Live and recent posts from across the Fediverse tagged #logical-graphs, aggregated by home.social.
Language Learning And Logical Modeling —
Wrote my first “Language Learning Module”, strictly speaking, a two‑level formal language learner, back in the 80s and it pretty much told me what every conceivable upscale of that ilk would be like. But it did not cross the threshold of logical reasoning, so I used Peirce's logical graphs for that. Et sic deinceps …
#Peirce #Logic #Mathematics #Semiotics #LogicalGraphs
#LanguageLearningAlgorithm #LogicalModelingAlgorithm
Animated Logical Graphs • 2
• https://inquiryintoinquiry.com/2015/01/14/animated-logical-graphs-2/
It's almost 50 years now since I first encountered the volumes of Peirce's “Collected Papers” in the math library at Michigan State, and shortly afterwards a friend called my attention to the entry for Spencer Brown's “Laws of Form” in the Whole Earth Catalog and I sent off for it right away. I would spend the next decade just beginning to figure out what either one of them was talking about in the matter of logical graphs and I would spend another decade after that developing a program, first in Lisp and then in Pascal, that turned graph‑theoretic data structures formed on their ideas to good purpose as the basis of its reasoning engine.
I thought it might contribute to a number of long‑running and ongoing discussions if I could articulate what I think I learned from that experience.
So I'll try to keep focused on that.
Resources —
Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2024/08/26/logical-graphs-first-impressions-a/
Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2024/09/12/logical-graphs-formal-development-b/
Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/
#Peirce #Logic #Mathematics #Semiotics #LogicalGraphs #GraphTheory
#SpencerBrown #LawsOfForm #PropositionalCalculus #ProofAnimations
Animated Logical Graphs • 1
• https://inquiryintoinquiry.com/2015/01/08/animated-logical-graphs-1/
For Your Musement …
Here are some animations I made up to illustrate several different styles of proof in an extended topological variant of Peirce's Alpha Graphs for propositional logic.
Proof Animations
• https://oeis.org/wiki/User:Jon_Awbrey/ANIMATION#Proof_Animations
Double Negation
• https://inquiryintoinquiry.com/wp-content/uploads/2026/04/proof-animation-e280a2-double-negation-2.0.gif
Peirce's Law
• https://inquiryintoinquiry.com/wp-content/uploads/2026/04/proof-animation-e280a2-peirces-law-2.0.gif
Praeclarum Theorema
• https://inquiryintoinquiry.com/wp-content/uploads/2026/04/proof-animation-e280a2-praeclarum-theorema-2.0.gif
Two‑Thirds Majority Function
• https://inquiryintoinquiry.com/wp-content/uploads/2026/04/proof-animation-e280a2-two-thirds-majority-function-2.0.gif
A full discussion of logical graphs can be found in the following article.
Logical Graphs
• https://oeis.org/wiki/Logical_Graphs
Resources —
Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2024/08/26/logical-graphs-first-impressions-a/
Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2024/09/12/logical-graphs-formal-development-b/
Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/
cc: https://www.academia.edu/community/ldzadj
cc: https://mathstodon.xyz/@Inquiry/116494097283214718
cc: https://www.researchgate.net/post/Animated_Logical_Graphs
cc: https://stream.syscoi.com/2026/04/30/animated-logical-graphs-1/
cc: https://groups.io/g/lawsofform/topic/animated_logical_graphs/119049814
#Peirce #Logic #Mathematics #Semiotics #LogicalGraphs #GraphTheory
#SpencerBrown #LawsOfForm #PropositionalCalculus #ProofAnimations
Animated Logical Graphs • 1
• https://inquiryintoinquiry.com/2015/01/08/animated-logical-graphs-1/
For Your Musement …
Here are some animations I made up to illustrate several different styles of proof in an extended topological variant of Peirce’s Alpha Graphs for propositional logic.
Proof Animations
• https://oeis.org/wiki/User:Jon_Awbrey/ANIMATION#Proof_Animations
See the following article for a full discussion of this type of logical graph.
Logical Graphs
• https://oeis.org/wiki/Logical_Graphs
Additional Resources —
Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2024/08/26/logical-graphs-first-impressions-a/
Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2024/09/12/logical-graphs-formal-development-b/
#Peirce #Logic #Mathematics #Semiotics #LogicalGraphs #GraphTheory
#SpencerBrown #LawsOfForm #PropositionalCalculus #ProofAnimations
Differential Logic • The Logic of Change and Difference
• https://inquiryintoinquiry.com/2026/03/14/differential-logic-the-logic-of-change-and-difference-a/
“Differential logic is the logic of variation — the logic of change and difference.”
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.
A simple case of a differential logical calculus is furnished by a “differential propositional calculus”, a formalism which augments ordinary propositional calculus in the same way the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.
