Differential Propositional Calculus • 21

Posted on December 12, 2023 by Jon Awbrey


There was never any more inception than there is now,
Nor any more youth or age than there is now;
And will never be any more perfection than there is now,
Nor any more heaven or hell than there is now.

— Walt Whitman • Leaves of Grass

A One‑Dimensional Universe

Let \mathcal{X} = \{ A \} be a logical basis containing one boolean variable or logical feature A.  The basis element A may be regarded as a simple proposition or coordinate projection A : \mathbb{B} \to \mathbb{B}.  Corresponding to the basis \mathcal{X} = \{ A \} is the alphabet \mathfrak{X} = \{ ``A" \} which serves whenever we need to make explicit mention of the symbols used in our formulas and representations.

The space X = \langle A \rangle = \{ \texttt{(} A \texttt{)}, A \} of points (cells, vectors, interpretations) has cardinality 2^n = 2^1 = 2 and is isomorphic to \mathbb{B} = \{ 0, 1 \}.  Moreover, X may be identified with the set of singular propositions \{ x : \mathbb{B} \xrightarrow{s} \mathbb{B} \}.

The space of linear propositions X^* = \{ \mathrm{hom} : \mathbb{B} \xrightarrow{\ell} \mathbb{B} \} = \{ 0, A \} is algebraically dual to X and also has cardinality 2.  Here, ``0" is interpreted as denoting the constant function 0 : \mathbb{B} \to \mathbb{B}, amounting to the linear proposition of rank 0, while A is the linear proposition of rank 1.

Last but not least we have the positive propositions \{ \mathrm{pos} : \mathbb{B} \xrightarrow{p} \mathbb{B} \} = \{ A, 1 \} of rank 1 and 0, respectively, where ``1" is understood as denoting the constant function 1 : \mathbb{B} \to \mathbb{B}.

All told there are 2^{2^n} = 2^{2^1} = 4 propositions in the universe of discourse \mathcal{X}^\bullet = [\mathcal{X}], collectively forming the set X^\uparrow = \{ f : X \to \mathbb{B} \} = \{ 0, \texttt{(} A \texttt{)}, A, 1 \} \cong (\mathbb{B} \to \mathbb{B}).

Resources

cc: FB | Differential Logic • Laws of Form • Mathstodon • Academia.edu
cc: Conceptual Graphs (1) (2) • Cybernetics • Structural Modeling • Systems Science

This entry was posted in Amphecks, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Computational Complexity, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Dynamical Systems, Equational Inference, Functional Logic, Gradient Descent, Graph Theory, Group Theory, Hologrammautomaton, Indicator Functions, Logic, Logical Graphs, Mathematical Models, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Propositional Equation Reasoning Systems, Time, Visualization and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

2 Responses to Differential Propositional Calculus • 21

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