antlerboy - Benjamin P Taylor 2:30 am on October 1, 2024
Tags: systems thinking ( 6 )

Updated rough draft systems | complexity | cybernetics reading list

See my post on LinkedIn (replicated below) and join the discussion there:
https://www.linkedin.com/posts/antlerboy_rough-draft-systemscomplexitycybernetics-activity-7246779585235664896-64Xz

pdf: https://www.dropbox.com/scl/fi/85zlt0t6ph8qarx7d7gic/2024-09-27-rough-draft-systems-thinking-reading-list-v1.1BT.pdf?rlkey=3rfavacsy4n6sl8j0pyedph1q&st=qagh1418&dl=0
Commentable Google Doc: https://docs.google.com/document/d/1Tt8GgQQj4Qw4HnR7DxKeF370o_HlDlpv/edit?usp=sharing&ouid=115526108239573817578&rtpof=true&sd=true

How do you get into systems | complexity | cybernetics?

Here’s my rough reading list.

There are a lot of answers to the question, many of them connecting with some kind of disjointing break from ‘normal’ ways of seeing and being. Anything from being bullied at school to being dyslexic. Being in an outsider group. Naively applying thinking from one domain to another. Studying a technical problem long enough to suddenly see it in a completely different light – then either have your breakthrough celebrated or rejected.

It isn’t some mystic thing and it doesn’t require to you break from polite society. But it is one of the richest, weirdest, most diverse and challenging, inspiring and confounding, confronting and validating things you can study.

I’m often asked for a reading list for people interested in the field, and I usually suck my teeth. Some of the books are engaging, insightful, humorous, relevant. Others are dry as old twigs but less likely to kindle a spark.

Really, it depends on you and your context – as David Ing says, it’s better to talk of the thinkers and their individual constellations of interests, history, learning, and personal tendencies than it is to talk of schools and fields and separate places.

And even presenting this reading list, I’d say that I’d recommend Terry Pratchett, Douglas Adams, Ursula K Le Guin, Italo Calvino, Jorge Luis Borges, Star Trek, old 20th Century Sci-Fi and Apartheid-era South African writing, art movies and music more – if you happen to be a bit like me. You’ll find your thing, if you’re interested.

But. The books are there – and many of them are *really good*. Top ones I’d recommend came out this decade

Hoverstadt’s Grammar of Systems
Jackson’s Critical Systems Thinking: A practitioner’s Guide
Opening the box – a slim little thing from SCiO colleagues
Essential Balances by Velitchkov

The attached list is a bit systems-practice focused. It is also too long and incomplete and partial simply for lack of time and energy.

There are *so many* flavours of systems thinking / complexity / cybernetics – do yourself a favour and don’t flog through stuff that doesn’t work for you, find things that bring your mind alive. Start with the articles and skim through.

But do start, because you will find in here the thinking and tools to find better ways of doing things for organisations, societies, the ecosystem, for people – and a lot of fun.

Tip: to save the pdf, hover over the image of the first page and find the rectangle bottom right – click that and it should go full screen. Top right you’ll have a download option, which when clicked will then resolve into a download button… (which might then open in your browser, but at least as a proper pdf you can save).

So… deep breath… what would you recommend? What do you think is missing?

#systems-thinking

antlerboy - Benjamin P Taylor 3:56 pm on February 16, 2026

NEEDED: SYSTEMS THINKING IN PUBLIC AFFAIRS – Conway (2024)

h/t Ivo Velitchkov

Introduction

What is systems thinking? The answer depends on whom you ask. Here are two commonperspectives from which you will get two different answers. Engineering. Here, systems thinking is what you need to build a system whose requirements go beyond current practice. Example: all stages in a plan to evolve into a national energy distribution system for low-emission transportation. Metapolitics (a neologism analogous to metamathematics). Here, systems thinking is what you need (1)to understand the ambient social systems in which we all have unconsciously long been embedded, and (2)to use that understanding to attempt to bring these systems into alignment with current needs, given some disruptive change such as newtechnology or increased scale. Example: modifying the global economy in response to climate change.

