynamic Systems}}

{{DISPLAYTITLE:Differential Logic and Dynamic Systems}}

[[Image:Tangent_Functor_Ferris_Wheel.gif]]

| [[Image:Tangent_Functor_Ferris_Wheel.gif]]

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For the sake of concreteness, let us suppose that we start with a continuous ''n''-dimensional vector space like ''X''&nbsp;=&nbsp;〈''x''₁,&nbsp;&hellip;,&nbsp;''x''_''n''〉 <math>\cong</math> '''R'''^''n''.  The coordinate

For the sake of concreteness, let us suppose that we start with a continuous ''n''-dimensional vector space like ''X''&nbsp;=&nbsp;〈''x''₁,&nbsp;&hellip;,&nbsp;''x''_''n''〉 <math>\cong</math> '''R'''^''n''.  The coordinate

system <font face=lucida calligraphy">X</font> = {''x''_''i''} is a set of maps ''x''_''i''&nbsp;:&nbsp;'''R'''_''n''&nbsp;&rarr;&nbsp;'''R''', also known as the coordinate projections.  Given a "dataset" of points ''x'' in '''R'''_''n'', there are numerous ways of sensibly reducing the data down to one bit for each dimension.  One strategy that is general enough for our present purposes is as follows.  For each ''i'' we choose an ''n''-ary relation ''L''_''i'' on '''R''', that is, a subset of '''R'''_''n'', and then we define the ''i''^th threshold map, or ''limen'' <u>''x''</u>_''i'' as follows:

system <font face=lucida calligraphy">X</font> = {''x''_''i''} is a set of maps ''x''_''i''&nbsp;:&nbsp;'''R'''^''n''&nbsp;&rarr;&nbsp;'''R''', also known as the coordinate projections.  Given a "dataset" of points ''x'' in '''R'''^''n'', there are numerous ways of sensibly reducing the data down to one bit for each dimension.  One strategy that is general enough for our present purposes is as follows.  For each ''i'' we choose an ''n''-ary relation ''L''_''i'' on '''R''', that is, a subset of '''R'''^''n'', and then we define the ''i''^th threshold map, or ''limen'' <u>''x''</u>_''i'' as follows:

: <u>''x''</u>_''i'' : '''R'''_''n'' &rarr; '''B''' such that:

: <u>''x''</u>_''i'' : '''R'''^''n'' &rarr; '''B''' such that:

: <u>''x''</u>_''i''(''x'') = 1 if ''x'' &isin; ''L''_''i'',

: <u>''x''</u>_''i''(''x'') = 1 if ''x'' &isin; ''L''_''i'',

Notice that, as defined here, there need be no actual relation between the ''n''-dimensional subsets {''L''_''i''} and the coordinate axes corresponding to {''x''_''i''}, aside from the circumstance that the two sets have the same cardinality.  In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, ''L''_''i'' is bounded by some hyperplane that intersects the ''i''^th axis at a unique threshold value ''r''_''i'' &isin; '''R'''.  Often, the hyperplane is chosen normal to the axis.  In recognition of this motive, let us make the following convention.  When the set ''L''_''i'' has points on the ''i''^th axis, that is, points of the form ‹0,&nbsp;&hellip;,&nbsp;0,&nbsp;''r''_''i'',&nbsp;0,&nbsp;&hellip;,&nbsp;0› where only the ''x''_''i'' coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation.  In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''.  For a knowledge based system ''X'', this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.

Notice that, as defined here, there need be no actual relation between the ''n''-dimensional subsets {''L''_''i''} and the coordinate axes corresponding to {''x''_''i''}, aside from the circumstance that the two sets have the same cardinality.  In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, ''L''_''i'' is bounded by some hyperplane that intersects the ''i''^th axis at a unique threshold value ''r''_''i'' &isin; '''R'''.  Often, the hyperplane is chosen normal to the axis.  In recognition of this motive, let us make the following convention.  When the set ''L''_''i'' has points on the ''i''^th axis, that is, points of the form ‹0,&nbsp;&hellip;,&nbsp;0,&nbsp;''r''_''i'',&nbsp;0,&nbsp;&hellip;,&nbsp;0› where only the ''x''_''i'' coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation.  In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''.  For a knowledge based system ''X'', this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.

