Inquiry Into Inquiry

Transformations of Logical Graphs • 8

Posted on May 12, 2024 by Jon Awbrey

Semiotic Transformations

Re: Transformations of Logical Graphs • (4) • (5) • (6) • (7)

Turning again to our Table of Orbits let’s see what we can learn about the structure of the sign relational system in view.

As we saw in Episode 2, the transformation group $T = \{ 1, t \}$ partitions the set $X$ of 16 logical graphs and also the set $O$ of 16 boolean functions into 10 orbits, all together amounting to 4 singleton orbits and 6 doubleton orbits.

Points in singleton orbits are called fixed points of the transformation group $T : X \to X$ since they are left unchanged, or changed into themselves, by all group actions. Viewed in the frame of the sign relation $L \subseteq O \times X \times X,$ where the transformations in $T$ are literally translations in the linguistic sense, these $T$ -invariant graphs have the same denotations in $O$ for both Existential Interpreters and Entitative Interpreters.

$\text{Interpretive Duality as Sign Relation} \stackrel{_\bullet}{} \text{Orbit Order}$

Resources

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

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 | 1 Comment

Transformations of Logical Graphs • 7

Posted on May 11, 2024 by Jon Awbrey

Semiotic Transformations

Re: Transformations of Logical Graphs • (4) • (5) • (6)

Our investigation has brought us to the point of seeing both a transformation group and a triadic sign relation in the duality between entitative and existential interpretations of logical graphs.

Given the level of the foregoing abstractions it helps to anchor them in concrete structural experience. In that spirit we’ve been pursuing the case of a group action $T : X \to X$ and a sign relation $L \subseteq O \times X \times X$ where $O$ is the set of boolean functions on two variables and $X$ is a set of logical graphs denoting those functions. We drew up a Table combining the aspects of both structures and sorted it according to the orbits $T$ induces on $X$ and consequently on $O.$

$\text{Interpretive Duality as Sign Relation} \stackrel{_\bullet}{} \text{Orbit Order}$

In the next few posts we’ll take up the orbits of logical graphs one by one, comparing and contrasting their syntax and semantics.

Resources

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

Transformations of Logical Graphs • 6

Posted on May 10, 2024 by Jon Awbrey

Semiotic Transformations

Re: Transformations of Logical Graphs • (4) • (5)

Our study of the duality between entitative and existential interpretations of logical graphs has brought to light its fully sign-relational character, casting the interpretive duality as a transformation of signs revolving about a common object domain. The overall picture is a triadic relation linking an object domain with two sign domains, whose signs denote the objects in two distinct ways.

By way of constructing a concrete example, we let our object domain consist of the 16 boolean functions on 2 variables and we let our sign domains consist of representative logical graphs for those 16 functions. Thus we arrived at the Table in the previous post, linked by its title below.

$\text{Interpretive Duality as Sign Relation}$

Column 1 shows the object domain $O$ as the set of 16 boolean functions on 2 variables.
Column 2 shows the sign domain $S$ as a representative set of logical graphs denoting the objects in $O$ according to the existential interpretation.
Column 3 shows the interpretant domain $I$ as the same set of logical graphs denoting the objects in $O$ according to the entitative interpretation.

Additional aspects of the sign relation’s structure can be brought out by sorting the Table in accord with the orbits induced on the object domain by the group action inherent in the interpretive duality. Performing that sort produces the following Table.

$\text{Interpretive Duality as Sign Relation} \stackrel{_\bullet}{} \text{Orbit Order}$

Resources

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

Transformations of Logical Graphs • 5

Posted on May 9, 2024 by Jon Awbrey

Semiotic Transformations

Re: Transformations of Logical Graphs • (4)

“I know what you mean but I say it another way” — it’s a thing I find myself saying often enough, if only under my breath, to rate an acronym for it ☞ IKWYMBISIAW ☜ and not too coincidentally it’s a rubric of relevance to many situations in semiotics where sundry manners of speaking and thinking converge, more or less, on the same patch of pragmata.