This essay is based on the Metapolitics perspective. In two Examples I explore perverse behavior patterns of two ambient social systems, a newoneandanolder one: 1. mass radicalization, disinformation, and other perverse social consequences secondary to new technologies that facilitate intensive everyone-to-everyone communication (for example, “social networking”), and 2. environmental destruction secondary to a compulsion to grow arising from the financing structures of public corporations. Analysis of both of these behavior patterns reveals a common element: Emergent behaviors, not anticipated in classical thinking, arise from highly intraconnected or coupled networks. This failure of classical thought leads to The Big Lesson I wish to communicate in this essay: THINK NETWORKSFIRST, ACTORS SECOND. Here is the importance of this lesson: Effective interventions will arise from altering interactions within networks. You cannot even see these interactions unless you focus on the network. This essay offers two examples that contradict the conventional understanding of Network Effects. We are living inside something we don’t understand.

Click to access ty.pdf

antlerboy - Benjamin P Taylor 6:22 pm on February 15, 2026

Peter Tuddenham on LinkedIn – mapping Claude’s eight conversation surfaces through the lens of Gordon Pask’s conversation theory

Peter writes:

“I’ve spent 40 years applying cybernetic frameworks to real organisations — from the U.S. Army War College to UNESCO to distributed educator networks spanning 18,000 participants. Recently, I’ve been working intensively with Claude (Anthropic’s AI), and something struck me: every interface Claude offers is a different kind of conversation, with different affordances and different costs.
So I wrote a practitioner’s guide mapping Claude’s eight conversation surfaces through the lens of Gordon Pask’s Conversation Theory (1975, 1976).
The core insight: every time you switch from chat to Claude Code, or from a Project to an Artifact, you’re not just changing tools — you’re changing the structure of the conversation itself. And that structure determines what kind of knowing is possible.
The guide introduces what I call the “re-education tax” — the real cost of re-establishing shared understanding when you switch surfaces or start fresh sessions. If you’ve ever felt frustrated explaining context to an AI again after switching tools, you’ve been paying this tax without naming it.”
(2) Post | LinkedIn

https://www.linkedin.com/posts/peterdtuddenham_claude-conversation-surfaces-a-practitioners-ugcPost-7425557491205246978-MB9C/?utm_source=share&utm_medium=member_desktop&rcm=ACoAAACuq-oBecVFDW6PCf3lkoG-peMeuLBeoho

Jon Awbrey 5:15 pm on February 15, 2026
Tags: Amphecks ( 10 ), Animata ( 12 ), Boolean Algebra ( 10 ), Boolean Functions ( 11 ), C.S. Peirce ( 30 ), Cactus Graphs ( 11 ), Change ( 9 ), cybernetics ( 22 ), Differential Calculus ( 9 ), Differential Logic ( 11 ), Discrete Dynamics ( 9 ), Equational Inference ( 11 ), Functional Logic ( 10 ), Gradient Descent ( 8 ), Graph Theory ( 13 ), Inquiry Driven Systems ( 14 ), logic ( 32 ), Logical Graphs ( 12 ), Mathematics ( 31 ), Minimal Negation Operators ( 11 ), Propositional Calculus ( 11 ), Time ( 8 ), Visualization ( 17 )

Differential Logic • 7

Differential Expansions of Propositions

Panoptic View • Enlargement Maps

The enlargement or shift operator $\mathrm{E}$ exhibits a wealth of interesting and useful properties in its own right, so it pays to examine a few of the more salient features playing out on the surface of our initial example, $f(p, q) = pq.$

A suitably generic definition of the extended universe of discourse is afforded by the following set‑up.

$\begin{array}{cccl} \text{Let} & X & = & X_1 \times \ldots \times X_k. \\[6pt] \text{Let} & \mathrm{d}X & = & \mathrm{d}X_1 \times \ldots \times \mathrm{d}X_k. \\[6pt] \text{Then} & \mathrm{E}X & = & X \times \mathrm{d}X \\[6pt] & & = & X_1 \times \ldots \times X_k ~\times~ \mathrm{d}X_1 \times \ldots \times \mathrm{d}X_k \end{array}$

For a proposition of the form $f : X_1 \times \ldots \times X_k \to \mathbb{B},$ the (first order) enlargement of $f$ is the proposition $\mathrm{E}f : \mathrm{E}X \to \mathbb{B}$ defined by the following equation.

