Differential Propositional Calculus • 8

Posted on March 7, 2020 by Jon Awbrey

Differential Extensions

An initial universe of discourse A^\bullet supplies the groundwork for any number of further extensions, beginning with the first order differential extension \mathrm{E}A^\bullet.  The construction of \mathrm{E}A^\bullet can be described in the following stages.

  • The initial alphabet \mathfrak{A} = \{ ``a_1", \ldots, ``a_n" \} is extended by a first order differential alphabet \mathrm{d}\mathfrak{A} = \{ ``\mathrm{d}a_1", \ldots, ``\mathrm{d}a_n" \} resulting in a first order extended alphabet \mathrm{E}\mathfrak{A} defined as follows.

    \mathrm{E}\mathfrak{A} ~=~ \mathfrak{A} ~\cup~ \mathrm{d}\mathfrak{A} ~=~ \{ ``a_1", \ldots, ``a_n", ``\mathrm{d}a_1", \ldots, ``\mathrm{d}a_n" \}.

  • The initial basis \mathcal{A} = \{ a_1, \ldots, a_n \} is extended by a first order differential basis \mathrm{d}\mathcal{A} = \{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \} resulting in a first order extended basis \mathrm{E}\mathcal{A} defined as follows.

    \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 \}.

  • The initial space A = \langle a_1, \ldots, a_n \rangle is extended by a first order differential space or tangent space \mathrm{d}A = \langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle at each point of A, resulting in a first order extended space or tangent bundle space \mathrm{E}A defined as follows.

    \mathrm{E}A ~=~ A ~\times~ \mathrm{d}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.

  • Finally, the initial universe A^\bullet = [ a_1, \ldots, a_n ] is extended by a first order differential universe or tangent universe \mathrm{d}A^\bullet = [ \mathrm{d}a_1, \ldots, \mathrm{d}a_n ] at each point of A^\bullet, resulting in a first order extended universe or tangent bundle universe \mathrm{E}A^\bullet defined as follows.

    \mathrm{E}A^\bullet ~=~ [ \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 gives \mathrm{E}A^\bullet a type defined as follows.

    [ \mathbb{B}^n \times \mathbb{D}^n ] ~=~ (\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 a differential extension of a universe of discourse is called a differential proposition and forms the analogue of a system of differential equations in ordinary calculus.  With these constructions, the first order extended universe \mathrm{E}A^\bullet and the first order differential propositions f : \mathrm{E}A \to \mathbb{B}, we arrive at the foothills of differential logic.

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

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