Differential Propositional Calculus • 18

The Extended Universe of Discourse

The extended basis $\mathrm{E}\mathcal{A}$ of a universe of discourse $[\mathcal{A}]$ is formed by taking the initial basis $\mathcal{A}$ together with the differential basis $\mathrm{d}\mathcal{A}.$ Thus we have the following formula.

$\mathrm{E}\mathcal{A} ~=~ \mathcal{A} \cup \mathrm{d}\mathcal{A} ~=~ \{ a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}$

This supplies enough material to construct the differential extension $\mathrm{E}A$ of the space $A,$ also called the tangent bundle of $A,$ in the following fashion.

$\mathrm{E}A ~=~ \langle \mathrm{E}\mathcal{A} \rangle ~=~ \langle \mathcal{A} \cup \mathrm{d}\mathcal{A} \rangle ~=~ \langle a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle$

and also

$\mathrm{E}A ~=~ A \times \mathrm{d}A ~=~ A_1 \times \ldots \times A_n \times \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n$

That gives $\mathrm{E}A$ the type $\mathbb{B}^n \times \mathbb{D}^n.$

Finally, the extended universe $\mathrm{E}A^\bullet = [\mathrm{E}\mathcal{A}]$ is the full collection of points and functions, or interpretations and propositions, based on the extended set of features $\mathrm{E}\mathcal{A},$ a fact summed up in the following notation.

$\mathrm{E}A^\bullet ~=~ [\mathrm{E}\mathcal{A}] ~=~ [a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n]$

That gives the extended universe $\mathrm{E}A^\bullet$ the following type.

$(\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B}) ~=~ (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}))$

A proposition in the extended universe $[\mathrm{E}\mathcal{A}]$ is called a differential proposition and forms the logical analogue of a system of differential equations, constraints, or relations in ordinary calculus.

With these constructions, the differential extension $\mathrm{E}A$ and the space of differential propositions $(\mathrm{E}A \to \mathbb{B}),$ we arrive at the launchpad of our space explorations.

Table 11 summarizes the notations needed to describe the first order differential extensions of propositional calculi in a systematic manner.

$\text{Table 11. Differential Extension} \stackrel{_\bullet}{} \text{Basic Notation}$

The adjective differential or tangent is systematically attached to every construct based on the differential alphabet $\mathrm{d}\mathfrak{A},$ taken by itself. In like fashion, the adjective extended or the substantive bundle is systematically attached to any construct associated with the full complement of ${2n}$ features.

Resources

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