$\mathrm{E}f(x_1, \ldots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k) ~=~ f(x_1 + \mathrm{d}x_1, \ldots, x_k + \mathrm{d}x_k) ~=~ f(\texttt{(} x_1 \texttt{,} \mathrm{d}x_1 \texttt{)}, \ldots, \texttt{(} x_k \texttt{,} \mathrm{d}x_k \texttt{)})$

The differential variables $\mathrm{d}x_j$ are boolean variables of the same type as the ordinary variables $x_j.$ Although it is conventional to distinguish the (first order) differential variables with the operational prefix $``\mathrm{d}",$ that way of notating differential variables is entirely optional. It is their existence in particular relations to the initial variables, not their names, which defines them as differential variables.

In the example of logical conjunction, $f(p, q) = pq,$ the enlargement $\mathrm{E}f$ is formulated as follows.

$\begin{matrix} \mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q) & = & (p + \mathrm{d}p)(q + \mathrm{d}q) & = & \texttt{(} p \texttt{,} \mathrm{d}p \texttt{)(} q \texttt{,} \mathrm{d}q \texttt{)} \end{matrix}$

Given that the above expression uses nothing more than the boolean ring operations of addition and multiplication, it is permissible to “multiply things out” in the usual manner to arrive at the following result.

$\begin{matrix} \mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q) & = & p~q & + & p~\mathrm{d}q & + & q~\mathrm{d}p & + & \mathrm{d}p~\mathrm{d}q \end{matrix}$

To understand what the enlarged or shifted proposition means in logical terms, it serves to go back and analyze the above expression for $\mathrm{E}f$ in the same way we did for $\mathrm{D}f.$ To that end, the value of $\mathrm{E}f_x$ at each $x \in X$ may be computed in graphical fashion as shown below.

Collating the data of that analysis yields a boolean expansion or disjunctive normal form (DNF) equivalent to the enlarged proposition $\mathrm{E}f.$

$\begin{matrix} \mathrm{E}f & = & pq \cdot \mathrm{E}f_{pq} & + & p(q) \cdot \mathrm{E}f_{p(q)} & + & (p)q \cdot \mathrm{E}f_{(p)q} & + & (p)(q) \cdot \mathrm{E}f_{(p)(q)} \end{matrix}$

Here is a summary of the result, illustrated by means of a digraph picture, where the “no change” element $\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}$ is drawn as a loop at the point $p~q.$

$\begin{array}{rcccccc} f & = & p & \cdot & q \\[4pt] \mathrm{E}f & = & p & \cdot & q & \cdot & \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{(} \mathrm{d}p \texttt{)} \texttt{~} \mathrm{d}q \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \mathrm{d}p \texttt{~} \texttt{(} \mathrm{d}q \texttt{)} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \mathrm{d}p \texttt{~~} \mathrm{d}q \end{array}$

We may understand the enlarged proposition $\mathrm{E}f$ as telling us all the ways of reaching a model of the proposition $f$ from the points of the universe $X.$

Resources

cc: Academia.edu • Cybernetics • Laws of Form • Ma^thstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#amphecks, #animata, #boolean-algebra, #boolean-functions, #c-s-peirce, #cactus-graphs, #change, #cybernetics, #differential-calculus, #differential-logic, #discrete-dynamics, #equational-inference, #functional-logic, #gradient-descent, #graph-theory, #inquiry-driven-systems, #logic, #logical-graphs, #mathematics, #minimal-negation-operators, #propositional-calculus, #time, #visualization

antlerboy - Benjamin P Taylor 4:12 pm on February 15, 2026

Jay Forrester and the Discipline That Learned to See Persistence – Damodaran (2026) (LinkedIn)

[Completing a trio of recent LinkedIn articles]

Sheila Damodaran
Global, National & Regional Strategy Development | Leadership Capacity, Systemic Research & Longitudinal Thinking Through The Fifth Discipline

February 1, 2026
(15) Jay Forrester and the Discipline That Learned to See Persistence | LinkedIn

https://www.linkedin.com/pulse/jay-forrester-discipline-learned-see-persistence-sheila-damodaran-u2d0f/?trackingId=UIPb3wu0%2FyqJRDbEm8XvPg%3D%3D

antlerboy - Benjamin P Taylor 1:43 pm on February 15, 2026

The Great Divide: Systems Thinking and Complexity Science – Aziz (2026) (LinkedIn)

[Another one where I have great sympathy with the author and intent, but don’t agree with the piece overall – however, lots of juicy debate!]