See —
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/
Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/
Differential Logic
• https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Overview
Differential Propositional Calculus
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Overview
Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview
cc: https://www.academia.edu/community/VXoNQ9
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference2
#Peirce #Logic #Mathematics #LogicalGraphs #DifferentialLogic #DynamicSystems
#Inquiry #PropositionalCalculus #BooleanFunctions #BooleanDifferenceCalculus
#EquationalInference #MinimalNegationOperators #CalculusOfLogicalDifferences
Differential Logic • 18
If we follow the classical line which singles out linear functions as ideals of simplicity then we may complete the analytic series of the proposition in the following way.
The next venn diagram shows the differential proposition we get by extracting the linear approximation to the difference map at each cell or point of the universe What results is the logical analogue of what would ordinarily be called the differential of but since the adjective differential is being attached to just about everything in sight the alternative name tangent map is commonly used for whenever it’s necessary to single it out.
To be clear about what’s being indicated here, it’s a visual way of summarizing the following data.
To understand the extended interpretations, that is, the conjunctions of basic and differential features which are being indicated here, it may help to note the following equivalences.
Capping the analysis of the proposition in terms of succeeding orders of linear propositions, the final venn diagram of the series shows the remainder map which happens to be linear in pairs of variables.
Reading the arrows off the map produces the following data.
In short, is a constant field, having the value at each cell.
Resources
cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi
Differential Logic • 17
Enlargement and Difference Maps
Continuing with the example the following venn diagram shows the enlargement or shift map in the same style of field picture we drew for the tacit extension
A very important conceptual transition has just occurred here, almost tacitly, as it were. Generally speaking, having a set of mathematical objects of compatible types, in this case the two differential fields and both of the type is very useful, because it allows us to consider those fields as integral mathematical objects which can be operated on and combined in the ways we usually associate with algebras.
In the present case one notices the tacit extension and the enlargement are in a sense dual to each other. The tacit extension indicates all the arrows out of the region where is true and the enlargement indicates all the arrows into the region where is true. The only arc they have in common is the no‑change loop at If we add the two sets of arcs in mod 2 fashion then the loop of multiplicity 2 zeroes out, leaving the 6 arrows of shown in the following venn diagram.
Resources
cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi
Differential Logic • 15
The structure of a differential field may be described as follows. With each point of there is associated an object of the following type: a proposition about changes in that is, a proposition In that frame of reference, if is the universe generated by the set of coordinate propositions then is the differential universe generated by the set of differential propositions The differential propositions and may thus be interpreted as indicating and respectively.
A differential operator of the first order type we are currently considering, takes a proposition and gives back a differential proposition In the field view of the scene, we see the proposition as a scalar field and we see the differential proposition as a vector field, specifically, a field of propositions about contemplated changes in
The field of changes produced by on is shown in the following venn diagram.
The differential field specifies the changes which need to be made from each point of in order to reach one of the models of the proposition that is, in order to satisfy the proposition
The field of changes produced by on is shown in the following venn diagram.
The differential field specifies the changes which need to be made from each point of in order to feel a change in the felt value of the field
Resources
cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi
Differential Logic • 14
Let us summarize the outlook on differential logic we’ve reached so far. We’ve been considering a class of operators on universes of discourse, each of which takes us from considering one universe of discourse to considering a larger universe of discourse An operator of that general type, namely, acts on each proposition of the source universe to produce a proposition of the target universe
The operators we’ve examined so far are the enlargement or shift operator and the difference operator The operators and act on propositions in that is, propositions of the form which amount to propositions about the subject matter of and they produce propositions of the form which amount to propositions about specified collections of changes conceivably occurring in
At this point we find ourselves in need of visual representations, suitable arrays of concrete pictures to anchor our more earthy intuitions and help us keep our wits about us as we venture into ever more rarefied airs of abstraction.
One good picture comes to us by way of the field concept. Given a space a field of a specified type over is formed by associating with each point of an object of type If that sounds like the same thing as a function from to the space of things of type — it is nothing but — and yet it does seem helpful to vary the mental images and take advantage of the figures of speech most naturally springing to mind under the emblem of the field idea.
In the field picture a proposition becomes a scalar field, that is, a field of values in
For example, consider the logical conjunction shown in the following venn diagram.
Each of the operators takes us from considering propositions here viewed as scalar fields over to considering the corresponding differential fields over analogous to what in real analysis are usually called vector fields over
Resources
cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi
Differential Logic • 13
Transforms Expanded over Ordinary and Differential Variables
Two views of how the difference operator acts on the set of sixteen functions are shown below. Table A5 shows the expansion of over the set of ordinary variables and Table A6 shows the expansion of over the set of differential variables.
Difference Map Expanded over Ordinary Variables
Difference Map Expanded over Differential Variables
Resources
cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi
Differential Logic • 2.2
• https://inquiryintoinquiry.com/2026/02/06/differential-logic-2-b/
Cactus Language for Propositional Logic (cont.)
The second kind of connective is a concatenated sequence of propositional expressions, written e₁ e₂ … eₖ₋₁ eₖ to mean all the propositions e₁, e₂, …, eₖ₋₁, eₖ 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 e₁, e₂, …, eₖ₋₁, eₖ as shown below.
