Differential Propositional Calculus • 30

Posted on December 22, 2023 by Jon Awbrey


I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.

— Michel Foucault • The Discourse on Language

Tacit Extensions

In viewing the previous Table of Differential Propositions it is important to notice the subtle distinction in type between a function f_i : X \to \mathbb{B} and its inclusion as a function g_j : \mathrm{E}X \to \mathbb{B}, even though they share the same logical expression.  Naturally, we want to maintain the logical equivalence of expressions representing the same proposition while appreciating the full diversity of a proposition’s functional and typical representatives.  Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.

Because this special circumstance points to a broader theme, it’s a good idea to discuss it more generally.  Whenever there arises a situation like that above, where one basis \mathcal{X} is a subset of another basis \mathcal{Y}, we say any proposition f : \langle \mathcal{X} \rangle \to \mathbb{B} has a tacit extension to a proposition \boldsymbol\varepsilon f : \langle \mathcal{Y} \rangle \to \mathbb{B} and we say the space (\langle \mathcal{X} \rangle \to \mathbb{B}) has an automatic embedding within the space (\langle \mathcal{Y} \rangle \to \mathbb{B}).

The tacit extension operator \boldsymbol\varepsilon is defined in such a way that \boldsymbol\varepsilon f puts the same constraint on the variables of \mathcal{X} within \mathcal{Y} as the proposition f initially put on \mathcal{X}, while it puts no constraint on the variables of \mathcal{Y} beyond \mathcal{X}, in effect, conjoining the two constraints.

Indexing the variables as \mathcal{X} = \{ x_1, \ldots, x_n \} and \mathcal{Y} = \{ x_1, \ldots, x_n, \ldots, x_{n+k} \} the tacit extension from \mathcal{X} to \mathcal{Y} may be expressed by the following equation.

\boldsymbol\varepsilon f(x_1, \ldots, x_n, \ldots, x_{n+k}) ~=~ f(x_1, \ldots, x_n).

On formal occasions, such as the present context of definition, the tacit extension from \mathcal{X} to \mathcal{Y} is explicitly symbolized by the operator \boldsymbol\varepsilon : (\langle \mathcal{X} \rangle \to \mathbb{B}) \to (\langle \mathcal{Y} \rangle \to \mathbb{B}), where the bases \mathcal{X} and \mathcal{Y} are set in context, but it’s normally understood the ``\boldsymbol\varepsilon" may be silent.

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 • 30

  1. Pingback: Survey of Differential Logic • 6 | Inquiry Into Inquiry

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