## Animated Logical Graphs • 68

Re: Animated Logical Graphs • (14)(15)(16)(17)(18)(19)(20)(21)
Re: Ontolog ForumMauro Bertani

Dear Mauro,

Let’s take a another look at the Table we reached at the end of Episode 21.

I call it a Formal Operation Table — rather than, say, a Truth Table — because it describes the operation of mathematical forms preceding the stage of logical interpretation.  I know the word formal tends to get overworked past the point of semantic fatigue but I can still hope to revive it a little.  We’ll use other labels for Table entries at other times but I tried this time to mitigate interpretive bias by choosing a mix of senses from both Peirce and Spencer Brown.

Entering the stage of logical interpretation, we arrive at the following two options.

• The entitative interpretation of $\texttt{(} a \texttt{,} b \texttt{)}$ produces the truth table for logical equality.

• The existential interpretation of $\texttt{(} a \texttt{,} b \texttt{)}$ produces the truth table for logical inequality, also known as exclusive disjunction.

### Resource

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## Animated Logical Graphs • 67

Re: Animated Logical Graphs • (14)(15)(16)(17)(18)(19)(20)
Re: Differential Propositional Calculus • Discussion • (4)(5)(6)
Re: Laws of FormLyle Anderson
Re: Peirce ListMauro Bertani

Dear Lyle,

Yes, the ability to work with functions as “first class citizens”, as we used to say, is one of the things making lambda calculus at the theoretical level and Lisp at the practical level so nice.  All of which takes us straight into Curry-Howard-ville …

Dear Mauro,

That is the right ball park, functional calculi and all that.  I haven’t been taking time out to mention all the players apart from Peirce and Spencer Brown — Boole, Frege, Schönfinkel, Curry, Howard, and others — because I’m still in the middle of tackling Helmut Raulien’s question about the link between cactus graphs and differential logic.  At any rate I’ll be focused on that for a while longer.

Regards,

Jon

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## Differential Propositional Calculus • Discussion 6

HR:
1. 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.
2. 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.
3. 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

## Differential Propositional Calculus • Discussion 5

HR:
1. 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.
2. 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.
3. 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.

## Differential Propositional Calculus • Discussion 4

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.

HR:
1. 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.
2. 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.
3. What about $\texttt{(} w \texttt{,} x \texttt{,} y \texttt{,} z \texttt{)}?$
4. 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?
5. 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.

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

## All Process, No Paradox • 9

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 ListHelmut 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.

## All Process, No Paradox • 8

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

Re: Laws of FormSeth • 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.

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.

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 Propositional Calculus • Discussion 3

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

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 …

## All Process, No Paradox • 7

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

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

## Animated Logical Graphs • 66

Re: Richard J. LiptonThe 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

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