Inquiry Into Inquiry

Transformations of Logical Graphs • 2

Posted on May 6, 2024 by Jon Awbrey

Re: Transformations of Logical Graphs • 1

Another way of looking at the dual interpretation of logical graphs from a group-theoretic point of view is provided by the following Table. In this arrangement we have sorted the rows of the previous Table to bring together similar graphs $\gamma$ belonging to the set $X,$ the similarity being determined by the action of the group $G = \{ 1, t \}.$ Transformation group theorists refer to the corresponding similarity classes as orbits of the group action under consideration. The orbits are defined by the group acting transitively on them, meaning elements of the same orbit can always be transformed into one another by some group operation while elements of different orbits cannot.

$\text{Interpretive Duality as Group Symmetry} \stackrel{_\bullet}{} \text{Orbit Order}$

Scanning the Table we observe the 16 points of $X$ fall into 10 orbits total, divided into 4 orbits of 1 point each and 6 orbits of 2 points each. The points in singleton orbits are called fixed points of the transformation group since they are not moved but mapped into themselves by all group actions. The bottom row of the Table tabulates the total number of fixed points for the group operations $1$ and $t$ respectively. The group identity $1$ always fixes all points, so its total is 16. The group action $t$ fixes only the four points in singleton orbits, giving a total of 4.

I leave it as an exercise for the reader to investigate the relationship between the group order $|G| = 2,$ the number of orbits $10,$ and the total number of fixed points $16 + 4 = 20.$

Resources

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Interpretive Duality, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Visualization | Tagged Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Interpretive Duality, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Visualization | 6 Comments

Transformations of Logical Graphs • 1

Posted on May 5, 2024 by Jon Awbrey

Re: Interpretive Duality in Logical Graphs • 1
Re: Mathematical Duality in Logical Graphs • 1

Anything called a duality is naturally associated with a transformation group of order 2, say a group $G$ acting on a set $X.$ Transformation groupies normally refer to $X$ as a set of “points” even when the elements have additional structure of their own, as they often do. A group of order two has the form $G = \{ 1, t \},$ where $1$ is the identity element and the remaining element $t$ satisfies the equation $t^2 = 1,$ being on that account self-inverse.

A first look at the dual interpretation of logical graphs from a group-theoretic point of view is provided by the following Table. The sixteen boolean functions $f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}$ on two variables are listed in Column 1. Column 2 lists the elements of the set $X,$ specifically, the sixteen logical graphs $\gamma$ giving canonical expression to the boolean functions in Column 1. Column 2 shows the graphs in existential order but the order is arbitrary since only the transformations of the set $X$ into itself are material in this setting. Column 3 shows the result $1 \gamma$ of the group element $1$ acting on each graph $\gamma$ in $X,$ which is of course the same graph $\gamma$ back again. Column 4 shows the result $t \gamma$ of the group element $t$ acting on each graph $\gamma$ in $X,$ which is the entitative graph dual to the existential graph in Column 2.

$\text{Interpretive Duality as Group Symmetry}$

The last Row of the Table displays a statistic of considerable interest to transformation group theorists. It is the total incidence of fixed points, in other words, the number of points in $X$ left invariant or unchanged by the respective group actions. I’ll explain the significance of the fixed point parameter next time.

Resources

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Mathematical Duality in Logical Graphs • Discussion 2

Posted on May 4, 2024 by Jon Awbrey

Re: Interpretive Duality in Logical Graphs • 1
Re: Mathematical Duality in Logical Graphs • 1
Re: Laws of Form • Lyle Anderson

LA:

Definition 1. A group $(G, *)$ is a set $G$ together with a binary operation $* : G \times G \to G$ satisfying the following three conditions.

Associativity. For any $x, y, z \in G,$ we have $(x * y) * z = x * (y * z).$
Identity. There is an identity element $e \in G$ such that $\forall g \in G,$
we have $e * g = g * e = g.$
Inverses. Each element has an inverse, that is, for each $g \in G,$
there is some $h \in G$ such that $g * h = h * g = e.$

Dear Lyle,

Thanks for supplying that definition of a mathematical group. It will afford us a wealth of useful concepts and notations as we proceed. As you know, the above three axioms define what is properly called an abstract group. Over the course of group theory’s history that definition was gradually abstracted from the more concrete examples of permutation groups and transformation groups initially arising in the theory of equations and their solvability.

