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

Differential Extension : Basic Notation

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