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'''Differential Propositions'''

====Differential Propositions====

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 <math>[\mathcal{A}]</math> 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 <math>[\mathcal{A}]</math> may change or move with respect to the features that are noted in the initial alphabet.

Viewed within a coordinate representation, spaces of type <math>\mathbb{B}^n</math> and <math>\mathbb{D}^n</math> may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.

Viewed within a coordinate representation, spaces of type <math>\mathbb{B}^n</math> and <math>\mathbb{D}^n</math> may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.

'''The Extended Universe of Discourse'''

====Extended Universe of Discourse====

Next, we define the so-called ''extended alphabet'' or ''bundled alphabet'' <math>\operatorname{E}\mathcal{A}</math> as:

Next, we define the so-called ''extended alphabet'' or ''bundled alphabet'' <math>\operatorname{E}\mathcal{A}</math> as:

: <math>\operatorname{E}\mathcal{A} = \mathcal{A} \cup \operatorname{d}\mathcal{A} = \{a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}.</math>

: <math>\operatorname{E}\mathcal{A} = \mathcal{A} \cup \operatorname{d}\mathcal{A} = \{a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}.</math>

This supplies enough material to construct the ''differential extension'' <math>\operatorname{E}A</math>, or the ''tangent bundle'' over the initial space <math>A\!</math>, in the following fashion:

This supplies enough material to construct the ''differential extension'' <math>\operatorname{E}A,</math> or the ''tangent bundle'' over the initial space <math>A\!</math>, in the following fashion:

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: <math>(\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B}) = (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})).</math>

: <math>(\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B}) = (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})).</math>

A proposition in the tangent universe [E<font face="lucida calligraphy">A</font>] is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.

A proposition in the tangent universe <math>[\operatorname{E}\mathcal{A}]</math> is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.

With these constructions, to be specific, the differential extension E''A'' and the differential proposition ''h''&nbsp;:&nbsp;E''A''&nbsp;&rarr;&nbsp;'''B''', we have arrived, in concept at least, at one of the major subgoals of this study.  At this juncture, I pause by way of summary to set another Table with the current crop of mathematical produce (Table 5).

With these constructions, to be specific, the differential extension <math>\operatorname{E}A</math> and the differential proposition <math>f : \operatorname{E}A \to \mathbb{B},</math> we have arrived, in concept at least, at one of the major subgoals of this study.  At this juncture, I pause by way of summary to set another Table with the current crop of mathematical produce (Table 5).

Table&nbsp;5 summarizes the basic notations that are needed to describe the (first order) differential extensions of propositional calculi in a corresponding manner.

Table&nbsp;5 summarizes the basic notations that are needed to describe the (first order) differential extensions of propositional calculi in a corresponding manner.