As it happens, the application of group theory I’ll be developing over the next several posts will be using the more concrete type of structure, where a transformation group $G$ is said to “act on” a set $X$ by permuting its elements among themselves. In the work we do here, each group $G$ we contemplate will act a set $X$ which may be viewed as either one of two things, either a canonical set of expressions in a formal language or the mathematical objects denoted by those expressions.

What you say about deriving arithmetic, algebra, group theory, and all the rest from the calculus of indications may well be true, but it remains to be shown if so, and that’s a ways down the road from here.

Resources

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Mathematical Duality in Logical Graphs • Discussion 1

Posted on May 4, 2024 by Jon Awbrey

Re: Mathematical Duality in Logical Graphs • 1
Re: Laws of Form • Lyle Anderson
Re: Brading, K., Castellani, E., and Teh, N., (2017), “Symmetry and Symmetry Breaking”, The Stanford Encyclopedia of Philosophy (Winter 2017), Edward N. Zalta (ed.). Online.

Dear Lyle,

Thanks for the link to the article on symmetry and symmetry breaking. I did once take a Master’s in Mathematics, specializing in combinatorics, graph theory, and group theory. When it comes to the bearing of symmetry groups on logical graphs and the calculus of indications, it will take careful attention to the details of the relationship between the two interpretations singled out by Peirce and Spencer Brown.

Both Peirce and Spencer Brown recognized the relevant duality, if they differed in what they found most convenient to use in their development and exposition, and most of us will emphasize one interpretation or the other as a matter of facility or taste in a chosen application, so it requires a bit of effort to keep the underlying unity in focus. I recently made another try at taking a more balanced view, drawing up a series of tables in parallel columns the way one commonly does with dual theorems in projective geometry, so I will shortly share more of that work.

Resources

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Mathematical Duality in Logical Graphs • 1

Posted on May 3, 2024 by Jon Awbrey

All other sciences without exception depend upon the principles of mathematics; and mathematics borrows nothing from them but hints.

C.S. Peirce • “Logic of Number”

A principal intention of this essay is to separate what are known as algebras of logic from the subject of logic, and to re‑align them with mathematics.

G. Spencer Brown • Laws of Form

The duality between entitative and existential interpretations of logical graphs tells us something important about the relation between logic and mathematics. It tells us the mathematical forms giving structure to reasoning are deeper and more abstract at once than their logical interpretations.

A formal duality points to a more encompassing unity, founding a calculus of forms whose expressions can be read in alternate ways by switching the meanings assigned to a pair of primitive terms. Spencer Brown’s mathematical approach to Laws of Form and the whole of Peirce’s work on the mathematics of logic shows both thinkers were deeply aware of this principle.

Peirce explored a variety of dualities in logic which he treated on analogy with the dualities in projective geometry. This gave rise to formal systems where the initial constants, and thus their geometric and graph‑theoretic representations, had no uniquely fixed meanings but could be given dual interpretations in logic.

It was in this context that Peirce’s systems of logical graphs developed, issuing in dual interpretations of the same formal axioms which Peirce referred to as entitative graphs and existential graphs, respectively. He developed only the existential interpretation to any great extent, since the extension from propositional to relational calculus appeared more natural in that case, but whether there is any logical or mathematical reason for the symmetry to break at that point is a good question for further research.

Resources

References

Peirce, C.S., [Logic of Number — Le Fevre] (MS 229), in Carolyn Eisele (ed., 1976),
The New Elements of Mathematics by Charles S. Peirce, vol. 2, 592–595. Excerpt.
Spencer Brown, G. (1969), Laws of Form, George Allen and Unwin, London, UK.

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Interpretive Duality in Logical Graphs • 8

Posted on May 1, 2024 by Jon Awbrey

Re: Interpretive Duality in Logical Graphs • 6

The last of our six ways of looking at interpretive duality is arrived at by taking the previous Table of Logical Graphs and Venn Diagrams and sorting it in Orbit Order.

