Differential Logic • Discussion 9

LA:
All I am asking is what is your definition of $\mathrm{d}p$ in relation to $p$‌.  So far I have $\mathrm{d}p$ is what one has to do to get from $p$ to $\texttt{(} p \texttt{)}$ or from $\texttt{(} p \texttt{)}$ to $p$‌.  Is that all there is to it?  If that is the case, then what you are really dealing with is some flavor of Lattice Theory.

Dear Lyle,

Standing back for a moment to take in the Big Picture, what we’re doing here is taking all the things we would normally do in a “calculus of many variables” setting with spaces like:

$\begin{matrix} \mathbb{R}, & \mathbb{R}^{j}, & \mathbb{R}^{j} \to \mathbb{R}, & \mathbb{R}^{j} \to \mathbb{R}^{k}, & \ldots \end{matrix}$

and functoring that whole business over to $\mathbb{B},$ in other words, cranking the analogies as far as we can push them to spaces like:

$\begin{matrix} \mathbb{B}, & \mathbb{B}^{j}, & \mathbb{B}^{j} \to \mathbb{B}, & \mathbb{B}^{j} \to \mathbb{B}^{k}, & \ldots \end{matrix}$

A few analogies are bound to break in transit through the Real-Bool barrier, once familiar constructions morph into new-fangled configurations, and other distinctions collapse or “condense” as Spencer Brown called it.  Still enough structure gets preserved overall to reckon the result a kindred subject.

To be continued …

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