Inquiry Into Inquiry

Differential Propositional Calculus • Discussion 6

Posted on March 22, 2021 by Jon Awbrey

Re: Oeis | Differential Propositional Calculus • Part 1 • Part 2 • Appendices
Re: Blog | Differential Propositional Calculus • Discussion • (3) • (4) • (5)

HR:

I think I like very much your Cactus Graphs. Meaning that I am in the process of understanding them, and finding it much better not to have to draw circles, but lines.
Less easy for me is the differential calculus. Where is the consistency between $\texttt{(} x \texttt{,} y \texttt{)}$ and $\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}?$ $\texttt{(} x \texttt{,} y \texttt{)}$ means that $x$ and $y$ are not equal and $\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}$ means that one of them is false. Unequality and truth/falsity for me are two concepts so different I cannot think them together or see a consistency between them.
What about $\texttt{(} w \texttt{,} x \texttt{,} y \texttt{,} z \texttt{)}?$

Dear Helmut,

Table 1 shows the cactus graphs, the corresponding cactus expressions in “traversal string” or plain text form, their logical meanings under the “existential interpretation”, and their translations into conventional notations for a number of common propositional forms. I’ll change variables to $\{ x, a, b, c \}$ instead of $\{ w, x, y, z \}$ at this point simply because I’ve already got a Table like that on hand.

As far as the consistency between $\texttt{(} a \texttt{,} b \texttt{)}$ and $\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}$ goes, that’s easy enough to see — if exactly one of two boolean variables is false then the two must have different values.

Out of time for today, so I’ll get to the rest of your questions next time.

Table 1. Syntax and Semantics of a Calculus for Propositional Logic

cc: Cybernetics • Laws of Form • Ontolog Forum • Peirce List (1) (2) (3)
cc: FB | Differential Logic • Structural Modeling • Systems Science

Posted in Amphecks, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Computational Complexity, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Dynamical Systems, Equational Inference, Functional Logic, Gradient Descent, Graph Theory, Group Theory, Hologrammautomaton, Indicator Functions, Logic, Logical Graphs, Mathematical Models, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Propositional Equation Reasoning Systems, Time, Visualization | Tagged Amphecks, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Computational Complexity, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Dynamical Systems, Equational Inference, Functional Logic, Gradient Descent, Graph Theory, Group Theory, Hologrammautomaton, Indicator Functions, Logic, Logical Graphs, Mathematical Models, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Propositional Equation Reasoning Systems, Time, Visualization | 8 Comments

Differential Propositional Calculus • Discussion 5

Posted on March 21, 2021 by Jon Awbrey

Re: Peirce List • Helmut Raulien
Re: Ontolog Forum • Mauro Bertani

HR:

I think I like very much your Cactus Graphs. Meaning that I am in the process of understanding them, and finding it much better not to have to draw circles, but lines.
Less easy for me is the differential calculus. Where is the consistency between $\texttt{(} x \texttt{,} y \texttt{)}$ and $\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}?$ $\texttt{(} x \texttt{,} y \texttt{)}$ means that $x$ and $y$ are not equal and $\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}$ means that one of them is false. Unequality and truth/falsity for me are two concepts so different I cannot think them together or see a consistency between them.
What about $\texttt{(} w \texttt{,} x \texttt{,} y \texttt{,} z \texttt{)}?$

MB:: So, if I want to transform a circle into a line I have to use a function $f : \mathbb{B}^n \to \mathbb{B}?$
This is the base of temporal logic? I’m using $f : \mathbb{N}^n \to \mathbb{N}.$

Dear Mauro,

If I understand what Helmut is saying about “circles” and “lines”, he is talking about the passage from forms of enclosure on plane sheets of paper — such as those used by Peirce and Spencer Brown — to their topological duals in the form of rooted trees. There is more discussion of this transformation at the following sites.

This is the first step in the process of converting planar maps to graph-theoretic data structures. Further transformations take us from trees to the more general class of cactus graphs, which implement a highly efficient family of logical primitives called minimal negation operators. These are described in the following article.

