Rudolf Kaehr Dr. phil^{@}

Copyright ThinkArt Lab ISSN 2041-4358

Abstract

The paper *"Diamond Calculus of Formation of Forms.* *A calculus of dynamic complexions of distinctions as an interplay of worlds and distinctions”* was mainly based on a deconstruction of the conditions of the calculus of indication, i.e. the assumption of a “*world*” and “*distinctions*” in it. The present paper “*Calculi of Indication and Differentiation*” opts for a graphematic turn in the understanding of calculi in general. This turn is exemplified with the George Spencer-Brown’s Calculus of Indication and the still to be discovered complementary Mersenne calculus of differentiations. The proposed study is restricted, mainly, to the mono-contextural case of dissemination of calculi.

First steps toward a graphematics had been presented with *“Interplay of Elementary Graphematic Calculi. Graphematic Fourfoldness of semiotics, Indication, Differentiation and Kenogrammatics".*

Graphematic calculi are not primarily related to a world or many worlds, like the CI and its diamondization. Graphematic calculi are studying the rules of the economy of kenomic inscriptions.

Graphematics was invented in the early 1970s as an interpretation of Gotthard Gunther’s keno- and morphogrammatics, inspired by Jaques Derrida’s grammatology and graphematics.

A new approach to a formalization of the calculus of indication and the calculus of differentiation too, is proposed.

Spencer-Brown’s calculus of indication has been extensively used to interpret human behavior in general.

The proposed new complementary calculus to the indicational calculus, the Mersenne calculus, might not be applicable to human beings, but there is a great chance that it will be a success for the interaction and study of non-human beings, such as robots, aliens, and Others.

SHORT VERSION (work in progress v.0.9)

Moshe Klein has given a simple introduction to George Spencer-Brown's calculus of indication (CI) as a special case of a bracket grammar. A context-free language with the grammar: S --> SS|(S)| λ is generating the proper paranthesis for formal languages.

What was an act of a genius becomes an *adhoc* decision to restrict the grammar of bracket production. Set the restriction of bracket rules to: (()) () = () (()) and you get the basic foundation of the famous CI as introduced by George Spencer-Brown. Nobody insists that this is an appropriate approach but it seems that it takes its legitimacy from the formal correctness of the elaboration.

**The law of complementarity***"There is no stronger mathematical law than the* law of complementarity*. A thing is defined by its complement, i.e. by what it is not. And its complement is defined by its uncomplement, i.e. by the thing itself, but this time thought of differently, as having got outside of itself to view itself as an object, i.e.`objectively', and then gone back into itself to see itself as the subject of its object, i.e.`subjectively' again.” *(George Spencer-Brown, Preface to the fifth English edition of LoF)

http://www.hyperkommunikation.ch/literatur/spencer-brown_form.htm

Now, with the same decisionism, albeit not pre-thought by a genius, I opt for an alternative restriction, (( )) = ( ). This decision is delivering the base system for a Mersenne calculus, interpreted as a calculus of differentiation, CD.

I stipulate that both calculi, the CI and the CD, are complementary. And both calculi have additionally their own internal *duality*, delivering the dual calculi, i.e. the dual-CI and the dual-CD.

It will be shown that, despite of its non-motivated *adhocism*, both calculi are well founded in graphematical systems, and are to be seen as *interpretations* of independent complementary graphematical calculi. In fact, they belong, with the identity system for semiotics to the only two non-kenogrammatic graphematical systems of the general architectonics of graphematics.

**LoF Mersenne**

distinction -- differentation (separation)

indication -- identification

A distinction of 2 distinctions is a distinction.

A distinction of a distinction is no distinction.

In Spencer-Browns wording:

“A1. The value of a call made again is the value of the call. *Calling*

A2. The value of a crossing made again is not the value of the crossing. *Crossing"*

*"In his* Laws of Form* (hereinafter LoF), in print since 1969, George Spencer-Brown proposed a minimalist formal system, called the primary arithmetic, arising from the primitive mental act of making a distinction. He reached the next rung on the ladder of abstraction by letting letters denote, indifferently, a distinction or its absence, resulting in the primary algebra. The primary arithmetic and algebra featured a single primitive symbol ‘ **’ in LoF.” *(Meguire)

A differentation between 2 differentiations is an absence of differentiation.

A differentiation of a differentiation is a differentiation.