Abdul Aziz
Strategy & Performance through Empathy, Architecture and Analytics

February 14, 2026
I recently developed a “Systems & Complexity Lifecycle” framework as a teaching device, treating systems theory, complexity science, chaos theory, and catastrophe theory as temporal stages in how entities evolve from stability through transformation.

The framework maps four stages:

Stage 1 – Systems: Stability and homeostasis (Bertalanffy’s General Systems Theory)
Stage 2 – Complexity: Emergence of higher-order properties (Holland’s Hidden Order)
Stage 3 – Chaos: Sensitivity to initial conditions (Gleick’s Chaos)
Stage 4 – Catastrophe: Discontinuous transformation (Thom’s catastrophe theory)

(4) The Great Divide: Systems Thinking and Complexity Science | LinkedIn

https://www.linkedin.com/pulse/great-divide-why-systems-thinking-complexity-science-abdul-aziz-7l8de/

antlerboy - Benjamin P Taylor 1:26 pm on February 15, 2026

Peter Senge: The Fifth Discipline at Thirty-Five — Lineage, Surge, and Scale – Damodaran (2026) (LinkedIn)

Sheila Damodaran

Global, National & Regional Strategy Development | Leadership Capacity, Systemic Research & Longitudinal Thinking Through The Fifth Discipline

February 15, 2026

Peter Senge: The Fifth Discipline at Thirty-Five — Lineage, Surge, and Scale

Sheila Damodaran
Global, National & Regional Strategy Development | Leadership Capacity, Systemic Research & Longitudinal Thinking Through The Fifth Discipline

February 15, 2026
(2) Peter Senge: The Fifth Discipline at Thirty-Five — Lineage, Surge, and Scale | LinkedIn

https://www.linkedin.com/pulse/peter-senge-fifth-discipline-thirty-five-lineage-surge-damodaran-35t4f/?trackingId=J8np3Qss09boTDn7Vs%2FmUA%3D%3D

antlerboy - Benjamin P Taylor 6:24 am on February 15, 2026

Free 90 Minute AI Modeling Tools Workshop with Gene Bellinger – 11am EST, 19 Feb 2026

Free 90 Minute AI Modeling Tools Workshop – Create diagrams such as the one below, completely documented, along with an Aha! Paradox, and emotional story embracing the relationships, usually in 10 min or less.
The Workshop will be at 11 am Eastern Time (New York) on Feb 19th, and I’ll send out the Zoom link info 1 day and 1 hour before the workshop. Just reply to this post, and I’ll put you on the list.
Post | Feed | LinkedIn

https://www.linkedin.com/feed/update/urn:li:activity:7428423481274413056/

Jon Awbrey 2:36 pm on February 14, 2026
Tags: Amphecks ( 10 ), Animata ( 12 ), Boolean Algebra ( 10 ), Boolean Functions ( 11 ), C.S. Peirce ( 30 ), Cactus Graphs ( 11 ), Change ( 9 ), cybernetics ( 22 ), Differential Calculus ( 9 ), Differential Logic ( 11 ), Discrete Dynamics ( 9 ), Equational Inference ( 11 ), Functional Logic ( 10 ), Gradient Descent ( 8 ), Graph Theory ( 13 ), Inquiry Driven Systems ( 14 ), logic ( 32 ), Logical Graphs ( 12 ), Mathematics ( 31 ), Minimal Negation Operators ( 11 ), Propositional Calculus ( 11 ), Time ( 8 ), Visualization ( 17 )

Differential Logic • 6

Differential Expansions of Propositions

Panoptic View • Difference Maps

In the previous post we computed what is variously described as the difference map, the difference proposition, or the local proposition $\mathrm{D}f_x$ of the proposition $f(p, q) = pq$ at the point $x$ where $p = 1$ and $q = 1.$

In the universe of discourse $X = P \times Q$ the four propositions $pq, \, p \texttt{(} q \texttt{)}, \, \texttt{(} p \texttt{)} q, \, \texttt{(} p \texttt{)(} q \texttt{)}$ can be taken to indicate the so‑called “cells” or smallest distinguished regions of the universe, otherwise indicated by their coordinates as the “points” $(1, 1), ~ (1, 0), ~ (0, 1), ~ (0, 0),$ respectively. In that regard the four propositions are called singular propositions because they serve to single out the minimal regions of the universe of discourse.

