## Differential Propositional Calculus • 8

### 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, $\mathfrak{A} = \{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime} \},$ is extended by a first order differential alphabet, $\mathrm{d}\mathfrak{A} = \{ {}^{\backprime\backprime} \mathrm{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \mathrm{d}a_n {}^{\prime\prime} \},$ resulting in a first order extended alphabet, $\mathrm{E}\mathfrak{A},$ defined as follows:

$\mathrm{E}\mathfrak{A} ~=~ \mathfrak{A} ~\cup~ \mathrm{d}\mathfrak{A} ~=~ \{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \mathrm{d}a_n {}^{\prime\prime} \}.$

• 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 $\mathrm{E}A^\bullet$ the type:

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

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 proposition $f : \mathrm{E}A \to \mathbb{B},$ we have arrived, in concept at least, 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.

Table 11.  Differential Extension : Basic Notation

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