**Paraphrase**

M1. The value of a call made again is not the value of the call. *Calling*

M2. The value of a crossing made again is the value of the crossing. *Crossing"*

*"In his Laws of Differentiation (hereinafter LoD), in print since 2011, Rudolf Kaehr proposed a minimalist complementary formal system to the LoF, called the primary complementary arithmetic, pca, arising from the primitive *scriptural* act of perceiving a *differentiation*. He reached the next rung on the ladder of *graphematic* abstraction, the Mersenne calculus, MC, by letting characters *inscribe*, differently, a differentiation or its absence, resulting in the primary complementary algebra, CD. The primary complementary arithmetic and algebra featured a single primitive symbol ‘ **’ in LoD.” *(Kaehr)

**A Morphic Turn for LoD**

*He reached the next rung on the ladder of *graphematic* abstraction considering the morphic structure of patterns of differentiations and absences of differentiations, the morphic Mersenne calculus, morphMC, by letting *patterns* of characters *inscribe*, differently, a differentiation or its absence, resulting in the primary complementary morphic *algebra of differentiation*, morphCD. *

*The groups of differentiations, called *situations*, are defined by the Mersenne distribution of elementary differentiations with the combinatorial formula: 2*^{n}-1*. Such groups are embedded into differential contexts. *

*The primary complementary *arithmetic* features two primitive symbols ‘ **’ and “∅ “ in LoD.**The pattern or morphic approach is motivated by the fact of the morphogrammatics of the MorphLoD as being a part of the graphematic writing systems.*

*While the primary complementary morphic *algebra* features a *group* of primitive constellations of symbols, ‘ **’ and ‘∅’, in morphLoD.” *(Kaehr)

**A Morphic Turn for LoF**

*The groups of distinctions, called *constellations*, are defined by the Brownian distribution of elementary distinctions with the combinatorial formula:*

Accepting the pattern-oriented approach as primordial, it is just a natural step to the development of an element-oriented approach of the classical CI and the proposed CD in its first presentation as a an element-oriented calculus too.

Some further introductions in German at:

http://www.thinkartlab.com/Memristics/Komplementaritaet/Komplementarität%20in%20der%20Graphematik.pdf

**Morphograms and sequences**A further difference between the Mersenne and the Brown calculus is naturally given by the difference of morphograms, MGs, and sequences, Seqs, or monomorphies and constellations of the configuration for m=n=2.

Another differentiation is achieved with the question of *second-level partitions *(Moshe Klein) for the graphematics systems of the different calculi. Because of the minimal complexity of the introduced CI and CD with m, n=2, an application of the question of second-level partitions is not delivering any useful results. Nevertheless, a reflection on the notion of partition, i.e. a partition of partition, is an interesting feature for any calculus. With an extension of the complexity even to just m=3, the covered features are appearing clearly.

It should be reflected that a *first*-level and a *second*-level operation of addition based on first-order and second-order distinctions has to be differentiated. In a term like “{1 +{{1+1}}", that is representing the bracket-notation for the CI, there are 2 different kinds of additions involved: a *first*-level addition “+_{1}” and a *second*-level addition "+_{2}”, hence the term is reflectionally: “{1 +_{1}{{1+_{2}1}}".

From the standpoint of a *theory of reflection* (Gunther) which takes into account the *difference* of the levels of reflection (distinction), it would be more interesting to study the reflectional properties of the *process* of “nivilation” of partitions towards a conglomerate of indistinguishable elements instead of the *results* of the elimination of the difference alone. This process second-level reflection is measured by Ord_{refl}(n).

**Example**

m = (3): {3}, {2 + 1}, {1 + 1 + 1} :first-level partition, P(n)

{3}, {2 + 1}, **{{1 +**_{2}** 1} +**_{1}**1}**, {1 + 1 + 1} : second-level partition, sp_{n}

Ord_{refl}(3) = 2

**Table**

Therefore, a calculus with complecity n=2, like the CI and CD, has no properties of second-level reflection. A calculus with n=3 has 1 second-level property, written as diff(n).

http://memristors.memristics.com/Interplay/Interplay%20of%20Elementary%20Graphematic%20Calculi.html

**Metaphors**

Brown’s *tabula rasa* world, cut by the act of a distinction. Mersenne’s streams of signs. Differentiated by the perception of a differentiation.

"Distinction without difference.”: or but != ?