Thus we can write $\mathrm{D}f_x = \mathrm{D}f|_x = \mathrm{D}f|_{(1, 1)} = \mathrm{D}f|_{pq},$ so long as we know the frame of reference in force.

In the example $f(p, q) = pq,$ the value of the difference proposition $\mathrm{D}f_x$ at each of the four points $x \in X$ may be computed in graphical fashion as shown below.

The easy way to visualize the values of the above graphical expressions is just to notice the following graphical equations.

Adding the arrows to the venn diagram gives us the picture of a differential vector field.

The Figure shows the points of the extended universe $\mathrm{E}X = P \times Q \times \mathrm{d}P \times \mathrm{d}Q$ indicated by the difference map $\mathrm{D}f : \mathrm{E}X \to \mathbb{B},$ namely, the following six points or singular propositions.

$\begin{array}{rcccc} 1. & p & q & \mathrm{d}p & \mathrm{d}q \\ 2. & p & q & \mathrm{d}p & \texttt{(} \mathrm{d}q \texttt{)} \\ 3. & p & q & \texttt{(} \mathrm{d}p \texttt{)} & \mathrm{d}q \\ 4. & p & \texttt{(} q \texttt{)} & \texttt{(} \mathrm{d}p \texttt{)} & \mathrm{d}q \\ 5. & \texttt{(} p \texttt{)} & q & \mathrm{d}p & \texttt{(} \mathrm{d}q \texttt{)} \\ 6. & \texttt{(} p \texttt{)} & \texttt{(} q \texttt{)} & \mathrm{d}p & \mathrm{d}q \end{array}$

The information borne by $\mathrm{D}f$ should be clear enough from a survey of these six points — they tell you what you have to do from each point of $X$ in order to change the value borne by $f(p, q),$ that is, the move you have to make in order to reach a point where the value of the proposition $f(p, q)$ is different from what it is where you started.

We have been studying the action of the difference operator $\mathrm{D}$ on propositions of the form $f : P \times Q \to \mathbb{B},$ as illustrated by the example $f(p, q) = pq$ which is known in logic as the conjunction of $p$ and $q.$ The resulting difference map $\mathrm{D}f$ is a (first order) differential proposition, that is, a proposition of the form $\mathrm{D}f : P \times Q \times \mathrm{d}P \times \mathrm{d}Q \to \mathbb{B}.$

The augmented venn diagram shows how the models or satisfying interpretations of $\mathrm{D}f$ distribute over the extended universe of discourse $\mathrm{E}X = P \times Q \times \mathrm{d}P \times \mathrm{d}Q.$ Abstracting from that picture, the difference map $\mathrm{D}f$ can be represented in the form of a digraph or directed graph, one whose points are labeled with the elements of $X = P \times Q$ and whose arrows are labeled with the elements of $\mathrm{d}X = \mathrm{d}P \times \mathrm{d}Q,$ as shown in the following Figure.

$\begin{array}{rcccccc} f & = & p & \cdot & q \\[4pt] \mathrm{D}f & = & p & \cdot & q & \cdot & \texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \end{array}$

Any proposition worth its salt can be analyzed from many different points of view, any one of which has the potential to reveal previously unsuspected aspects of the proposition’s meaning. We will encounter more and more such alternative readings as we go.

Resources

cc: Academia.edu • Cybernetics • Laws of Form • Ma^thstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Jon Awbrey 11:32 am on February 12, 2026
Tags: Amphecks ( 10 ), Animata ( 12 ), Boolean Algebra ( 10 ), Boolean Functions ( 11 ), C.S. Peirce ( 30 ), Cactus Graphs ( 11 ), Change ( 9 ), cybernetics ( 22 ), Differential Calculus ( 9 ), Differential Logic ( 11 ), Discrete Dynamics ( 9 ), Equational Inference ( 11 ), Functional Logic ( 10 ), Gradient Descent ( 8 ), Graph Theory ( 13 ), Inquiry Driven Systems ( 14 ), logic ( 32 ), Logical Graphs ( 12 ), Mathematics ( 31 ), Minimal Negation Operators ( 11 ), Propositional Calculus ( 11 ), Time ( 8 ), Visualization ( 17 )

Differential Logic • 5

Differential Expansions of Propositions

Worm’s Eye View

Let’s run through the initial example again, keeping an eye on the meanings of the formulas which develop along the way. We begin with a proposition or a boolean function $f(p, q) = pq$ whose venn diagram and cactus graph are shown below.