Node Connective
• https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-ej-node-connective.jpg
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 (…) may be used for the logical operators.
Resources —
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/
Minimal Negation Operator
• https://oeis.org/wiki/Minimal_negation_operator
Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/
Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/
#Peirce #Logic #Mathematics #LogicalGraphs #DifferentialLogic #DynamicSystems
#Inquiry #PropositionalCalculus #BooleanFunctions #BooleanDifferenceCalculus
#EquationalInference #MinimalNegationOperators #CalculusOfLogicalDifferences
Differential Logic • 2.1
• https://inquiryintoinquiry.com/2026/02/06/differential-logic-2-b/
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 (e₁, e₂, …, eₖ₋₁, eₖ) to mean exactly one of the propositions e₁, e₂, …, eₖ₋₁, eₖ 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 e₁, e₂, …, eₖ₋₁, eₖ as shown below.
Lobe Connective
• https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-ej-lobe-connective.jpg
Resources —
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/
Minimal Negation Operator
• https://oeis.org/wiki/Minimal_negation_operator
Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/
Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/
#Peirce #Logic #Mathematics #LogicalGraphs #DifferentialLogic #DynamicSystems
#Inquiry #PropositionalCalculus #BooleanFunctions #BooleanDifferenceCalculus
#EquationalInference #MinimalNegationOperators #CalculusOfLogicalDifferences
Differential Logic • 1
• https://inquiryintoinquiry.com/2026/02/05/differential-logic-1-b/
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 —
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/
Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/
#Peirce #Logic #Mathematics #LogicalGraphs #DifferentialLogic #DynamicSystems
#Inquiry #PropositionalCalculus #BooleanFunctions #BooleanDifferenceCalculus
#EquationalInference #MinimalNegationOperators #CalculusOfLogicalDifferences
Differential Logic • Overview
• https://inquiryintoinquiry.com/2026/02/03/differential-logic-overview-b/
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.
Resources —
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/
Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/
Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/
#Peirce #Logic #Mathematics #LogicalGraphs #DifferentialLogic #DynamicSystems
#Inquiry #PropositionalCalculus #BooleanFunctions #BooleanDifferenceCalculus
#EquationalInference #MinimalNegationOperators #CalculusOfLogicalDifferences
Propositions As Types Analogy • 1
• https://inquiryintoinquiry.com/2013/01/29/propositions-as-types-analogy-1/
One of my favorite mathematical tricks — it almost seems too tricky to be true — is the Propositions As Types Analogy. And I see hints the 2‑part analogy can be extended to a 3‑part analogy, as follows.
Proof Hint ∶ Proof ∶ Proposition
∷
Untyped Term ∶ Typed Term ∶ Type
or
Proof Hint ∶ Untyped Term
∷
Proof ∶ Typed Term
∷
Proposition ∶ Type
See my working notes on the Propositions As Types Analogy —
• https://oeis.org/wiki/Propositions_As_Types_Analogy
#Mathematics #CategoryTheory #ProofTheory #TypeTheory
#Logic #Analogy #Isomorphism #PropositionalCalculus
#CombinatorCalculus #CombinatoryLogic #LambdaCalculus
#Peirce #LogicalGraphs #GraphTheory #RelationTheory
This thing all things devours:
Birds, beasts, trees, flowers;
Gnaws iron, bites steel;
Grinds hard stones to meal;
Slays king, ruins town,
And beats high mountain down.
— Tolkien • The Hobbit
Talking about time is a waste of time. Time is merely an abstraction from process and what is needed are better languages and better pictures for describing process in all its variety. In the sciences the big breakthrough in describing process came with the differential and integral calculus, that made it possible to shuttle between quantitative measures of state and quantitative measures of change. But every inquiry into a new phenomenon begins with the slimmest grasp of its qualitative features and labors long and hard to reach as far as a tentative logical description. What can avail us in the mean time, still tuning up before the first measure, to reason about change in qualitative terms?
Et sic deinceps … (So it begins …)
#Animata, #CSPeirce, #Change, #Cybernetics, #DifferentialLogic, #GraphTheory, #LawsOfForm, #Logic, #LogicalGraphs, #Mathematics, #Paradox, #Peirce, #Process, #ProcessThinking, #SpencerBrown, #SystemsTheory, #Time, #Tolkien
Cactus Language • Overview 3.2
• https://inquiryintoinquiry.com/2025/03/07/cactus-language-overview-3/
Given a body of conceivable propositions we need a way to follow the threads of their indications from their object domain to their values for the mind and a way to follow those same threads back again. Moreover, we need to implement both ways of proceeding in computational form. Thus we need programs for tracing the clues sentences provide from the universe of their objects to the signs of their values and, in turn, from signs to objects. Ultimately, we need to render propositions so functional as indicators of sets and so essential for examining the equality of sets as to give a rule for the practical conceivability of sets. Tackling that task requires us to introduce a number of new definitions and a collection of additional notational devices, to which we now turn.