"Difference without distinction.”

The motivation for Brownian distinctions are founded in the Mersennian process of differentiation; both are interacting simultaneously together. This interplay of “*constructivist” *and “*recognicist” *actions is marked by the quadralectics of both calculi.

"Discuss the distinction between indicational and differential calculi. Is there a difference or is it a distinction?”

"How to draw a difference between a distinction and a differentiation? How is it indicated?”

At first, the graphematic complementarity approach is in no way forced to establish a *hierarchy* between the concepts and strategies of distinction and differentiation.

Secondly, the involved calculi are not forced to reduce their space of realizations to mono-contexturality.

http://www.cpsa-acsp.ca/paper-2003/gaon.pdf

**Operator-operand interplay: The “Cross” as an operator and as an operand**

*"The true nature of the distinction between the *pa* on the one hand, and 2 and sentential logic on the other, now emerges. In the latter formalisms, *complementation/negation* operating on "nothing" is not well-formed. But an empty *Cross

Thus the

The Cross as an *operand*: ,

The Cross as an *operator*: .

Obviously, the *functionality* of the Cross is depending on the contextual use of the Cross. It is lazy thinking to tell that the Cross *is* both an operator and an operand, i.e.* ”merely different facets of a single fundamental action”*. The Cross is acting, functioning or playing *as* an operator or (exclusively) *as* an operand. But the Cross *is* not acting as both at once.

A similar situation is well known for the calculus of *Combinatory Logic* (Schönfinkel, Curry, Fey, Rosenbloom, Barendregt).

**Bracket interpretation of graphematic constellations**

**Interpretations, again****LoF**

Following Wolfram’s statement, according to M. Schreiber:

"A kind of *form* is all you need to compute. A system can emulate rule 110 if it can distinguish: *‘More than one is one but one inside one is none.’ *

Simple distinctions can be configured into forms which are able to perform universal computations.”

**LoD**

"A kind of structuration is part of what you need to transpute. A system can inscribe the complementary rule 110” if it can differentiate: *‘More than one is none but one inside one is one.’ *

Complex differentiations can be inscribed into structurations which are able to create pluri-versal transputations.”

**LoF & LoD**

Both together, and set into a polycontextural framework, are staging polyversal co-creations in the doman of distinctions, differentiations and worlds.

**CI-BrownT9: **”

**CD-MersenneDT9: **

**Theorem DT13**

Let p, q, r stand for any expressions. Then in any case,

Duality is well known as an interesting property of formal systems with structural and economic advantages; *”Two for one.*” (Herrlich)

**Duality for indication calculi**P, A ∈ CI ,

P: property and op= duality operation

1. (A

2. Pholds if and only if P(A

**Example**

X, Y ∈ CI:

X = Y --> X^{op} = Y^{op}

For X = AB, Y = BA:

AB = BA --> =

**Duality of Mersenne calculi**CM: p = q --> =

Both characterizations are referring to the distinctions (differentiations) made, and inscribed, by its marks. There is no reference to an outside world which has to be divided and marked needed.

Mersenne calculi are not referring to a world to be divided but to the media they are inscribed. Hence, they are reflecting and referring to themselves instead of an external world which needs observer who are making distinctions in this world.

Also the basic “beginnings" (initials, axioms) of the complementary Brownian and the Mersennian calculi look intriguingly simple and more like duals of each other than highly different complementary approaches, the consequences of the differences and similarities becomes surprisingly decisive and clear in the “algebra” of both calculi.

**Spencer-Brown**

The most fundamental activity is to draw a distinction and to mark it.

This marks the *constructivitic* appraoch of a subject to the world (Kant).

*“Draw a distinction! Mark it!*” refers to a so called world, and implies an actor of the distinction.

"*Distinctions splits the world into two parts, [...]. One of the most fundamental of all human activities is the making of distinctions.*” (Principles of Biological Autonomy, Varela, 1979, p. 84)

There is no need to privilege a starting point as the ultimate beginning and determination of human activities as the Brownians are insisting. There are other possibilities too, and there is also no chance to legitimate or to prove the correctness of such a statement about any beginnings.

http://www.vordenker.de/ics/downloads/logik-second-order.pdf

**Mersenne**

The most fundamental activity is to separate and to identify the separated.

"Without separation no identification, and *vice versa*."

