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A '''differential propositional calculus''' is a [[propositional calculus]] extended by a set of terms for describing aspects of change and difference, for example, processes that take place in a [[universe of discourse]] or transformations that map a source universe into a target universe.
A '''differential propositional calculus''' is a [[propositional calculus]] extended by a set of terms for describing aspects of change and difference, for example, processes that take place in a [[universe of discourse]] or transformations that map a source universe into a target universe.
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Table 5 summarizes the basic notations that are needed to describe the (first order) differential extensions of propositional calculi in a corresponding manner.
Table 5 summarizes the basic notations that are needed to describe the (first order) differential extensions of propositional calculi in a corresponding manner.
==Expository examples==
==Expository examples==
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'''…'''
=Work Area=
==Formal development==
===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.
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> (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 <math>A\!</math> at a point <math>x,\!</math> frequently denoted as <math>T_x(A),\!</math> can be characterized as having the generic construction <math>\operatorname{d}A = \langle \operatorname{d}\mathcal{A} \rangle = \langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle.</math> 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.
Proceeding as we did before with the base space <math>A,\!</math> we can analyze the individual tangent space at a point of <math>A\!</math> as a product of distinct and independent factors:
: <math>\operatorname{d}A = \prod_{i=1}^n \operatorname{d}A_i = \operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n.</math>
Here, <math>\operatorname{d}\mathcal{A}_i</math> is an alphabet of two symbols, <math>\operatorname{d}\mathcal{A}_i = \{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \},</math> where <math>\overline{\operatorname{d}a_i}</math> is a symbol with the logical value of <math>\operatorname{not}\ \operatorname{d}a_i.</math> Each component <math>\operatorname{d}A_i</math> has the type <math>\mathbb{B},</math> under the correspondence <math>\{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \} \cong \{ 0, 1 \}.</math> However, clarity is often served by acknowledging this differential usage with a superficially distinct type <math>\mathbb{D}, </math> whose intension may be indicated as follows:
: <math>\mathbb{D} = \{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \} = \{ \operatorname{same}, \operatorname{different} \} = \{ \operatorname{stay}, \operatorname{change} \} = \{ \operatorname{stop}, \operatorname{step} \}.</math>
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.
===Extended Universe of Discourse===
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>
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:
:{| cellpadding=2
| <math>\operatorname{E}A</math>
| =
| <math>A \times \operatorname{d}A</math>
|-
|
| =
| <math>\langle \operatorname{E}\mathcal{A} \rangle</math>
|-
|
| =
| <math>\langle \mathcal{A} \cup \operatorname{d}\mathcal{A} \rangle</math>
|-
|
| =
| <math>\langle a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle,</math>
|}
thus giving <math>\operatorname{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math>
Finally, the tangent universe <math>\operatorname{E}A^\circ = [\operatorname{E}\mathcal{A}]</math> is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features <math>\operatorname{E}\mathcal{A}:</math>
: <math>\operatorname{E}A^\circ = [\operatorname{E}\mathcal{A}] = [a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n],</math>
thus giving the tangent universe <math>\operatorname{E}A^\circ</math> the type:
: <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 <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 <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).
=Materials from "Dif Log Dyn Sys" for Reuse Here=
=Materials from "Dif Log Dyn Sys" for Reuse Here=
