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
Jon Awbrey
Inquiry Into Inquiry