This hints to the transcendental-phenomenological turn to the “*Sachen selbst*” (Husserl, Heidegger) as a complementary approach to cognitivistic constructions.

There is no need to privilege a starting point as the ultimate beginning and determination of human activities as the Brownians are insisting. There are other possibilities too, and there is also no chance to legitimate or to prove the correctness of such a statement about any beginnings. There is also no need to insist on a Mersenne complementary statement about the importance of beginnings.

**Spencer-Brown/Mersenne**The

The “

Both approaches are denying their “*blind spot*”, unmasked by the graphematical approach as the graphematic “*gaps*” of the calculi. Both gaps are complementary to each other.

Instead of the constructivist activity to draw a distinction in a presumed world as demanded by the CI, the CD ask to accept the perceived or encountered difference in the world as a separation or differentiation of something, especially something written or inscribed, independent of an active and constructivist observer but depending on someone who is able to accept what is given. The given is not a pre-given entity from nowhere but a cultural event of other cultural events, i.e. cultural agents. With this turn, miseries of solipsism are results of blindness towards what is given as encountered in an happenstance of encounter.

In other words, the CI is drawing distinctions, the CD is encountering events.

For the CI, the ‘patron’ might be Immanuel Kant, for the CD, a reference to Alfred North Withehead might be accurate.

Hence, the interpretation of the initials of the CD are becoming:

M1: The iteration (of the acceptance) of an event is the absence of (an acceptance of) an event. And,

M2: The (acceptance of an) event of an event is (the acceptance of) an event.

**Graphematics**

The most fundamental activity is to live the “Schied” of the “Unterschied”, i.e. the ‘tinction’ of the dis-tinction, distinction and difference, between the two complementary ‘tinctions’ of Brown and Mersenne and their dual forms of inscriptions as part of the general system of graphematics. The “Schied” of the “Unterschied” is in the history and ressemblance with the process of “différance” as close as possible to the movement of identity.

Both, the Brownian calculus of indication and and the Mersennian calculus of differentation might be based on the graphematic system of inscription. Both appears as interpretations of different graphematic systems, belonging to the general system of graphematics.

How is the system of graphematics introduced? Is there any transparency or is it as obscure as the introduction of the Calculus of Indication?

Graphematics are an interpretation based on the classification of partitions of a set of signs. It is proved that this classification system of sets of signs is complete. With the *Stirling Turn*, graphematics is based on kenogrammatics and not anymore exclusively on semiotics.

**Gaps**

Gaps appear in the interaction between different calculi, i.e. CI, CD and semiotics. There is no direct access for a calculus to its own gap. Hence, a gap is a *blind spot* of a calculus. An *interactional* calculus of indication and differentiation is including the interactivity of calculi and gaps. Gaps are a third category to the “mark”, “unmark” , ⌀, and differentiation and absence of differentiation, ∅.

**Algebraic and Co-algebraic characterizations**There is also no single point of beginning in graphematical scriptures, like “

**Algebraic CI:***”In the beginning there is a space, normally a plane surface, that is featureless but upon which symbols (a primitive notion) may be inscribed.”* (Meguire)

**Co-algebraic graphematics**

"There are always symbols to encounter that are defining a space of inscriptions. Such a scriptural space is never featureless, and symbols, marks and signs (complex kenoms) are always distinguishable and are always further differentiated by coordinated inscriptions.” (paraphrase)

*"Interaction of actors has no specific beginning or end. It goes on forever. Since it does so it has very peculiar properties.”* (Gordon Pask, 1996)

What might be presupposed for the CI as minimal conditions, is a 2-dimensional open field of kenomic marks. Then, to each identified kenomic mark of the two dimensions the questions leading to the initials “J1” and “J2” might be asked and answered by the indicational rules of J1 and J2 of the Laws of Form.

Calculi, like the CI and the MC, arise as “interpretations” of graphematic streams of kenomic inscriptions.

A formalization of the behavior of the graphematic systems implied by the calculi CI and CD would have to be realized as an *interplay* between algebraic and co-algebraic tectonics.

Also both calculi are, in their complementarity, very similar, they are based on strictly distinct graphematic systems.

The whole wording introduced by GSB to characterize the process of distinction and indication related to a space and time, is a relict from a subordination of writing to the aim of representation in the tradition of sign-related ontology. From a graphematic point of view, only the inscriptions and their laws are of relevance.