Resources —
Cactus Language • Overview
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview
Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/
Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/
#Peirce #Logic #Semiotics #LogicalGraphs #DifferentialLogic
#AutomataTheory #FormalLanguages #FormalGrammars #GraphTheory
Cactus Language • Overview 3.1
• https://inquiryintoinquiry.com/2025/03/07/cactus-language-overview-3/
In the development of Cactus Language to date the following two species of graphs have been instrumental.
• Painted And Rooted Cacti (PARCAI).
• Painted And Rooted Conifers (PARCOI).
It suffices to begin with the first class of data structures, developing their properties and uses in full, leaving discussion of the latter class to a part of the project where their distinctive features are key to developments at that stage. Partly because the two species are so closely related and partly for the sake of brevity, we'll always use the genus name “PARC” to denote the corresponding cacti.
To provide a computational middle ground between sentences seen as syntactic strings and propositions seen as indicator functions the language designer must not only supply a medium for the expression of propositions but also link the assertion of sentences to a means for inverting the indicator functions, that is, for computing the “fibers” or “inverse images” of the propositions.
#Peirce #Logic #Semiotics #LogicalGraphs #DifferentialLogic
#AutomataTheory #FormalLanguages #FormalGrammars #GraphTheory
Cactus Language • Overview 2
• https://inquiryintoinquiry.com/2025/03/06/cactus-language-overview-2/
In order to facilitate the use of propositions as indicator functions it helps to acquire a flexible notation for referring to propositions in that light, for interpreting sentences in a corresponding role, and for negotiating the requirements of mutual sense between the two domains. If none of the formalisms readily available or in common use meet all the design requirements coming to mind then it is necessary to contemplate the design of a new language especially tailored to the purpose.
In the present application, there is a pressing need to devise a general calculus for composing propositions, computing their values on particular arguments, and inverting their indications to arrive at the sets of things in the universe which are indicated by them.
For computational purposes it is convenient to have a middle ground or an intermediate language for negotiating between the “koine” of sentences regarded as strings of literal characters and the realm of propositions regarded as objects of logical value, even if that makes it necessary to introduce an artificial medium of exchange between the two domains.
If the necessary computations are to be carried out in an organized fashion, and ultimately or partially by familiar classes of machines, then the strings expressing logical propositions are likely to find themselves parsed into tree‑like data structures at some stage of the game. As far as their abstract structures as graphs are concerned, there are several species of graph‑theoretic data structures fitting the task in a reasonably effective and efficient way.
#Peirce #Logic #Semiotics #LogicalGraphs #DifferentialLogic
#FormalLanguages
Cactus Language • Overview 1.2
• https://inquiryintoinquiry.com/2025/03/01/cactus-language-overview-1/
Resource —
For readers interested and intrepid enough to read ahead, here’s an outline of my work in progress on the OEIS Wiki, which I’ll be revising and serializing to my Inquiry blog.
Part 1
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Syntax
Part 2
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_2
Generalities About Formal Grammars
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_2#Generalities
Part 3
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3
Cactus Language • Stylistics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Stylistics
Cactus Language • Mechanics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Mechanics
Cactus Language • Semantics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Semantics
Stretching Exercises
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Stretching_Exercises
References
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_References
Document History
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Document_History
#Peirce #Logic #Semiotics #LogicalGraphs #DifferentialLogic
#Automata #FormalLanguages #FormalGrammars #GraphTheory
Cactus Language • Overview 1.1
• https://inquiryintoinquiry.com/2025/03/01/cactus-language-overview-1/
❝Thus, what looks to us like a sphere of scientific knowledge more accurately should be represented as the inside of a highly irregular and spiky object, like a pincushion or porcupine, with very sharp extensions in certain directions, and virtually no knowledge in immediately adjacent areas. If our intellectual gaze could shift slightly, it would alter each quill’s direction, and suddenly our entire reality would change.❞
— Herbert J. Bernstein • “Idols of Modern Science”
The following report describes a calculus for representing propositions as sentences, that is, as syntactically defined sequences of signs, and for working with those sentences in light of their semantically defined contents as logical propositions. In their computational representation the expressions of the calculus parse into a class of graph‑theoretic data structures whose underlying graphs are called “painted cacti”.
Painted cacti are a specialization of what graph‑theorists refer to as “cacti”, which are in turn a generalization of what they call “trees”. The data structures corresponding to painted cacti have especially nice properties, not only useful in computational terms but interesting from a theoretical standpoint. The remainder of the present Overview is devoted to motivating the development of the indicated family of formal languages, going under the generic name of Cactus Language.
#Peirce #Logic #Semiotics #LogicalGraphs #DifferentialLogic
#Automata #FormalLanguages #FormalGrammars #GraphTheory
Charles Sanders Peirce, George Spencer Brown, and Me • 4
• https://inquiryintoinquiry.com/2017/08/06/charles-sanders-peirce-george-spencer-brown-and-me-4/
• https://bsky.app/profile/inquiryintoinquiry.bsky.social/post/3lh5fsszkmk23
Two things impacting my studies of Peirce and Spencer Brown over the years were my parallel studies in mathematics and computer science. In the overlap between those areas came courses in logic, mathematical linguistics, and the theory of formal languages, grammars, and automata.
