Tag Archives: Logical Graphs

Unknown's avatar Jon Awbrey 11:32 am on February 12, 2026
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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.

Venn Diagram f = pq

Cactus Graph f = pq

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.

Venn Diagram p q dp dq

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}

Cactus Graph Ef = (p,dp)(q,dq)

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}

Cactus Graph Df = ((p,dp)(q,dq),pq)

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

Venn Diagram Difference pq @ pq

Cactus Graph Difference pq @ pq

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

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Unknown's avatar Jon Awbrey 4:32 pm on February 8, 2026
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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.

Cactus Graph Existential p and q

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.

Venn Diagram p and q

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:

Cactus Graph (p,dp)(q,dq)

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.

Cactus Graph (p,dp)

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.

Cactus Graph ((p,dp)(q,dq),pq)

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.

Cactus Graph (( ,dp)( ,dq), )

And this is equivalent to the following graph.

Cactus Graph ((dp)(dq))

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}

Cactus Graph pq → Diff → ((dp)(dq))

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

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Unknown's avatar Jon Awbrey 5:20 pm on February 7, 2026
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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

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

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Unknown's avatar Jon Awbrey 10:45 am on February 6, 2026
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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.

Lobe Connective

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.

Node Connective

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.

Resources

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Unknown's avatar Jon Awbrey 4:00 pm on February 5, 2026
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Differential Logic • 1

Introduction

Differential logic is the component of logic whose object is the description of variation — focusing on the aspects of change, difference, distribution, and diversity — in universes of discourse subject to logical description.  A definition that broad naturally incorporates any study of variation by way of mathematical models, but differential logic is especially charged with the qualitative aspects of variation pervading or preceding quantitative models.

To the extent a logical inquiry makes use of a formal system, its differential component governs the use of a differential logical calculus, that is, a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.

Simple examples of differential logical calculi are furnished by differential propositional calculi.  A differential propositional calculus is a propositional calculus extended by a set of terms for describing aspects of change and difference, for example, processes taking place in a universe of discourse or transformations mapping a source universe to a target universe.  Such a calculus augments ordinary propositional calculus in the same way the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.

Resources

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Unknown's avatar Jon Awbrey 12:48 pm on February 3, 2026
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Differential Logic • Overview

A reader once told me “venn diagrams are obsolete” and of course we all know how unwieldy they become as our universes of discourse expand beyond four or five dimensions.  Indeed, one of the first lessons I learned when I set about implementing Peirce’s graphs and Spencer Brown’s forms on the computer is that 2‑dimensional representations of logic quickly become death traps on numerous conceptual and computational counts.

Still, venn diagrams do us good service at the outset in visualizing the relationships among extensional, functional, and intensional aspects of logic.  A facility with those connections is critical to the computational applications and statistical generalizations of propositional logic commonly used in mathematical and empirical practice.

All things considered, then, it is useful to make the links between various styles of imagery in logical representation as visible as possible.  The first few steps in that direction are set out in the sketch of Differential Logic to follow.

Part 1

Introduction

Cactus Language for Propositional Logic

Differential Expansions of Propositions

Bird’s Eye View

Worm’s Eye View

Panoptic View • Difference Maps

Panoptic View • Enlargement Maps

Part 2

Propositional Forms on Two Variables

Transforms Expanded over Ordinary and Differential Variables

Enlargement Map Expanded over Ordinary Variables

Enlargement Map Expanded over Differential Variables

Difference Map Expanded over Ordinary Variables

Difference Map Expanded over Differential Variables

Operational Representation

Part 3

Field Picture

Differential Fields

Propositions and Tacit Extensions

Enlargement and Difference Maps

Tangent and Remainder Maps

Least Action Operators

Goal-Oriented Systems

Further Reading

Document History

Document History

Differential Logic • Ontology List 2002

Dynamics And Logic • Inquiry List 2004

Dynamics And Logic • NKS Forum 2004

Resources

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Unknown's avatar Jon Awbrey 7:16 am on November 20, 2025
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Survey of Theme One Program • 7