Minimal Negation Operators

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cc: FB | Differential Logic • Structural Modeling • Systems Science

Differential Propositional Calculus • Discussion 4

Posted on March 20, 2021 by Jon Awbrey

It is one of the rules of my system of general harmony, that the present is big with the future, and that he who sees all sees in that which is that which shall be.

Leibniz • Theodicy

Re: R.J. Lipton • Anti-Social Networks
Re: Lou Kauffman • Iterants, Imaginaries, Matrices
Re: Logical Cactus Graphs @ All Process, No Paradox • 6
Re: Differential Propositional Calculus • Discussion 3
Re: Peirce List • Helmut Raulien

HR:

I think I like very much your Cactus Graphs. Meaning that I am in the process of understanding them, and finding it much better not to have to draw circles, but lines.
Less easy for me is the differential calculus. Where is the consistency between $\texttt{(} x \texttt{,} y \texttt{)}$ and $\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}?$ $\texttt{(} x \texttt{,} y \texttt{)}$ means that $x$ and $y$ are not equal and $\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}$ means that one of them is false. Unequality and truth/falsity for me are two concepts so different I cannot think them together or see a consistency between them.
What about $\texttt{(} w \texttt{,} x \texttt{,} y \texttt{,} z \texttt{)}?$
Can you give a grammar, like, what does a comma mean, what do brackets mean, what does writing letters following each other with an empty space but no comma mean, and so on?
Same with Cactus Graphs, though I think, they might be self-explaining for me — everything is self-explaining, depending on intellectual capacity, but mine is limited.

Dear Helmut,

Many thanks for your detailed comments and questions. They help me see the places where more detailed explanation is needed. I added numbers to your points above for ease of reference and possible future reference in case I can’t get to them all in one pass.

I’m glad you found the cactus graphs to your liking. It was a critical transition for me when I passed from trees to cacti in my graphing and programming and it came about by recursively applying a trick of thought I learned from Peirce himself. These days I call it a “Meta-Peircean Move” to apply one of Peirce’s heuristics of choice or standard operating procedures to the state of development resulting from previous applications. All that makes for a longer story I made a start at telling in the following series of posts.

Animated Logical Graphs • (14) • (15) • (16) • (17) • (18) • (19) • (20) • (21) • (22) • (23) • (24) • (25) • (26) • (27)

Well, the clock in the hall struck time for lunch some time ago, so I think I’ll heed its call and continue later …

Regards,

Jon

cc: Cybernetics • Laws of Form • Ontolog Forum • Peirce List (1) (2) (3)
cc: FB | Differential Logic • Structural Modeling • Systems Science

All Process, No Paradox • 9

Posted on March 18, 2021 by Jon Awbrey

In the midst of this strife, whereat the halls of Ilúvatar shook and a tremor ran out into the silences yet unmoved, Ilúvatar arose a third time, and his face was terrible to behold. Then he raised up both his hands, and in one chord, deeper than the Abyss, higher than the Firmament, piercing as the light of the eye of Ilúvatar, the Music ceased.

Tolkien • Ainulindalë

Re: Objects, Models, Theories • (1) • (2) • (3)
Re: Peirce List • Helmut Raulien

Continuing my review of previous discussions concerned with various proposals to extend bivalent logic to encompass sundry dimensions of alterity, change, diversity, dynamics, imagination, indefinability, indeterminacy, information, interpretation, intuitionism, likelihood, mutability, probability, quantity, relativity, time, uncertainty, and so on.

For continuity’s sake I’m recycling my replies to a comment by Helmut Raulien on the Peirce List which raised a host of questions about Peirce’s categories, logic, and semiotics in the light of Spencer Brown’s Laws of Form.

Comment 1

George Spencer Brown’s Laws of Form tends to be loved XOR hated by most folks, with few coming down in between. I ran across the book early in my undergrad years, shortly after encountering C.S. Peirce, so I recognized the way it revived Peirce’s logical graphs, emphasizing the entitative interpretation of the abstract formal calculus immanent in Peirce’s “Alpha” graphs. It took me a decade to gain a modicum of clarity about all that “imaginary truth value” and “re-entry” folderol. I’ll say some things about that later on.