We encountered just such a situation in our exploration of the duality between entitative and existential interpretations of logical graphs. The two interpretations afford distinct but equally adequate ways of reasoning about a shared objective domain. To cut our teeth on a simple but substantial example of an object domain, we picked the space of boolean functions or propositional forms on two variables. That brought us to the following Table, highlighting the sign relation $L \subseteq O \times S \times I$ involved in switching between existential and entitative interpretations of logical graphs.

$\text{Interpretive Duality as Sign Relation}$

Column 1 shows the object domain $O$ as the set of 16 boolean functions on 2 variables.
Column 2 shows the sign domain $S$ as a representative set of logical graphs denoting the objects in $O$ according to the existential interpretation.
Column 3 shows the interpretant domain $I$ as the same set of logical graphs denoting the objects in $O$ according to the entitative interpretation.

Resources

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

Transformations of Logical Graphs • 4

Posted on May 8, 2024 by Jon Awbrey

Semiotic Transformations

Once we bring the dual interpretations of logical graphs to the same Table and relate their parleys to the same objects, it is clear we are dealing with a triadic sign relation of the sort taken up in C.S. Peirce’s semiotics or theory of signs.

A sign relation $L \subseteq O \times S \times I,$ as a set $L$ embedded in a cartesian product $O \times S \times I,$ tells how the signs in $S$ and the interpretant signs in $I$ correlate with the objects or objective situations in $O.$

There are many ways of using sign relations to model various types of sign‑theoretic situations and processes. The following cases are often seen.

Some sign relations model co‑referring signs or transitions between signs within a single language or symbol system. In that event $L \subseteq O \times S \times I$ has $S = I.$
Other sign relations model translations between different languages or different interpretations of the same language, in other words, different ways of referring the same set of signs to a shared object domain.

The next Table extracts the sign relation $L \subseteq O \times S \times I$ involved in switching between existential and entitative interpretations of logical graphs.

Column 1 shows the object domain $O$ as the set of 16 boolean functions on 2 variables.
Column 2 shows the sign domain $S$ as a representative set of logical graphs denoting the objects in $O$ according to the existential interpretation.
Column 3 shows the interpretant domain $I$ as the same set of logical graphs denoting the objects in $O$ according to the entitative interpretation.

$\text{Interpretive Duality as Sign Relation}$

Resources

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

Transformations of Logical Graphs • 3

Posted on May 7, 2024 by Jon Awbrey

Re: Transformations of Logical Graphs • (1) • (2)

We’ve been using the duality between entitative and existential interpretations of logical graphs to get a handle on the mathematical forms pervading logical laws. A few posts ago we took up the tools of groups and symmetries and transformations to study the duality and we looked to the space of 2-variable boolean functions as a basic training grounds. On those grounds the translation between interpretations presents as a group $G$ of order two acting on a set $X$ of sixteen logical graphs denoting boolean functions.

Last time we arrived at a Table showing how the group $G$ partitions the set $X$ into ten orbits of logical graphs. Here again is that Table.

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

I invited the reader to investigate the relationship between the group order $|G| = 2,$ the number of orbits $10,$ and the total number of fixed points $16 + 4 = 20.$ In the present case the product of the group order (2) and the number of orbits (10) is equal to the sum of the fixed points (20) — Is that just a fluke? If not, why so? And does it reflect a general rule?

We can make a beginning toward answering those questions by inspecting the incidence relation of fixed points and orbits in the Table above. Each singleton orbit accumulates two hits, one from the group identity and one from the other group operation. But each doubleton orbit also accumulates two hits, since the group identity fixes both of its two points. Thus all the orbits are double-counted by counting the incidence of fixed points and orbits. In sum, dividing the total number of fixed points by the order of the group brings us back to the exact number of orbits.

Resources

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

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

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

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