My intellectual wanderings over a nine‑year undergraduate career would take me through a cycle of majors from math and physics, to communication, psychology, philosophy, and a cross‑cultural liberal arts program, then back to grad school in mathematics.
The puzzles Peirce and Spencer Brown beset my brain with were a big part of what drove me back to math, since I could see I had no chance of resolving them without learning a lot more algebra, logic, and topology than I had learned till then.
Charles Sanders Peirce, George Spencer Brown, and Me • 1
• https://inquiryintoinquiry.com/2017/07/20/charles-sanders-peirce-george-spencer-brown-and-me-1/
• https://bsky.app/profile/inquiryintoinquiry.bsky.social/post/3lgxtd6z3nk2t
It’s almost 50 years now since I first encountered the volumes of Peirce’s Collected Papers in the math library at Michigan State, and shortly afterwards a friend called my attention to the entry for Spencer Brown’s Laws of Form in the Whole Earth Catalog and I sent off for it right away.
I would spend the next decade just beginning to figure out what either one of them was talking about in the matter of logical graphs and I would spend another decade after that developing a program, first in Lisp and then in Pascal, converting graph-theoretic data structures formed on their ideas to good purpose in the mechanics of its propositional reasoning engine. I thought it might contribute to a number of ongoing discussions if I could articulate what I think I learned from that experience.
Differential Propositional Calculus • 10
Special Classes of Propositions (cont.)
Let’s pause at this point and get a better sense of how our special classes of propositions are structured and how they relate to propositions in general. We can do this by recruiting our visual imaginations and drawing up a sufficient budget of venn diagrams for each family of propositions. The case for 3 variables is exemplary enough for a start.
Linear Propositions
The linear propositions, may be written as sums:
One thing to keep in mind about these sums is that the values in are added “modulo 2”, that is, in such a way that
In a universe of discourse based on three boolean variables, the linear propositions take the shapes shown in Figure 8.
At the top is the venn diagram for the linear proposition of rank 3, which may be expressed by any one of the following three forms.
Next are the venn diagrams for the three linear propositions of rank 2, which may be expressed by the following three forms, respectively.
Next are the three linear propositions of rank 1, which are none other than the three basic propositions,
At the bottom is the linear proposition of rank 0, the everywhere false proposition or the constant function, which may be expressed by the form or by a simple
Resources
cc: Academia.edu • Cybernetics • Structural Modeling • Systems Science
cc: Conceptual Graphs • Laws of Form • Mathstodon • Research Gate
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Differential Propositional Calculus • 8
• https://inquiryintoinquiry.com/2024/12/07/differential-propositional-calculus-8-b/
Formal Development (cont.)
Before moving on, let's unpack some of the assumptions, conventions, and implications involved in the array of concepts and notations introduced above.
A universe of discourse A° = [a₁, …, aₙ] qualified by the logical features a₁, …, aₙ is a set A plus the set of all functions from the space A to the boolean domain B = {0, 1}. There are 2ⁿ elements in A, often pictured as the cells of a venn diagram or the nodes of a hypercube. There are 2^(2ⁿ) possible functions from A to B, accordingly pictured as all the ways of painting the cells of a venn diagram or the nodes of a hypercube with a palette of two colors.
A logical proposition about the elements of A is either true or false of each element in A, while a function f : A → B evaluates to 1 or 0 on each element of A. The analogy between logical propositions and boolean-valued functions is close enough to adopt the latter as models of the former and simply refer to the functions f : A → B as propositions about the elements of A.
Resources —
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/
Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7
#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics
Differential Propositional Calculus • 7.2
• https://inquiryintoinquiry.com/2024/12/06/differential-propositional-calculus-7-b/
Formal Development (cont.)
Table 7 summarizes the basic notations needed to describe ordinary propositional calculi in a systematic fashion.
Table 7. Propositional Calculus • Basic Notation
• https://inquiryintoinquiry.files.wordpress.com/2020/02/propositional-calculus-basic-notation.png
Resources —
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/
Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7
#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics
Differential Propositional Calculus • 7.1
• https://inquiryintoinquiry.com/2024/12/06/differential-propositional-calculus-7-b/
Note. Please see the blog post linked above for the proper formats of the notations used below.
Formal Development —
The preceding discussion outlined the ideas leading to the differential extension of propositional logic. The next task is to lay out the concepts and terminology needed to describe various orders of differential propositional calculi.
Elementary Notions —
Logical description of a universe of discourse begins with a collection of logical signs. For simplicity in a first approach we assume the signs are collected in the form of a finite alphabet, ‡A‡ = {“a₁”, …, “aₙ”}. The signs are interpreted as denoting logical features, for example, properties of objects in the universe of discourse or simple propositions about those objects. Corresponding to the alphabet ‡A‡ there is then a set of logical features, †A† = {a₁, …, aₙ}.