States of knowledge about the location of a system or about the distribution of a population of systems in a state space ''X'' = '''R'''^''n'' can now be expressed by taking the set <font face="lucida calligraphy"><u>X</u></font>&nbsp;=&nbsp;{<u>''x''</u>_''i''} as a basis of logical features.  In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the ''i''^th threshold map.  This can

States of knowledge about the location of a system or about the distribution of a population of systems in a state space ''X'' = '''R'''^''n'' can now be expressed by taking the set <font face="lucida calligraphy"><u>X</u></font>&nbsp;=&nbsp;{<u>''x''</u>_''i''} as a basis of logical features.  In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the ''i''^th threshold map.  This can help to remind us that the ''threshold operator'' (<u>&nbsp;</u>)_''i'' acts on ''x'' by setting up a kind of a "hurdle" for it.  In this interpretation, the coordinate proposition <u>''x''</u>_''i'' asserts that the representative point ''x'' resides ''above'' the ''i''^th threshold.

help to remind us that the ''threshold operator'' <u>&nbsp;</u>)_''i'' acts on ''x'' by setting up a kind of a "hurdle" for it.  In this interpretation, the coordinate proposition <u>''x''</u>_''i'' asserts that the representative point ''x'' resides ''above'' the ''i''^th threshold.

Primitive assertions of the form <u>''x''</u>_''i''(''x'') can then be negated and joined by means of propositional connectives in the usual ways to provide information about the state ''x'' of a contemplated system or a statistical ensemble of systems.  Parentheses "(&nbsp;)" may be used to indicate negation.  Eventually one discovers the usefulness of the ''k''-ary ''just one false'' operators of the form "(&nbsp;,&nbsp;,&nbsp;,&nbsp;)", as treated in earlier reports.  This much tackle generates a space of points (cells, interpretations), <u>''X''</u>&nbsp;=&nbsp;〈<font face="lucida calligraphy"><u>X</u></font>〉&nbsp;<math>\cong</math>&nbsp;'''B'''^''n'', and

Primitive assertions of the form <u>''x''</u>_''i''(''x'') can then be negated and joined by means of propositional connectives in the usual ways to provide information about the state ''x'' of a contemplated system or a statistical ensemble of systems.  Parentheses "(&nbsp;)" may be used to indicate negation.  Eventually one discovers the usefulness of the ''k''-ary ''just one false'' operators of the form "(&nbsp;,&nbsp;,&nbsp;,&nbsp;)", as treated in earlier reports.  This much tackle generates a space of points (cells, interpretations), <u>''X''</u>&nbsp;=&nbsp;〈<font face="lucida calligraphy"><u>X</u></font>〉&nbsp;<math>\cong</math>&nbsp;'''B'''^''n'', and

Figure 70-b shows another way to picture the action of the tangent functor on the logical transformation ''F''‹''u'',&nbsp;''v''›&nbsp;=&nbsp;‹((''u'')(''v'')),&nbsp;((''u'',&nbsp;''v''))›, roughly in the style of the ''bundle of universes'' type of diagram.

Figure 70-b shows another way to picture the action of the tangent functor on the logical transformation ''F''‹''u'',&nbsp;''v''›&nbsp;=&nbsp;‹((''u'')(''v'')),&nbsp;((''u'',&nbsp;''v''))›, roughly in the style of the ''bundle of universes'' type of diagram.

[[Image:Tangent_Functor_Ferris_Wheel.gif|frame|<font size="3">'''Figure 70-b.  Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›'''</font>]]

<p>[[Image:Tangent_Functor_Ferris_Wheel.gif|center|frame|<font size="3">'''Figure 70-b.  Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›'''</font>]]</p>

* '''Nota Bene.'''  The original Figure&nbsp;70-b lost some of its labeling in a succession of platform metamorphoses over the years, so I have included an Ascii version below to indicate where the missing labels go.

* '''Nota Bene.'''  The original Figure&nbsp;70-b lost some of its labeling in a succession of platform metamorphoses over the years, so I have included an Ascii version below to indicate where the missing labels go.

Recoded:  03 Jun 2007

Recoded:  03 Jun 2007

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