A function like $f$ has an abstract type and a concrete type. The abstract type is what we invoke when we write things like $f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}$ or $f : \mathbb{B}^2 \to \mathbb{B}.$ The concrete type takes into account the qualitative dimensions or “units” of the case, which can be explained as follows.

Let $P$ be the set of values $\{ \texttt{(} p \texttt{)},~ p \} ~=~ \{ \mathrm{not}~ p,~ p \} ~\cong~ \mathbb{B}.$
Let $Q$ be the set of values $\{ \texttt{(} q \texttt{)},~ q \} ~=~ \{ \mathrm{not}~ q,~ q \} ~\cong~ \mathbb{B}.$

Then interpret the usual propositions about $p, q$ as functions of the concrete type $f : P \times Q \to \mathbb{B}.$

We are going to consider various operators on these functions. An operator $\mathrm{F}$ is a function which takes one function $f$ into another function $\mathrm{F}f.$

The first couple of operators we need are logical analogues of two which play a founding role in the classical finite difference calculus, namely, the following.

The difference operator $\Delta,$ written here as $\mathrm{D}.$
The enlargement operator, written here as $\mathrm{E}.$

These days, $\mathrm{E}$ is more often called the shift operator.

In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of discourse. Starting from the initial space $X = P \times Q,$ its (first order) differential extension $\mathrm{E}X$ is constructed according to the following specifications.

$\begin{array}{rcc} \mathrm{E}X & = & X \times \mathrm{d}X \end{array}$

where:

$\begin{array}{rcc} X & = & P \times Q \\[4pt] \mathrm{d}X & = & \mathrm{d}P \times \mathrm{d}Q \\[4pt] \mathrm{d}P & = & \{ \texttt{(} \mathrm{d}p \texttt{)}, ~ \mathrm{d}p \} \\[4pt] \mathrm{d}Q & = & \{ \texttt{(} \mathrm{d}q \texttt{)}, ~ \mathrm{d}q \} \end{array}$

The interpretations of these new symbols can be diverse, but the easiest option for now is just to say $\mathrm{d}p$ means “change $p$ ” and $\mathrm{d}q$ means “change $q$ ”.

Drawing a venn diagram for the differential extension $\mathrm{E}X = X \times \mathrm{d}X$ requires four logical dimensions, $P, Q, \mathrm{d}P, \mathrm{d}Q,$ but it is possible to project a suggestion of what the differential features $\mathrm{d}p$ and $\mathrm{d}q$ are about on the 2‑dimensional base space $X = P \times Q$ by drawing arrows crossing the boundaries of the basic circles in the venn diagram for $X,$ reading an arrow as $\mathrm{d}p$ if it crosses the boundary between $p$ and $\texttt{(} p \texttt{)}$ in either direction and reading an arrow as $\mathrm{d}q$ if it crosses the boundary between $q$ and $\texttt{(} q \texttt{)}$ in either direction, as indicated in the following figure.

Propositions are formed on differential variables, or any combination of ordinary logical variables and differential logical variables, in the same ways propositions are formed on ordinary logical variables alone. For example, the proposition $\texttt{(} \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{))}$ says the same thing as $\mathrm{d}p \Rightarrow \mathrm{d}q,$ in other words, there is no change in $p$ without a change in $q.$

Given the proposition $f(p, q)$ over the space $X = P \times Q,$ the (first order) enlargement of $f$ is the proposition $\mathrm{E}f$ over the differential extension $\mathrm{E}X$ defined by the following formula.

$\begin{matrix} \mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q) & = & f(p + \mathrm{d}p,~ q + \mathrm{d}q) & = & f( \texttt{(} p, \mathrm{d}p \texttt{)},~ \texttt{(} q, \mathrm{d}q \texttt{)} ) \end{matrix}$

In the example $f(p, q) = pq,$ the enlargement $\mathrm{E}f$ is computed as follows.

