Differential Propositional Calculus • 18

Posted on December 8, 2023 by Jon Awbrey

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}
Differential Extension • 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

cc: FB | Differential Logic • Laws of Form • Mathstodon • Academia.edu
cc: Conceptual Graphs (1) (2) • Cybernetics • Structural Modeling • Systems Science

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