A set of logical features †A† = {a₁, …, aₙ} affords a basis for generating an n‑dimensional universe of discourse, written A° = [†A†] = [a₁, …, aₙ]. It is useful to consider a universe of discourse as a categorical object incorporating both the set of points A = <a₁, …, aₙ> and the set of propositions A↑ = {f : A → B} implicit with the ordinary picture of a venn diagram on n features.
Accordingly, the universe of discourse A° may be regarded as an ordered pair (A, A↑) bearing the type (Bⁿ, (Bⁿ → B)), which type designation may be abbreviated as Bⁿ +→ B or even more succinctly as [Bⁿ]. For convenience, the data type of a finite set on n elements may be indicated by either one of the equivalent notations [n] or *n*.
Differential Propositional Calculus • 6.2
• https://inquiryintoinquiry.com/2024/12/04/differential-propositional-calculus-6-b/
Cactus Calculus (cont.)
The briefest expression for logical truth is the empty word, denoted ε or λ in formal languages, where it forms the identity element for concatenation. It may be given visible expression in textual settings by means of the logically equivalent form (()), 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. For example, we have the following paraphrases of algebraic expressions.
• x + y = (x, y)
• x + y + z = ((x, y), z) = (x, (y, z))
It is important to note the last expressions are not equivalent to the triple bracket (x, y, z).
Resources —
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/
Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7
#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics
Differential Propositional Calculus • 6.1
• https://inquiryintoinquiry.com/2024/12/04/differential-propositional-calculus-6-b/
Cactus Calculus —
Table 6 outlines a syntax for propositional calculus based on two types of logical connectives, both of variable k‑ary scope.
• A bracketed sequence of propositional expressions (e₁, e₂, …, eₖ) is taken to mean exactly one of the propositions e₁, e₂, …, eₖ is false, in other words, their “minimal negation” is true.
• A concatenated sequence of propositional expressions e₁ e₂ … eₖ is taken to mean every one of the propositions e₁, e₂, …, eₖ is true, in other words, their “logical conjunction” is true.
Table 6. Syntax and Semantics of a Calculus for Propositional Logic
• https://inquiryintoinquiry.files.wordpress.com/2022/10/syntax-and-semantics-of-a-calculus-for-propositional-logic-4.0.png
All other propositional connectives may be obtained through combinations of the above two forms. As it happens, the concatenation form is dispensable in light of the bracket form but it is convenient to maintain it as an abbreviation for more complicated bracket expressions. While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for bracket forms. In contexts where parentheses are needed for other purposes “teletype” parentheses (…) or barred parentheses (|…|) may be used for logical operators.
#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics
Differential Propositional Calculus • 5
• https://inquiryintoinquiry.com/2024/12/03/differential-propositional-calculus-5-b/
Casual Introduction (concl.)
Table 5 exhibits the rules of inference responsible for giving the differential proposition dq its meaning in practice.
Table 5. Differential Inference Rules
• https://inquiryintoinquiry.files.wordpress.com/2020/02/differential-propositional-calculus-e280a2-differential-inference-rules.png
If the feature q is interpreted as applying to an object in the universe of discourse X then the differential feature dq may be taken as an attribute of the same object which tells it is changing “significantly” with respect to the property q — as if the object bore an “escape velocity” with respect to the condition q.
For example, relative to a frame of observation to be made more explicit later on, if q and dq are true at a given moment, it would be reasonable to assume ¬q will be true in the next moment of observation. Taken all together we have the fourfold scheme of inference shown above.
Resources —
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/
Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7
#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics
Systems of Interpretation • 2
Re: Peirce List • Mike Bergman • Valentine Daniel
Let’s start as simply as possible. The following Figure is typical of many I have used to illustrate sign relations from the time I first began studying Peirce’s theory of signs.
The above variant comes from a paper Susan Awbrey and I presented at a conference in 1999, a revised version of which was published in 2001.
As the drafter of that drawing I can speak with authority about the artist’s intentions in drawing it and also about the conventions of interpretation forming the matrix of its conception and delivery.
Just by way of refreshing my own memory, here is how we set it up —
Figure 2 represents an “elementary sign relation”. It is a single transaction taking place among three entities, the object the sign and the interpretant sign the association of which is typically represented by means of the ordered triple
One of the interpretive conventions implied in that setup is hallowed by long tradition, going back to the earliest styles of presentation in mathematics. In it one draws a figure intended as “representative” of many figures. Regarded as a concrete drawing the figure is naturally imperfect, individual, peculiar, and special but it’s meant to be taken purely as a representative of its class — generic, ideal, and typical. That is the main convention of interpretation which goes into giving diagrams and figures their significant power.