$\begin{matrix} \mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q) & = & (p + \mathrm{d}p)(q + \mathrm{d}q) & = & \texttt{(} p, \mathrm{d}p \texttt{)(} q, \mathrm{d}q \texttt{)} \end{matrix}$

Given the proposition $f(p, q)$ over $X = P \times Q,$ the (first order) difference of $f$ is the proposition $\mathrm{D}f$ over $\mathrm{E}X$ defined by the formula $\mathrm{D}f = \mathrm{E}f - f,$ or, written out in full:

$\begin{matrix} \mathrm{D}f(p, q, \mathrm{d}p, \mathrm{d}q) & = & f(p + \mathrm{d}p,~ q + \mathrm{d}q) - f(p, q) & = & \texttt{(} f( \texttt{(} p, \mathrm{d}p \texttt{)},~ \texttt{(} q, \mathrm{d}q \texttt{)} ),~ f(p, q) \texttt{)} \end{matrix}$

In the example $f(p, q) = pq,$ the difference $\mathrm{D}f$ is computed as follows.

$\begin{matrix} \mathrm{D}f(p, q, \mathrm{d}p, \mathrm{d}q) & = & (p + \mathrm{d}p)(q + \mathrm{d}q) - pq & = & \texttt{((} p, \mathrm{d}p \texttt{)(} q, \mathrm{d}q \texttt{)}, pq \texttt{)} \end{matrix}$

This brings us by the road meticulous to the point we reached at the end of the previous post. There we evaluated the above proposition, the first order difference of conjunction $\mathrm{D}f,$ at a single location in the universe of discourse, namely, at the point picked out by the singular proposition $pq,$ in terms of coordinates, at the place where $p = 1$ and $q = 1.$ That evaluation is written in the form $\mathrm{D}f|_{pq}$ or $\mathrm{D}f|_{(1, 1)},$ and we arrived at the locally applicable law which may be stated and illustrated as follows.

$f(p, q) ~=~ pq ~=~ p ~\mathrm{and}~ q \quad \Rightarrow \quad \mathrm{D}f|_{pq} ~=~ \texttt{((} \mathrm{dp} \texttt{)(} \mathrm{d}q \texttt{))} ~=~ \mathrm{d}p ~\mathrm{or}~ \mathrm{d}q$

The venn diagram shows the analysis of the inclusive disjunction $\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}$ into the following exclusive disjunction.

$\begin{matrix} \mathrm{d}p ~\texttt{(} \mathrm{d}q \texttt{)} & + & \texttt{(} \mathrm{d}p \texttt{)}~ \mathrm{d}q & + & \mathrm{d}p ~\mathrm{d}q \end{matrix}$

The differential proposition $\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}$ may be read as saying “change $p$ or change $q$ or both”. And this can be recognized as just what you need to do if you happen to find yourself in the center cell and require a complete and detailed description of ways to escape it.

Resources

cc: Academia.edu • Cybernetics • Laws of Form • Ma^thstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

antlerboy - Benjamin P Taylor 4:27 am on February 12, 2026

Request for votes for interactive skills training workshops at Systems Thinking Systems Practice conference, 24-26 March 2026, University of Hull

At Systems Thinking Systems Practice, 24-26 March 2026, University of Hull, we will again run Skills Training Workshops.

These workshops were a huge success at SysPrac25, with many of them oversubscribed.

They will take the form of interactive workshops, which will further develop your skills or introduce you to new approaches you may not have encountered before.

To vote visit https://docs.google.com/forms/d/e/1FAIpQLSfsLoKstfeA5BwI8pK5kAb25abdUAhLeKe2s51Xp4Gmxw_FAg/viewform before 20 February 2026!

The conference: https://stream.syscoi.com/2026/01/25/2026-conference-systems-thinking-and-systems-practice-hosted-by-the-university-of-hull-centre-for-systems-studies-css-systems-and-complexity-in-organisation-scio-and-the-or-society-24-26-march/

2026 Conference: Systems Thinking and Systems Practice, Hosted by the University of Hull Centre for Systems Studies (CSS), Systems and Complexity in Organisation (SCiO) and The OR Society, 24-26 March 2026 | University of Hull, UK

antlerboy - Benjamin P Taylor 1:10 pm on February 9, 2026

Complex Systems Frameworks Collection

Complex Systems Frameworks Collection
Frameworks A-Z – Complex Systems Frameworks Collection