This is a Survey of resources relating to the Theme One Program I worked on all through the 1980s.  The aim was to develop fundamental algorithms and data structures for integrating empirical learning with logical reasoning.  I had earlier developed separate programs for basic components of those tasks, in particular, two‑level formal language learning and propositional constraint satisfaction, the latter using an extension of C.S. Peirce’s logical graphs as a syntax for propositional logic.  Thus arose the question of how well it might be possible to get “empiricist” and “rationalist” modes of operation to cooperate.  The long‑term vision is the implementation of an Automated Research Tool able to double as a platform for Inquiry Driven Education.

Wiki Hub

Documentation

Blog Series

Blog Dialogs

Applications

References

  • Awbrey, S.M., and Awbrey, J.L. (May 1991), “An Architecture for Inquiry • Building Computer Platforms for Discovery”, Proceedings of the Eighth International Conference on Technology and Education, Toronto, Canada, pp. 874–875.  Online.
  • Awbrey, J.L., and Awbrey, S.M. (January 1991), “Exploring Research Data Interactively • Developing a Computer Architecture for Inquiry”, Poster presented at the Annual Sigma Xi Research Forum, University of Texas Medical Branch, Galveston, TX.
  • Awbrey, J.L., and Awbrey, S.M. (August 1990), “Exploring Research Data Interactively • Theme One : A Program of Inquiry”, Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training, Society for Applied Learning Technology, Washington, DC, pp. 9–15.  Online.

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Unknown's avatar Jon Awbrey 11:30 am on November 15, 2025
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Survey of Animated Logical Graphs • 8

This is a Survey of blog and wiki posts on Logical Graphs, encompassing several families of graph‑theoretic structures originally developed by Charles S. Peirce as graphical formal languages or visual styles of syntax amenable to interpretation for logical applications.

Beginnings

Elements

Examples

Blog Series

  • Logical Graphs • Interpretive Duality • (1) • (2) • (3) • (4)
  • Logical Graphs, Iconicity, Interpretation • (1) • (2)
  • Genus, Species, Pie Charts, Radio Buttons • (1)

Excursions

Applications

Anamnesis

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Unknown's avatar Jon Awbrey 11:48 am on November 9, 2025
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Survey of Relation Theory • 9

In the present Survey of blog and wiki resources for Relation Theory, relations are viewed from the perspective of combinatorics, in other words, as a topic in discrete mathematics, with special attention to finite structures and concrete set‑theoretic constructions, many of which arise quite naturally in applications.  This approach to relation theory is distinct from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other.

Elements

Relational Concepts

Relation Composition Relation Construction Relation Reduction
Relative Term Sign Relation Triadic Relation
Logic of Relatives Hypostatic Abstraction Continuous Predicate

Illustrations

Information‑Theoretic Perspective

  • Mathematical Demonstration and the Doctrine of Individuals • (1) • (2)

Blog Series

Peirce’s 1870 “Logic of Relatives”

Peirce’s 1880 “Algebra of Logic” Chapter 3

Peirce’s 1885 “Algebra of Logic”

  • C.S. Peirce • Algebra of Logic ∫ Philosophy of Notation • (1) • (2)
  • C.S. Peirce • Algebra of Logic 1885 • Selections • (1) • (2) • (3) • (4)

Resources

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Unknown's avatar Jon Awbrey 11:20 am on November 7, 2025
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Survey of Differential Logic • 8

This is a Survey of work in progress on Differential Logic, resources under development toward a more systematic treatment.

Differential logic is the component of logic whose object is the description of variation — the aspects of change, difference, distribution, and diversity — in universes of discourse subject to logical description.  A definition as broad as that naturally incorporates any study of variation by way of mathematical models, but differential logic is especially charged with the qualitative aspects of variation pervading or preceding quantitative models.  To the extent a logical inquiry makes use of a formal system, its differential component treats the use of a differential logical calculus — a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.

Elements

Blog Series

Architectonics

Applications

Blog Dialogs

Explorations

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