References
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science
cc: FB | Semeiotics • Mathstodon • Laws of Form • Academia.edu
Systems of Interpretation • 1
Re: Peirce List • Mike Bergman • Valentine Daniel
Questions have arisen about the different styles of diagrams and figures used to represent triadic sign relations in Peircean semiotics. What do they mean? Which style is best? Among the most popular pictures some use geometric triangles while others use the three‑pronged graphs Peirce used in his logical graphs to represent triadic relations.
Diagrams and figures, like any signs, can serve to communicate the intended interpretants and thus to coordinate the conduct of interpreters toward the intended objects — but only in communities of interpretation where the conventions of interpretation are understood. Conventions of interpretation are by comparison far more difficult to communicate.
That brings us to the first question we have to ask about the possibility of communication in this area, namely, what conventions of interpretation are needed to make sense of these diagrams, figures, and graphs?
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science
cc: FB | Semeiotics • Mathstodon • Laws of Form • Academia.edu
Logical Graphs, Truth Tables, Venn Diagrams • 1
Re: Peirce List • Mauro Bertani • Helmut Raulien
Dear Mauro, Helmut,
I’ll be focusing on logical graphs, especially the duality between entitative and existential interpretations, for quite a while longer, so this doesn’t address your questions about modal logic, but you might find it useful to compare the representations of logical operators by means of truth tables with those using logical graphs.
You could start with the top eight entries in the section headed “Logical Operators” on the following page.
There’s also a page bringing all eight of those Truth Tables together in one place.
I had been meaning to include the corresponding Logical Graphs and Venn Diagrams — I’ll spend some of my pandemic time working on that — It looks like it would be worth the candle reviewing their properties as representations of basic operations and going over their relative utilities for various logical purposes.
The following two pages also contain useful synopses of the boolean basics.
Resource
cc: Cybernetics • Ontolog Forum • Peirce List • Structural Modeling • Systems Science
cc: FB | Logical Graphs • Laws of Form
The Difference That Makes A Difference That Peirce Makes • 29
Re: Ontolog Forum • Jon Awbrey • John Sowa
We are called to recall Peirce’s deeper meanings
of ideas like Form (think Platonic Ideas and the
way Aristotle compounded Form and Matter). When
we come to Sentiment, by any other word, Feeling,
that is the medium of Aesthetics, which concerns
Beauty in no merely skin-deep sense but all that
embodies and manifests ‘the admirable in itself’,
thus every form of life worth living. So Peirce
stands the normative science of Logic on grounds
within the pale of Ethics and fixes the sight of
Ethics on the prize Aesthetics picks to steer by.
cc: Systems Science • Structural Modeling • Ontolog Forum • Laws of Form • Cybernetics
#Analogy #CSPeirce #Communication #DescriptiveScience #FixationOfBelief #FormalSystems #Information #Inquiry #Logic #LogicOfRelatives #LogicOfScience #LogicalGraphs #Mathematics #NormativeScience #Paradigms #Peirce #PragmaticMaxim #Pragmatism #RelationTheory #Semiotics #SignRelations #TriadicRelations #Triadicity
Icon Index Symbol • 15
Questions Concerning Certain Faculties Claimed For Signs
I put down the cup and turn to my mind. It is up to my mind to find the truth. But how? What grave uncertainty, whenever the mind feels overtaken by itself; when it, the seeker, is also the obscure country where it must seek and where all its baggage will be nothing to it. Seek? Not only that: create. It is face to face with something that does not yet exist and that only it can accomplish, and bring into its light.
— Proust • In Search of Lost Time
Re: Peirce List (1) (2) • Tom Gollier • Jerry Rhee • John Sowa
That passage from Proust epitomizes for me one of the most distinctive features of the inquiry process, the fact that its object is a state of information that the inquisiturus, the agent of inquiry, may never have known before.
I have thought of inquiry and intelligence in terms of cybernetic or system-theoretic processes ever since my first encounters with the works of Arbib, Ashby, Bateson, McCulloch, Wiener, Young, and others during my undergrad years. In the early 90s I returned to grad school in a Systems Engineering program with the idea that I might be able to string together many loose threads of unfinished business that continued to tug at my brain. Here’s a bit I wrote at the outset of that project, that comes to mind in this context:
Prospects for Inquiry Driven Systems • Architecture of Inquiry
It is important to remember that knowledge is a different sort of goal from the run-of-the-mill setpoints a system might have. The typical goal is a state that a system has actually experienced many times before, like normal body temperature for a human being. But a particular state of knowledge an intelligent system moves toward may be a state it has never been through before. The fundamental equivocation on this point expressed in Plato’s Meno, whether learning is functionally equivalent to remembering, was discussed above. In spite of this quibble, it still seems necessary to regard states of knowledge as a distinctive class. The reasons for this may lie in the fact that a useful definition of inquiry for human beings necessarily involves a whole community of inquiry.
Resources
Icon Index Symbol • 14
Questions Concerning Certain Faculties Claimed For Signs
Re: Peirce List (1) (2) (3) • Helmut Raulien
Re: Icon Index Symbol • (10) (11) (12) (13)
Let me sum up the main points of the above exchange before moving on.