Comment 2

I mulled the matter over for a fair spell of days and nights and decided it wouldn’t be good to jump into the middle of the muddle about re-entry and imaginary truth values right off the bat, that it would be better in the long run to get a solid grip on what is going on with the propositional level of Peirce’s logical graphs and how Spencer Brown’s elaborations can be seen to manifest the same spirit of reasoning, if they are read the right way. Toward that end I’ll append a list of resources to break the ice on this approach.

Resources

Logic Syllabus
Semeiotic, Semiotics
Precursors Of Category Theory
Logical Graphs • Introduction • Formal Development
Differential Propositional Calculus • Part 1 • Part 2

cc: Cybernetics • Laws of Form • Ontolog Forum • Peirce List
cc: FB | Cybernetics • Structural Modeling • Systems Science

Posted in Animata, C.S. Peirce, Category Theory, Cybernetics, Differential Logic, Laws of Form, Logic, Logical Graphs, Mathematics, Paradox, Peirce, Process, Semiotics, Spencer Brown, Systems Theory, Tertium Quid, Time, Tolkien | Tagged Animata, C.S. Peirce, Category Theory, Cybernetics, Differential Logic, Laws of Form, Logic, Logical Graphs, Mathematics, Paradox, Peirce, Process, Semiotics, Spencer Brown, Systems Theory, Tertium Quid, Time, Tolkien | 7 Comments

All Process, No Paradox • 8

Posted on March 16, 2021 by Jon Awbrey

These are the forms of time, which imitates eternity and revolves according to a law of number.

Plato • Timaeus

Re: Laws of Form • Seth • James Bowery (1) (2) (3) • Lyle Anderson

Dear Seth, James, Lyle,

Nothing about calling time an abstraction makes it a nullity. I’m too much a realist about mathematical objects to ever think that. As a rule, on the other hand, I try to avoid letting abstractions leave us so absent-minded as to forget the concrete realities from which they are abstracted. Keeping time linked to process, especially the orders of standard process we call “clocks”, is just part and parcel of that practice.

Synchronicity being what it is, this very issue came up just last night in a very amusing Facebook discussion about “windshield wipers slappin’ time …”

At any rate, this thread is already moving too fast for the pace I keep these days but maybe I can resolve remaining confusions about the game afoot by recycling a post I shared to the old Laws of Form list. This was originally a comment on Lou Kauffman’s blog back when he first started it. Sadly, he wrote only a few more entries there in the time since.

Re: Lou Kauffman • Iterants, Imaginaries, Matrices

As serendipity would have it, Lou Kauffman, who knows a lot about the lines of inquiry Charles Sanders Peirce and George Spencer Brown pursued into graphical syntaxes for logic, just last month opened a blog and his very first post touched on perennial questions of logic and time — Logos and Chronos — puzzling the wits of everyone who has thought about them for as long as anyone can remember. Just locally and recently these questions have arisen in the following contexts.

Alpha Now, Omega Later • (1) • (2) • (3) • (4) • (5) • (6) • (7)
Objects, Models, Theories • (1) • (2) • (3) • (4)
Peirce List • (1) • (2) • (3)

Kauffman’s treatment of logic, paradox, time, and imaginary truth values led me to make the following comments I think are very close to what I’d been struggling to say before.

Let me get some notational matters out of the way before continuing.

I use $\mathbb{B}$ for a generic 2-point set, usually $\{ 0, 1 \}$ and typically but not always interpreted for logic so that $0 = \mathrm{false}$ and $1 = \mathrm{true}.$ I use “teletype” parentheses $\texttt{(} \ldots \texttt{)}$ for negation, so that $\texttt{(} x \texttt{)} = \lnot x$ for $x ~\text{in}~ \mathbb{B}.$ Later on I’ll be using teletype format lists $\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)}$ for minimal negation operators.

