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===Differential Propositions : The Qualitative Analogues of Differential Equations===

===Differential Propositions : The Qualitative Analogues of Differential Equations===

In order to define the differential extension of a universe of discourse [<font face="lucida calligraphy">A</font>], the initial alphabet <font face="lucida calligraphy">A</font> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in [<font face="lucida calligraphy">A</font>]. Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in [<font face="lucida calligraphy">A</font>] may change or move with respect to the features that are noted in the initial alphabet.

In order to define the differential extension of a universe of discourse <math>[\mathcal{A}],</math> the initial alphabet <math>\mathcal{A}</math> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in <math>[\mathcal{A}].</math>  Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in the universe change or move with respect to the features that are noted in the initial alphabet.

Hence, let us define the corresponding ''differential alphabet'' or ''tangent alphabet'' as d<font face="lucida calligraphy">A</font>&nbsp;=&nbsp;{d''a''₁,&nbsp;&hellip;,&nbsp;d''a''_''n''}, in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <font face="lucida calligraphy">A</font>&nbsp;=&nbsp;{''a''₁,&nbsp;&hellip;,&nbsp;''a''_''n''}, that is intended to be interpreted in the way just indicated.  It only remains to be understood that the precise interpretation of the symbols in d<font face="lucida calligraphy">A</font> is often conceived to be changeable from point to point of the underlying space ''A''.  (For all we know, the state space ''A'' might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by <font face="lucida calligraphy">A</font> and d<font face="lucida calligraphy">A</font>.)

Hence, let us define the corresponding ''differential alphabet'' or ''tangent alphabet'' as <math>\operatorname{d}\mathcal{A} = \{\operatorname{d}a_1, \ldots, \operatorname{d}a_n\},</math> in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <math>\mathcal{A} = \{a_1, \ldots, a_n\},</math> that is intended to be interpreted in the way just indicated.  It only remains to be understood that the precise interpretation of the symbols in <math>\operatorname{d}\mathcal{A}</math> is often conceived to be changeable from point to point of the underlying space <math>A.\!</math> (Indeed, for all we know, the state space <math>A\!</math> might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by <math>\mathcal{A}</math> and <math>\operatorname{d}\mathcal{A}.</math>)

In the above terms, a typical tangent space of ''A'' at a point ''x'', frequently denoted as T_''x''(''A''), can be characterized as having the generic construction d''A''&nbsp;=&nbsp;〈d<font face="lucida calligraphy">A</font>〉&nbsp;=&nbsp;〈d''a''₁,&nbsp;&hellip;,&nbsp;d''a''_''n''〉.  Strictly speaking, the name ''cotangent space'' is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.

In the above terms, a typical tangent space of ''A'' at a point ''x'', frequently denoted as T_''x''(''A''), can be characterized as having the generic construction d''A''&nbsp;=&nbsp;〈d<font face="lucida calligraphy">A</font>〉&nbsp;=&nbsp;〈d''a''₁,&nbsp;&hellip;,&nbsp;d''a''_''n''〉.  Strictly speaking, the name ''cotangent space'' is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.