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 ( 25 ), Animata ( 27 ), Boolean Algebra ( 25 ), Boolean Functions ( 26 ), C.S. Peirce ( 56 ), Cactus Graphs ( 26 ), Change ( 22 ), cybernetics ( 35 ), Differential Calculus ( 22 ), Differential Logic ( 26 ), Discrete Dynamics ( 22 ), Equational Inference ( 24 ), Functional Logic ( 23 ), Gradient Descent ( 21 ), Graph Theory ( 28 ), Inquiry Driven Systems ( 33 ), logic ( 58 ), Logical Graphs ( 27 ), Mathematics ( 56 ), Minimal Negation Operators ( 26 ), Propositional Calculus ( 26 ), Time ( 21 ), Visualization ( 38 )

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

#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

Jon Awbrey 11:32 am on February 12, 2026
Tags: Amphecks ( 25 ), Animata ( 27 ), Boolean Algebra ( 25 ), Boolean Functions ( 26 ), C.S. Peirce ( 56 ), Cactus Graphs ( 26 ), Change ( 22 ), cybernetics ( 35 ), Differential Calculus ( 22 ), Differential Logic ( 26 ), Discrete Dynamics ( 22 ), Equational Inference ( 24 ), Functional Logic ( 23 ), Gradient Descent ( 21 ), Graph Theory ( 28 ), Inquiry Driven Systems ( 33 ), logic ( 58 ), Logical Graphs ( 27 ), Mathematics ( 56 ), Minimal Negation Operators ( 26 ), Propositional Calculus ( 26 ), Time ( 21 ), Visualization ( 38 )

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 ( 25 ), Animata ( 27 ), Boolean Algebra ( 25 ), Boolean Functions ( 26 ), C.S. Peirce ( 56 ), Cactus Graphs ( 26 ), Change ( 22 ), cybernetics ( 35 ), Differential Calculus ( 22 ), Differential Logic ( 26 ), Discrete Dynamics ( 22 ), Equational Inference ( 24 ), Functional Logic ( 23 ), Gradient Descent ( 21 ), Graph Theory ( 28 ), Inquiry Driven Systems ( 33 ), logic ( 58 ), Logical Graphs ( 27 ), Mathematics ( 56 ), Minimal Negation Operators ( 26 ), Propositional Calculus ( 26 ), Time ( 21 ), Visualization ( 38 )

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.

Resources

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

antlerboy - Benjamin P Taylor 1:22 pm on February 8, 2026

Harish’s Notebook – When the Map Becomes More Coherent Than the Territory – Jose (2026)

When the Map Becomes More Coherent Than the Territory:

Jon Awbrey 5:20 pm on February 7, 2026
Tags: Amphecks ( 25 ), Animata ( 27 ), Boolean Algebra ( 25 ), Boolean Functions ( 26 ), C.S. Peirce ( 56 ), Cactus Graphs ( 26 ), Change ( 22 ), cybernetics ( 35 ), Differential Calculus ( 22 ), Differential Logic ( 26 ), Discrete Dynamics ( 22 ), Equational Inference ( 24 ), Functional Logic ( 23 ), Gradient Descent ( 21 ), Graph Theory ( 28 ), Inquiry Driven Systems ( 33 ), logic ( 58 ), Logical Graphs ( 27 ), Mathematics ( 56 ), Minimal Negation Operators ( 26 ), Propositional Calculus ( 26 ), Time ( 21 ), Visualization ( 38 )

Differential Logic • 3

Cactus Language for Propositional Logic (cont.)

Table 1 shows the cactus graphs, the corresponding cactus expressions, their logical meanings under the so‑called existential interpretation, and their translations into conventional notations for a sample of basic propositional forms.

Table 1. Syntax and Semantics of a Calculus for Propositional Logic

The simplest expression for logical truth is the empty word, typically denoted by $\boldsymbol\varepsilon$ or $\lambda$ in formal languages, where it is the identity element for concatenation. To make it visible in context, it may be denoted by the equivalent expression $``\texttt{(())}"$ or, especially if operating in an algebraic context, by a simple $``1".$ Also when working in an algebraic mode, the plus sign $``+"$ may be used for exclusive disjunction. Thus we have the following translations of algebraic expressions into cactus expressions.

$\begin{matrix} a + b \quad = \quad \texttt{(} a \texttt{,} b \texttt{)} \\[8pt] a + b + c \quad = \quad \texttt{(} a \texttt{,(} b \texttt{,} c \texttt{))} \quad = \quad \texttt{((} a \texttt{,} b \texttt{),} c \texttt{)} \end{matrix}$

It is important to note the last expressions are not equivalent to the 3‑place form $\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}.$

Resources

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

antlerboy - Benjamin P Taylor 4:18 pm on February 7, 2026

Laws of Form Conference 2026 (LoF26), 10 – 14 August 2026, University of Cambridge, UK

Call for Abstracts

Submissions for presentations of papers, panel discussions, workshops, performance sessions, and creative contributions, inspired by George Spencer-Brown’s work and life, are now warmly invited for the Laws of Form 2026 Conference (LoF26).

