Differential Logic • Discussion 9

Posted on July 12, 2021 by Jon Awbrey

Re: Laws of Form • Lyle Anderson

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 …

cc: Category Theory • Cybernetics • Ontolog • Structural Modeling • Systems Science
cc: FB | Differential Logic • Laws of Form

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