Mathematics is useful in our present endeavor because it treats relations in general. In addition — and multiplication, too — mathematics is chock full of well‑studied examples of triadic relations. When it comes to the job of analyzing sign relations and teasing out their structures we can save ourselves a lot of trouble and trial and error by examining this record of prior art and adapting its methods to handle sign relations.
On the other hand, there are hints in Peirce’s work of how triadic relations extend across a threshold of complexity in such a way that relations of all higher adicities can be analyzed in terms of 1‑adic, 2‑adic, and 3‑adic relations. This is where the analogy between Peirce’s category theory and mathematical category theory both forms and breaks. The utility of categories in mathematical theory derives from the ubiquity of functions in mathematical practice, and functions are dyadic relations. Still, triadic relations pervade the background of mathematical category theory, being visible in the triadic composition relation and in the concept of “natural transformations”, the clarification of which is one of the original motivations for the subject. Bringing the triadic roots of category theory into higher relief is one of my motives for bringing about an encounter with Peirce’s categories, an effort to which I have given not a few years of thought.
That brings us to the case of sign relations proper. These types of triadic relations form our first stepping stones and also our first stumbling blocks in the inquiry into inquiry and I gave some indications why that might be true. I don’t know if I can do better than that at this time, but I’ll think on it further after that all‑essential secondness of caffeination.
Resources
Animated Logical Graphs • 6
Re: Peirce List • Jim Willgoose
At root we are dealing with a genre of very abstract formal systems. They have grammars that determine their well-formed expressions and rules that determine the permissible transformations among expressions, but they lack all logical meaning until we supply an interpretation.
The formal system Peirce developed for propositional logic and Spencer Brown resurrected for his Laws of Form admits a formal duality which allows it to be fleshed out with logical meanings in two distinct ways. The two interpretations are employed in Peirce’s entitative graphs and existential graphs, respectively. It is clear from everything they write that both authors are well aware of both interpretations, but Peirce would come to found his later developments on the existential sense while Spencer Brown favored the entitative sense in his expositions.
See the following readings for further discussion.
References
Animated Logical Graphs • 3
Re: Peirce List • Helmut Raulien
I have a little more leisure now to start climbing back into the saddle, so let me see where we left off …
Try looking into the article I linked before:
Or my first couple of blog posts on Logical Graphs:
There are literally decades of thought and work that went into those, and if they do not engage the reader in the excitement of possible future developments then I would sorely appreciate any feedback on where and why they fail to do so.
George Spencer Brown is one of the few writers I’ve run across in the time since my first encounter with Peirce’s logical graphs who truly grasped the full depth of Peirce’s insight into logic, a vision that pierced the veil of logical interpretations, entitative and existential, to the deep formal unity between them. That is one of the reasons I’ve made an effort to treat the two interpretations in parallel as far as I was able. It is an extremely enticing research question to me whether that symmetry is necessarily broken as we pass from propositional to quantificational logic, or whether there is some way it may be or need be maintained.
But that is a question for the future …
Resource
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Animated Logical Graphs • 2
Re: Peirce List • Jim Willgoose
It’s almost 50 years now since I first encountered the volumes of Peirce’s Collected Papers in the math library at Michigan State, and shortly afterwards a friend called my attention to the entry for Spencer Brown’s Laws of Form in the Whole Earth Catalog and I sent off for it right away. I would spend the next decade just beginning to figure out what either one of them was talking about in the matter of logical graphs and I would spend another decade after that developing a program, first in Lisp and then in Pascal, that turned graph‑theoretic data structures formed on their ideas to good purpose as the basis of its reasoning engine.
I thought it might contribute to a number of long‑running and ongoing discussions if I could articulate what I think I learned from that experience.
So I’ll try to keep focused on that.
Resources
cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi
Animated Logical Graphs • 1
For Your Musement …
Here are some animations I made up to illustrate several different styles of proof in an extended topological variant of Peirce’s Alpha Graphs for propositional logic.
A full discussion of logical graphs can be found in the following article.
Resources
cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi
Praeclarum Theorema
Introduction
The praeclarum theorema, or splendid theorem, is a theorem of propositional calculus noted and named by G.W. Leibniz, who stated and proved it in the following manner.
If a is b and d is c, then ad will be bc.
This is a fine theorem, which is proved in this way:
a is b, therefore ad is bd (by what precedes),
d is c, therefore bd is bc (again by what precedes),
ad is bd, and bd is bc, therefore ad is bc. Q.E.D.
— Leibniz • Logical Papers, p. 41.
Expressed in contemporary logical notation, the theorem may be written as follows.
Expressed as a logical graph under the existential interpretation, the theorem takes the shape of the following formal equivalence or propositional equation.
And here’s a neat proof of that nice theorem —
The steps of the proof are replayed in the following animation.
Reference
Readings
Resources
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cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science