As long as we’re reading $x$ as a boolean variable $(x \in \mathbb{B})$ the equation $x = \texttt{(} x \texttt{)}$ is not paradoxical but simply false. As an algebraic structure $\mathbb{B}$ can be extended in many ways but it remains a separate question what sort of application, if any, such extensions might have to the normative science of logic.

On the other hand, the assignment statement $x := \texttt{(} x \texttt{)}$ makes perfect sense in computational contexts. The effect of the assignment operation on the value of the variable $x$ is commonly expressed in time series notation as $x' = \texttt{(} x \texttt{)}$ and the same change is expressed even more succinctly by defining $\mathrm{d}x = x' - x$ and writing $\mathrm{d}x = 1.$

Now suppose we are observing the time evolution of a system $X$ with a boolean state variable $x : X \to \mathbb{B}$ and what we observe is the following time series.

Computing the first differences we get:

Computing the second differences we get:

This leads to thinking of the system $X$ as having an extended state $(x, \mathrm{d}x, \mathrm{d}^2 x, \ldots, \mathrm{d}^k x),$ and this additional language gives us the facility of describing state transitions in terms of the various orders of differences. For example, the rule $x' = \texttt{(} x \texttt{)}$ can now be expressed by the rule $\mathrm{d}x = 1.$

The following article has a few more examples along these lines.

Differential Analytic Turing Automata

Resources

Differential Logic • Introduction
Differential Logic • Part 1 • Part 2 • Part 3
Differential Propositional Calculus • Part 1 • Part 2
Differential Logic and Dynamic Systems • Part 1 • Part 2 • Part 3 • Part 4 • Part 5

cc: Cybernetics • Laws of Form • Ontolog Forum • Peirce List
cc: FB | Cybernetics • Structural Modeling • Systems Science

Posted in Animata, Boolean Functions, C.S. Peirce, Cybernetics, Differential Logic, Discrete Dynamics, Laws of Form, Logic, Logical Graphs, Lou Kauffman, Mathematics, Paradox, Peirce, Plato, Process, Spencer Brown, Timaeus, Time | Tagged Animata, Boolean Functions, C.S. Peirce, Cybernetics, Differential Logic, Discrete Dynamics, Laws of Form, Logic, Logical Graphs, Lou Kauffman, Mathematics, Paradox, Peirce, Plato, Process, Spencer Brown, Timaeus, Time | 7 Comments

Differential Propositional Calculus • Discussion 3

Posted on March 15, 2021 by Jon Awbrey

That mathematics, in common with other art forms, can lead us beyond ordinary existence, and can show us something of the structure in which all creation hangs together, is no new idea. But mathematical texts generally begin the story somewhere in the middle, leaving the reader to pick up the thread as best he can. Here the story is traced from the beginning.

G. Spencer Brown • Laws of Form

Re: Laws of Form • Lyle Anderson (1) (2)

Dear Lyle,

Charles S. Peirce, with his x-ray vision, revealed for the first time in graphic detail the mathematical forms structuring our logical organon. Spencer Brown broadened that perspective in two directions, tracing more clearly than Peirce’s bare foreshadowings the infrastructure of primary arithmetic and hypothesizing the existence of imaginary logical values in a larger algebraic superstructure.

Spencer Brown explored the algebraic extension of the boolean domain $\mathbb{B}$ to a superset equipped with logical imaginaries, operating on analogy with the algebraic extension of the real line $\mathbb{R}$ to the complex plane $\mathbb{C}.$ Seeing as how complex variables are frequently used to model time domains in physics and engineering, that will continue to be a likely and natural direction of exploration.

My own work, however, led me in a different direction. There are many different ways of fruitfully extending a given domain. Aside from the above class of algebraic extensions there is a class of differential extensions and when that proverbial road diverged I took the differential one.

Who knows? maybe on through that undergrowth the roads converge again …

Resources

Differential Logic • Introduction
Differential Logic • Part 1 • Part 2 • Part 3
Differential Propositional Calculus • Part 1 • Part 2
Differential Logic and Dynamic Systems • Part 1 • Part 2 • Part 3 • Part 4 • Part 5

cc: Cybernetics • Laws of Form • Ontolog Forum • Peirce List (1) (2) (3)
cc: FB | Differential Logic • Structural Modeling • Systems Science

All Process, No Paradox • 7

Posted on March 13, 2021 by Jon Awbrey

Unlike more superficial forms of expertise, mathematics is a way of saying less and less about more and more. A mathematical text is thus not an end in itself, but a key to a world beyond the compass of ordinary description.