Frameworks A-Z

antlerboy - Benjamin P Taylor 1:10 pm on February 9, 2026

The Game

THE GAME
The Game – Birmingham Food Council

The Game

antlerboy - Benjamin P Taylor 4:00 am on February 9, 2026

Storytelling for Systems Change

New website

Storytelling for Systems Change
A story-kit for changemakers
Storytelling for Systems Change – Dusseldorp Forum

Storytelling for Systems Change

Jon Awbrey 4:32 pm on February 8, 2026
Tags: Amphecks ( 10 ), Animata ( 12 ), Boolean Algebra ( 10 ), Boolean Functions ( 11 ), C.S. Peirce ( 30 ), Cactus Graphs ( 11 ), Change ( 9 ), cybernetics ( 22 ), Differential Calculus ( 9 ), Differential Logic ( 11 ), Discrete Dynamics ( 9 ), Equational Inference ( 11 ), Functional Logic ( 10 ), Gradient Descent ( 8 ), Graph Theory ( 13 ), Inquiry Driven Systems ( 14 ), logic ( 32 ), Logical Graphs ( 12 ), Mathematics ( 31 ), Minimal Negation Operators ( 11 ), Propositional Calculus ( 11 ), Time ( 8 ), Visualization ( 17 )

Differential Logic • 4

Differential Expansions of Propositions

Bird’s Eye View

An efficient calculus for the realm of logic represented by boolean functions and elementary propositions makes it feasible to compute the finite differences and the differentials of those functions and propositions.

For example, consider a proposition of the form $``p ~\mathrm{and}~ q"$ graphed as two letters attached to a root node, as shown below.

Written as a string, this is just the concatenation $p~q$ .

The proposition $pq$ may be taken as a boolean function $f(p, q)$ having the abstract type $f : \mathbb{B} \times \mathbb{B} \to \mathbb{B},$ where $\mathbb{B} = \{ 0, 1 \}$ is read in such a way that $0$ means $\mathrm{false}$ and $1$ means $\mathrm{true}.$

Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition $pq$ is true, as shown in the following Figure.

Now ask yourself: What is the value of the proposition $pq$ at a distance of $\mathrm{d}p$ and $\mathrm{d}q$ from the cell $pq$ where you are standing?

Don’t think about it — just compute:

The cactus formula $\texttt{(} p \texttt{,} \mathrm{d}p \texttt{)(} q \texttt{,} \mathrm{d}q \texttt{)}$ and its corresponding graph arise by replacing $p$ with $p + \mathrm{d}p$ and $q$ with $q + \mathrm{d}q$ in the boolean product or logical conjunction $pq$ and writing the result in the two dialects of cactus syntax. This follows because the boolean sum $p + \mathrm{d}p$ is equivalent to the logical operation of exclusive disjunction, which parses to a cactus graph of the following form.

Next question: What is the difference between the value of the proposition $pq$ over there, at a distance of $\mathrm{d}p$ and $\mathrm{d}q$ from where you are standing, and the value of the proposition $pq$ where you are, all expressed in the form of a general formula, of course? The answer takes the following form.

There is one thing I ought to mention at this point: Computed over $\mathbb{B},$ plus and minus are identical operations. This will make the relation between the differential and the integral parts of the appropriate calculus slightly stranger than usual, but we will get into that later.

Last question, for now: What is the value of this expression from your current standpoint, that is, evaluated at the point where $pq$ is true? Well, replacing $p$ with $1$ and $q$ with $1$ in the cactus graph amounts to erasing the labels $p$ and $q,$ as shown below.

And this is equivalent to the following graph.

We have just met with the fact that the differential of the and is the or of the differentials.

$\begin{matrix} p ~\mathrm{and}~ q & \quad & \xrightarrow{\quad\mathrm{Diff}\quad} & \quad & \mathrm{d}p ~\mathrm{or}~ \mathrm{d}q \end{matrix}$

It will be necessary to develop a more refined analysis of that statement directly, but that is roughly the nub of it.

If the form of the above statement reminds you of De Morgan’s rule, it is no accident, as differentiation and negation turn out to be closely related operations. Indeed, one can find discussion of logical difference calculus in the personal correspondence between Boole and De Morgan and Peirce, too, made use of differential operators in a logical context, but the exploration of those ideas has been hampered by a number of factors, not the least of which has been the lack of a syntax adequate to handle the complexity of expressions evolving in the process.