Differential Propositional Calculus • 29

Posted on December 25, 2024 by Jon Awbrey


I guess it must be the flag of my disposition, out of hopeful
     green stuff woven.

— Walt Whitman • Leaves of Grass

Back to the Feature

Let’s assume the sense intended for differential features is well enough established in the intuition for now to continue outlining the structure of the differential extension [\mathrm{E}\mathcal{X}] = [A, \mathrm{d}A].

The extended alphabet \mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \} of cardinality 2^n = 2 generates the terms of description for the extended space \mathrm{E}X of cardinality 2^{2n} = 4 according to the following series of equations.

\begin{array}{lll} \mathrm{E}X & = & \langle A, \mathrm{d}A \rangle \\[4pt] & = & \{ \texttt{(} A \texttt{)}, A \} ~\times~ \{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \} \\[4pt] & = & \{ \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},~ \texttt{(} A \texttt{)} \mathrm{d}A,~ A \texttt{(} \mathrm{d}A \texttt{)},~ A ~ \mathrm{d}A \}. \end{array}

The space \mathrm{E}X may be given the nominal type \mathbb{B} \times \mathbb{D}, at root isomorphic to \mathbb{B} \times \mathbb{B} = \mathbb{B}^2.  An element of \mathrm{E}X may be regarded as a disposition at a point or a situated direction, in effect, a singular mode of change occurring at a single point in the universe of discourse.  In practice the modality of those changes may be interpreted in various ways, for example, as expectations, intentions, or observations with respect to the behavior of a system.

To construct the extended universe of discourse \mathrm{E}X^\bullet = [A, \mathrm{d}A] the basic dispositions in \mathrm{E}X need to be extended to the full set of differential propositions \mathrm{E}X^\uparrow = \{ g : \mathrm{E}X \to \mathbb{B} \}, each of type \mathbb{B} \times \mathbb{D} \to \mathbb{B}.  There are 2^{2^{2n}} = 16 propositions in \mathrm{E}X^\uparrow, as detailed in the following Table.

\text{Differential Propositions}
Differential Propositions

Aside from changing the names of variables and shuffling the order of rows, the Table follows the format previously used for boolean functions of two variables.  The rows are grouped to reflect natural similarity classes holding among the propositions.  In a future discussion the classes will be given additional explanation and motivation as the orbits of a certain transformation group acting on the set of 16 propositions.  Notice that four of the propositions, in their logical expressions, resemble those given in the table for X^\uparrow.  Thus the first set of propositions \{ f_i \} is automatically embedded in the present set \{ g_j \} and the corresponding inclusions are indicated at the far left margin of the Table.

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 • 29

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