$\text{Logical Graphs} \stackrel{_\bullet}{} \text{Entitative and Existential Venn Diagrams} \stackrel{_\bullet}{} \text{Orbit Order}$

Resources

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Interpretive Duality in Logical Graphs • 7

Posted on April 30, 2024 by Jon Awbrey

Re: Interpretive Duality in Logical Graphs • 2

Dualities are symmetries of order two and symmetries bear on complexity by reducing its measure in proportion to their order. The inverse relationship between symmetry and the usual dissymmetries from dispersion and diversity to entropy and uncertainty is governed in cybernetics by the Law of Requisite Variety, the medium of whose exchange C.S. Peirce invested in the formula Information = Comprehension × Extension.

The duality between entitative and existential interpretations of logical graphs is one example of a mathematical symmetry but it’s not unusual to find symmetries within symmetries and it’s always rewarding to find them where they exist. To that end let’s take up our Table of Venn Diagrams and Logical Graphs on Two Variables and sort the rows to bring together diagrams and graphs having similar shapes. What defines their similarity is the action of a mathematical group whose operations transform the elements of each class among one another but intermingle no dissimilar elements. In the jargon of transformation groups those classes are called orbits. We find the sixteen rows partition into seven orbits, as shown below.

$\text{Venn Diagrams and Logical Graphs on Two Variables} \stackrel{_\bullet}{} \text{Orbit Order}$

Resources

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Interpretive Duality in Logical Graphs • 6

Posted on April 29, 2024 by Jon Awbrey

Re: Interpretive Duality in Logical Graphs • 2

A more graphic picture of interpretive duality is given by the next Table, showing how logical graphs map to venn diagrams under entitative and existential interpretations. Column 1 shows the logical graphs for the sixteen boolean functions on two variables. Column 2 shows the venn diagrams associated with the entitative interpretation and Column 3 shows the venn diagrams associated with the existential interpretation.

$\text{Logical Graphs} \stackrel{_\bullet}{} \text{Entitative and Existential Venn Diagrams}$

Resources

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Interpretive Duality in Logical Graphs • 5

Posted on April 26, 2024 by Jon Awbrey

Re: Interpretive Duality in Logical Graphs • 1

Another way of looking at interpretive duality in logical graphs is given by the following Table, showing how logical graphs denote the sixteen boolean functions on two variables under entitative and existential interpretations, respectively.

$\text{Logical Graphs} \stackrel{_\bullet}{} \text{Entitative and Existential Interpretations}$

Resources

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Interpretive Duality in Logical Graphs • 4

Posted on April 25, 2024 by Jon Awbrey

Re: Interpretive Duality in Logical Graphs • (1) • (2) • (3)

Last time we took up Peirce’s law, $((p \Rightarrow q) \Rightarrow p) \Rightarrow p,$ and saw how it might be expressed in two different ways, under the entitative and existential interpretations, respectively. The next thing to do is see how our choice of interpretation bears on the patterns of proof we might find. A sense of the possibilities may be gotten by displaying the two styles of proof in parallel columns, as shown below.

$\text{Peirce's Law} \stackrel{_\bullet}{} \text{Parallel Proofs}$

For convenience, the formal axioms and a few theorems of frequent use are linked below.

Resources

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

	Survey of Differenti… on Differential Logic • 4
	Differential Logic •… on Survey of Animated Logical Gra…
	Differential Logic •… on Survey of Differential Logic •…
	Differential Logic •… on Logic Syllabus
	Survey of Differenti… on Differential Logic • 3
	Survey of Differenti… on Differential Logic • 2
	Survey of Differenti… on Differential Logic • 1
	Survey of Differenti… on Differential Logic • Over…
	Differential Logic •… on Survey of Animated Logical Gra…
	Differential Logic •… on Survey of Differential Logic •…

M	T	W	T	F	S	S
						1
2	3	4	5	6	7	8
9	10	11	12	13	14	15
16	17	18	19	20	21	22
23	24	25	26	27	28

Resources

Share this:

Resources

Share this:

Resources

Share this:

Resources

Share this:

Resources

References

Share this:

Resources

Share this:

Resources

Share this:

Resources

Share this:

Resources

Share this:

Resources

Share this:

Elsewhere

Blogroll

Follow Blog via Email

Recent Posts

Recent Comments

Archives

Blog Stats

Categories

Logic, Mathematics, Philosophy

Current Events

Meta