Differential Propositional Calculus • 30

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

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cc: Conceptual Graphs • Laws of Form • Ma^thstodon • Research Gate

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