G. Spencer Brown • Laws of Form

Re: Laws of Form • James Bowery

Dear James,

Sorry for the sluggish response … but I’ve been slogging through a mass of mindless link repair due to the slew of url-extinctions and url-mutations afflicting our web of maya over the last few years. I’ve been working to recover-revise my better contributions to the old LoF list along the lines of what Spencer Brown wrote about time and imaginary logical values and the impact it had on my own work with logical graphs from the early days on.

There was a time when I spent a lot of time thinking about the “phenomenology of internal time consciousness” and so on but that was a long time passing. I think I first learned the word phenomenology from early readings in Bachelard and Sartre but my current take on it is more heavily influenced by subsequent experiences in physics labs and libraries.

Physicists speak of the need to reflect on the circumstance that even our most exalted theories get their first leg up from our “naked eye” perception of “pointer readings”, that is, from the superposition in our visual field of a needle on a graduated dial, or the analogous incidentals in other sensory modes. As a rule, a working physicist would never think of taking that “observation of obvious” truth in too reductive a sense, since that would lead to sheer sensationalism, and even the purest experimentalist has a better appreciation for the role of theoretical conception than that.

Well, I didn’t know I was going to write this much when I opened the page, but I started remembering experiences and thoughts from the earliest days. At any rate, I think I’ll blog this on my series about Process and Paradox since that is occupying my mind at present and I wouldn’t want to sidetrack the time-phenomenology line.

Regards,

Jon

cc: Cybernetics • Laws of Form • Ontolog Forum • Peirce List
cc: FB | Cybernetics • Structural Modeling • Systems Science

Posted in Animata, C.S. Peirce, Change, Cybernetics, Differential Logic, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Paradox, Peirce, Process, Process Thinking, Spencer Brown, Systems Theory, Time, Tolkien | Tagged Animata, C.S. Peirce, Change, Cybernetics, Differential Logic, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Paradox, Peirce, Process, Process Thinking, Spencer Brown, Systems Theory, Time, Tolkien | 8 Comments

Animated Logical Graphs • 66

Posted on March 9, 2021 by Jon Awbrey

Re: Richard J. Lipton • The Art Of Math
Re: Animated Logical Graphs • (57) • (58) • (59) • (60) • (61) • (62) • (63) • (64) • (65)

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.

$\text{Peirce Duality as Sign Relation}$

Resources

cc: Cybernetics (1) (2) • Peirce List (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)
cc: Ontolog Forum (1) (2) • Structural Modeling (1) (2) • Systems Science (1) (2)
cc: FB | Logical Graphs • Laws of Form

Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Peirce, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Theorem Proving, Visualization | Tagged Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Peirce, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Theorem Proving, Visualization | 15 Comments

Animated Logical Graphs • 65

Posted on March 4, 2021 by Jon Awbrey

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”

Re: Laws of Form • Lyle Anderson
Re: Richard J. Lipton • The Art Of Math

Dear Lyle,

Thanks for the link to the Wikipedia article on Cactus Graphs, which I found surprisingly good for that venue. I was pleased to see it mentioned the role my own first teacher in graph theory, Frank Harary, played in the history of cactus graphs. Frank co-authored Graphical Enumeration and many papers with Ed Palmer, my second teacher in graph theory and later my advisor in grad school.

Synchronicity being what it is, one of the jobs I worked on between my undergrad decade and my first crack at grad school was scanning and measuring particle interactions on bubble-chamber filmstrips in a high-energy physics lab, so I got a gadshillion gammas engrammed in my brain from that time.

