Inquiry Into Inquiry

Transforms Expanded over Ordinary and Differential Variables

As promised in Episode 10, in the next several posts we’ll extend our scope to the full set of boolean functions on two variables and examine how the differential operators and act on that set.  There being some advantage to singling out the enlargement or shift operator in its own right, we’ll begin by computing for each of the functions

Enlargement Map Expanded over Ordinary Variables

We first encountered the shift operator in Episode 4 when we imagined being in a state described by the proposition and contemplated the value of that proposition in various other states, as determined by the differential propositions and   Those thoughts led us from the boolean function of two variables to the boolean function of four variables as shown in the entry for in the first three columns of Table A3.  (Let’s catch a breath here and discuss what the rest of the Table shows next time.)

cc: Category Theory • Cybernetics • Ontolog • Structural Modeling • Systems Science
cc: FB | Differential Logic • Laws of Form • Peirce (1) (2) (3) (4)

Re: Differential Logic • Discussion • (12) (13)
Re: FB | Peirce Society • Χριστο Φόρος

Another bit of work I did toward a Psych M.A. was applying my Theme One program to a real-live dataset on family dynamics.  A collection of notes on that project is linked below.

• Exploratory Qualitative Analysis of Sequential Observation Data

Resources

• Survey of Theme One Program

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Re: Differential Logic • Discussion 12
Re: FB | Peirce Society • Χριστο Φόρος

Χριστο Φόρος asked whether the difference between qualitative and quantitative information was really all that much of a problem, especially in view of mixed datasets.  As I have encountered it in practice the rub is not so much between different types of data as between the two cultures of quantitative and qualitative research paradigms.

As it happens, my mix of backgrounds often found me employed consulting on statistics at the interface between quantitative and qualitative researchers.  On the qual side back in the 80s and 90s we were just beginning to develop software for ethographic methods, massaging linguistic, narrative, and verbal protocols toward categorical variables and non‑parametric statistics.  I worked a lot on concepts and software bridging the gap between qual and quant paradigms.

The program I spent the 80s developing and eventually submitted toward a Master’s in Psych integrated a Learning module (Slate) and a Reasoning module (Chalk).  The first viewed its input stream as a two-level formal language (“words” and “phrases”) and sought to induce a grammar for the language its environment was speaking to it.  The second was given propositional expressions describing universes of discourse and had to find all the conjunctions of basic qualitative features (boolean variables) satisfying those descriptions.  There’s a report on this work in the following paper.

• Exploring Research Data Interactively. Theme One : A Program of Inquiry

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Re: Category Theory • John Baez (1) (2)

JB:

One thing I’m interested in is functorially relating purely qualitative models to quantitative ones, or mixed quantitative-qualitative models where you have some numerical information of the sort you describe, but not all of it.  That’s a situation we often find ourselves in:  having a mixture of quantitative and qualitative information about what’s going on in a complicated system.

When I say “functorially”, I mean for starters:  there should be a functor from “quantitative models” of system dynamics to “qualitative models”.

Dear John,

This is something I’ve been working on.  In a turn of phrase I once concocted, it’s like passing from the qualitative theory of differential equations to the differential theory of qualitative equations.  I’m posting a few notes on the Chategory topic Differential Logic.

Regards,

Jon

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cc: FB | Differential Logic • Laws of Form

Re: Differential Logic • Discussion 9

Let’s look more closely at the “functor” from to and the connection it makes between real and boolean hierarchies of types.  There’s a detailed discussion of this analogy in the article and section linked below.

• Differential Logic and Dynamic Systems
• The Analogy Between Real and Boolean Types

Assorted types of mathematical objects which turn up in practice often enough to earn themselves common names, along with their common isomorphisms, are shown in the following Table.

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cc: FB | Differential Logic • Laws of Form

Re: Laws of Form • Lyle Anderson

Let’s say we’re observing a system at discrete intervals of time and testing whether its state satisfies or falsifies a given predicate or proposition at each moment.  Then and are two state variables describing the time evolution of the system.  In logical conception and are independent variables, even if empirical discovery finds them bound by law.

What gives the differential variable its meaning in relation to the ordinary variable is not the conventional notation used here but a class of temporal inference rules, in the present example, the fourfold scheme of inference shown below.

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cc: FB | Differential Logic • Laws of Form