This is a step further towards a graphematic understanding of calculi compared to the sketch presented by the *Diamond Calculus* approach which is still emphasizing the proemiality of ”world” and “distinction”.

**Qualitative characterizations of the graphematics of Brownian and Mersennian configurations**Following Matzka’s charcterizations of semiotic, indicational and trito-systems (or strings), but omitting Mersenne systems, the formal approach of Schadach gets some intuitive support additionally to its strictly combinatorial treatement.

**General assumption**(A) If the two given tokens of strings have different lengths, then they are different.

If they have equal lengths, then go to (B).

x = (a_{1} a_{2} a_{3}... a_{n}),

y = (b_{1} b_{2} b_{3} ... b_{m}),

m=n or m!=n.

**Identity**

(B) For each position i from 1 to the common length, check whether the atom at the i-th position of x equals the atom at the i-th position of y. If this is true for all positions i, then the given tokens are equal, otherwise they are different.

**Spencer-Brown**

(B') Check whether each atom appears equally often in both string-tokens. If this is the case, then they are equal, otherwise they are different.

Here we have enlarged the abstractive distance between string-token and string-type, by including the abstraction from the order of the atoms into the abstraction from token to type.

**Trito-stucture**(B'') For each pair i, k, i<k, of positions, check whether within x there is equality between position i and k, and check whether within y there is equality between position i and k. If within both x and y there is equality, or if within both x and y there is inequality, then state equality for this pair of positions, otherwise state inequality for this pair of positions. If for each pair of positions there is equality, then x and y are equal. Otherwise they are not.

**Mersenne**

(B“‘) For each pair i, k, i<k, of positions, check whether within x there is equality between position i and k, and check whether wihin y there is equality between position i and k. If within both x and y there is equality for all i, k, and additionally (B) holds, then x and y are equal. Otherwise they are not.

The graphematic complementarity of CI and CD is reflected in the complementarity of their arithmetics and algebras.

**Graphematic constellations**

How are the graphematic situations mirrored in the formulas of the calculi?

For the CI it seems easy to show the property of *commutativity* of the terms, like

=

=

How is the *non-commutativity* in the CM represented?

= --> = --> ∅ = ∅.

**States and constellations: **CD = {, ∅}

CD-constellations = {, , }.

CI = { , }

CI-constellations = {, , ∅∅}

**Concatenation, combination and superpositionCI: **

X ∈ CI, Y ∈ CI => X^Y ∈ CI : concatenation

X ∈ CI, Y ∈ CI => X(Y) ∈ CI : superposition

**CD: **

X ∈ CD, Y ∈ CD => X*Y ∈ CD : combination

X ∈ CD, Y ∈ CD => X(Y) ∈ CD : superposition

**Equality in CI and CM**

Equality in the CI is defined in a traditional way, i.e. an equality relation is reflexive, symmetric transitive.

Therefore, x = x and x != , hence:

CI: != --> != .

This corresponds in the CI properly the graphematic situation: (aa) != (bb), confirmed with:

CI: != --> != ⌀ --> != .

The corresponding situation for the CD is given with (aa) = (bb) and confirmed with:

CD: = --> ∅ = --> ∅ = ∅.

And in particular: (a) = (b):

CD: = --> = .

**Trans-classical concept of equality**

As introduced in earlier papers, the *equality* relation gets a deconstruction into a system of different types of equality:*equality, equivalence, similarity, bisimilarity and metamorphosis*. All those concepts or paradigms of “equality” are applicable to a general theory of graphematically based calculi.

http://memristors.memristics.com/Church-Rosser%20Morphogrammatics/Church-Rosser%20in%20Morphogrammatics.pdf

CI : = , corresponds to: (aa) !=_{Brown} (bb).

CD: , corresponds to: (aa) =_{Mers} (bb).

CI: , corresponds to: (ab) =_{Brown} (ba).

CD: = , corresponds to: (ab) !=_{Mers} (ba).

Hence, the formula of "exchange" in the calculus of differentiation CD is *specific *for the CD. It doesn’t hold in the calculus of indication, CI, nor in a Boolean logical interpretation with negation and conjunction (disjunction). Therefore, the calculus of differentiation is not modeled by a Boolean algebra as this is the case for the calculus of indication (minus some differences).

**"Negation” in CD?**

How to define “negation” in CD if the calculus of differentiation, CD, is negation-invariant?