Regards,

Jon

Resources

cc: Cybernetics (1) (2) • Laws of Form • FB | Logical Graphs • Ontolog Forum (1) (2)
• Peirce List (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) • Structural Modeling (1) (2)
• Systems Science (1) (2)

Animated Logical Graphs • 64

Posted on March 4, 2021 by Jon Awbrey

If exegesis raised a hermeneutic problem, that is, a problem of interpretation, it is because every reading of a text always takes place within a community, a tradition, or a living current of thought, all of which display presuppositions and exigencies — regardless of how closely a reading may be tied to the quid, to “that in view of which” the text was written.

Paul Ricoeur • The Conflict of Interpretations

Re: Laws of Form • John Mingers
Re: Richard J. Lipton • The Art Of Math

Dear John,

It occurred to me a picture might save a few thousand words. A good place to start is the following Table from an earlier post on my blog.

The smart way to deal with parens + character strings in computing is to parse them into graph-theoretic data structures and then work on those instead of the strings themselves. Usually one gets some sort of tree structures for the parse graphs. In my work on logical graphs I eventually came to use the more general species of structure graph theorists call cactus graphs or cacti.

Referring to the Table —

Column 1 shows the logical graphs I use for the sixteen boolean functions on two variables, with the string forms underneath. The cactus string obtained by traversing the cactus graph uses parens + commas + variables in forms like $\texttt{(} x \texttt{,} y \texttt{)}$ and $\texttt{((} x \texttt{,} y \texttt{))}.$
Column 2 shows the venn diagram associated with the entitative interpretation of the graph in Column 1. This is the interpretation C.S. Peirce used in his earlier work on entitative graphs and the one Spencer Brown used in his Laws of Form.
Column 3 shows the venn diagram associated with the existential interpretation of the graph in Column 1. This is the interpretation C.S. Peirce used in his later work on existential graphs.

Logical Graphs • Entitative and Existential Venn Diagrams
$\text{Logical Graph}$	$\text{Entitative Interpretation}$	$\text{Existential Interpretation}$

$\texttt{(} ~ \texttt{)}$	$\text{true}$ $f_{15}$	$\text{false}$ $f_{0}$

$\texttt{(} x \texttt{)(} y \texttt{)}$	$\lnot x \lor \lnot y$ $f_{7}$	$\lnot x \land \lnot y$ $f_{1}$

$\texttt{(} x \texttt{)} y$	$x \Rightarrow y$ $f_{11}$	$x \nLeftarrow y$ $f_{2}$

$\texttt{(} x \texttt{)}$	$\lnot x$ $f_{3}$	$\lnot x$ $f_{3}$

$x \texttt{(} y \texttt{)}$	$x \Leftarrow y$ $f_{13}$	$x \nRightarrow y$ $f_{4}$

$\texttt{(} y \texttt{)}$	$\lnot y$ $f_{5}$	$\lnot y$ $f_{5}$

$\texttt{(} x \texttt{,} y \texttt{)}$	$x = y$ $f_{9}$	$x \ne y$ $f_{6}$

$\texttt{(} x y \texttt{)}$	$\lnot (x \lor y)$ $f_{1}$	$\lnot (x \land y)$ $f_{7}$

$x y$	$x \lor y$ $f_{14}$	$x \land y$ $f_{8}$

$\texttt{((} x \texttt{,} y \texttt{))}$	$x \ne y$ $f_{6}$	$x = y$ $f_{9}$

$y$	$y$ $f_{10}$	$y$ $f_{10}$

$\texttt{(} x \texttt{(} y \texttt{))}$	$x \nLeftarrow y$ $f_{2}$	$x \Rightarrow y$ $f_{11}$

$x$	$x$ $f_{12}$	$x$ $f_{12}$

$\texttt{((} x \texttt{)} y \texttt{)}$	$x \nRightarrow y$ $f_{4}$	$x \Leftarrow y$ $f_{13}$

$\texttt{((} x \texttt{)(} y \texttt{))}$	$x \land y$ $f_{8}$	$x \lor y$ $f_{14}$

	$\text{false}$ $f_{0}$	$\text{true}$ $f_{15}$

Take a gander at all that and I’ll discuss more tomorrow …

Regards,

Jon

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