There is no charge to attend or present at the conference.

Submission Guidelines

Please submit an extended abstract (up to 300 words) outlining the content and structure of your proposed contribution. Please include:

• Title of your presentation

• Name, affiliation, and contact email address

• Format preference (paper presentation, panel discussion, workshop, creative, etc.)

• Short biographical note (≤ 150 words)

• Any AV / technical / access requirements

• Submission deadline Sunday 1^st March 2026

• Notification of acceptance 31^st March 2026

Facilities for remote video presentations will be available for those unable to attend in person.

Please send submissions as a single PDF or Word file to: conferences@lof50.com

Subject line: “LoF26 Submission – [Your Name]”

If you have a Google account you may prefer to upload your submission here:

https://forms.gle/zknFvXWQXzmfQtn2A

As with previous conferences, and subject to peer review, contributions may be published in Distinction: Journal of Form (College Publications Ltd) or in future volumes of the Spencer-Brown Society book series Marked States.

Venue

University of Cambridge

Faculty of Education

184 Hills Road

Cambridge CB2 8PQ, United Kingdom

Monday 10 August – Friday 14 August 2026

Social & Cultural Events

In addition to the conference, optional events will include:

• Punting on the River Cam

• Evensong at King’s College Chapel

• A meal at the Eagle pub, where, on 28 February 1953, Francis Crick dramatically announced that he and James Watson had “discovered the secret of life.”

Support

LoF26 is entirely free to attend, made possible through the generosity of the Faculty of Education, Cambridge University, sponsors, and individual supporters.

Donations

Contributions toward the costs of running the conference and sharing its results are deeply appreciated and help ensure that participation remains open to all. Every contribution — large or small — directly sustains the continuation of this unique, open, and evolving forum dedicated to the work and life of George Spencer-Brown. If you are able to support this ongoing work, please make a donation through our website: https://lof50.com/

For enquiries concerning LoF26 please email: conferences@lof50.com

Join the Spencer-Brown Society

Membership of the Spencer-Brown Society is open to all and free of charge. To join and receive updates on conferences and publications, please visit https://lof50.com

We look forward to welcoming you to Cambridge in August for LoF26!

Jon Awbrey 10:45 am on February 6, 2026
Tags: Amphecks ( 25 ), Animata ( 27 ), Boolean Algebra ( 25 ), Boolean Functions ( 26 ), C.S. Peirce ( 56 ), Cactus Graphs ( 26 ), Change ( 22 ), cybernetics ( 35 ), Differential Calculus ( 22 ), Differential Logic ( 26 ), Discrete Dynamics ( 22 ), Equational Inference ( 24 ), Functional Logic ( 23 ), Gradient Descent ( 21 ), Graph Theory ( 28 ), Inquiry Driven Systems ( 33 ), logic ( 58 ), Logical Graphs ( 27 ), Mathematics ( 56 ), Minimal Negation Operators ( 26 ), Propositional Calculus ( 26 ), Time ( 21 ), Visualization ( 38 )

Differential Logic • 2

Cactus Language for Propositional Logic

The development of differential logic is facilitated by having a moderately efficient calculus in place at the level of boolean‑valued functions and elementary logical propositions. One very efficient calculus on both conceptual and computational grounds is based on just two types of logical connectives, both of variable $k$ -ary scope. The syntactic formulas of that calculus map into a family of graph-theoretic structures called “painted and rooted cacti” which lend visual representation to the functional structures of propositions and smooth the path to efficient computation.

The first kind of connective is a parenthesized sequence of propositional expressions, written $\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)}$ to mean exactly one of the propositions $e_1, e_2, \ldots, e_{k-1}, e_k$ is false, in short, their minimal negation is true. An expression of that form is associated with a cactus structure called a lobe and is “painted” with the colors $e_1, e_2, \ldots, e_{k-1}, e_k$ as shown below.

The second kind of connective is a concatenated sequence of propositional expressions, written $e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k$ to mean all the propositions $e_1, e_2, \ldots, e_{k-1}, e_k$ are true, in short, their logical conjunction is true. An expression of that form is associated with a cactus structure called a node and is “painted” with the colors $e_1, e_2, \ldots, e_{k-1}, e_k$ as shown below.

All other propositional connectives can be obtained through combinations of the above two forms. As it happens, the parenthesized form is sufficient to define the concatenated form, making the latter formally dispensable, but it’s convenient to maintain it as a concise way of expressing more complicated combinations of parenthesized forms. While working with expressions solely in propositional calculus, it’s easiest to use plain parentheses for logical connectives. In contexts where ordinary parentheses are needed for other purposes an alternate typeface $\texttt{(} \ldots \texttt{)}$ may be used for the logical operators.