Differential Propositional Calculus • 13

Posted on December 11, 2024 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 ].

    That 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 the construction of 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.

Resources

cc: Academia.edu • Cybernetics • Structural Modeling • Systems Science
cc: Conceptual Graphs • Laws of Form • Mathstodon • Research Gate

This entry was posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Equational Inference, Functional Logic, Graph Theory, Hologrammautomaton, Indicator Functions, Inquiry Driven Systems, Leibniz, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Propositional Calculus, Time, Topology, Visualization and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

4 Responses to Differential Propositional Calculus • 13

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