Again,

M1: *A repetition of a quotation is the absence of a quotation.*

M2: *A quotation of a quotation is a quotation.*

C1: A differentiation of a differentiation of a situation p is a differentiation:

C2: The differentiation of 2 situations, p and q, combined with the second situation is the

differentiation of the first situation p combined with the second situation q:

C3: A differentiation of a differentiation combined with a situation is a differentiation:

C4: A differentiation of a situation p combined with a differentiation of a combination of a situation

q combined with the differentiation of the situation p is a differentiation of p: .

C5: The repetition of a differentiation is the absence of a differentiation: .

(The iteration of the same differentiation is not a differentiation.)

C5’: The differentation of the repetiton of a situations p, is a differentiation *per se*: = .

: To differentiate and to differentiate and to differentiate again is to differentiate: =

, i.e. p p = ∅, but p p p = p, based on M1, i.e. = .

Algebraic proofs are not referring anymore to states and meta-states of a formula. They refer solely to some selected formulas, proven as correct and used as *axioms*, and the rules of *equality* and *substitutions* applicable in the domain.

**Example**Proof based on (GSB) the CI initials J1, J2 and equality.

1.

2.

3. J1, 2

4.

5. a J2, 4

6. a J1, 5.

**Examples**

Proof based on N1, N2 and equality.

**Rules for concatenation**Concatenation is either disjunctive or conjunctive exclusively.

F ( ) = T.

**Example**

log2:** **** = ** disjunctive log2:** **** = ** conjunctive

1. F (0 1. F (0

2. T (0 2. T (0

3. F 3. F | F q (1

4. F q (1 4. T p q | T p q (3

5. T p q (3 5. T p | T p (4

6. T | T q (2 6. T q | T q (4

7. F p | # (6 7. T | # (2

8. T p | T q | (5 8. F p | (7

9. # # 9. #

**Rules for concatenation**Concatenation is disjunctive or conjunctive.

**Rules for signatures**T ( ) = F

F ( ) = T.

e1: e2:** **** = ** disj.

1. F 1’. F p 1. F (0

2. T p 2’. T 2. T (0

3. T 3’. F 3. F (1

4. F p 4’. T p 4. F q (1

5. # 5’. # 5. T p q (3

6. T p | T q (5

7. F p | T q **|** # (2

8. # | # **|** ** ** (5

**Rules for signatures**

T ( ) = T but T ( ) = F

F ( ) = F and F ( ) = T.

**Contradiction set:**

# = {, } for CD

value set = {T, F} for signatures.

**Rules for concatenation**Concatenation is disjunctive or conjunctive.

Hence,

T p q F p q

/\ |

T p T q F p

F q

**Contradiction****1. **Proof by contradiction of signatures, based on Mersenne M2: .

Equality “=" is taken as double implication.

**Examples**

d1 =

Tableau (d1):

1. F (0. 1. F (0.

2. T (0. 2. T (0.

3. T p (2. 3* F p (1

4. F (1. 4* T p (2

5. F p (4. 5* #

6. #

d2 = ** ****= **** **

Tableau (d2): Stripped of differentiation operation:

1. F (0. 1*. F (0.

2. T (0. 2*. T (0.

3. F (1. 3*. F p q (1

4. F q (1. 4*. T q p (2

5. F p (3. 5*. F p (3

6. T q | T (2. 6*. F q (3

7. | T p (6. 7*. T q | T p (4

7. # # (6, 4; 7, 5 8*. # | #

**2.** Proof of annulation: p p = ∅, based on Mersenne M1:

d3: **p p = ∅**

1. F p p (0.) 1’. F ∅

2. T ∅ (0.) 2’. T p p

3. F p (1.) 3’. T p | T p

4. F p (1.) 4’. # | # (1’, 3')

5. #

d4: ** ** **= **** **

Tableau (d4):

1. F (0

2. T (0

3. T (2

4. F (1

5. F (4

6. F r (4

7. F p (5

8. F q (5

9. T **|** Tq r (3

10. T p r **|** T q | T r (9

11. T p | T r **|** (10

12. # # # #

**Comment**

The formulae ** ** = , = ** **are generally provable in CI, CD and PC.

A comparison of Spencer-Brown’s calculus of indication, CI, and the postulated complementary Mersenne calculus of differentiation, CD, is not emphasizing properly enough its *differences* based on its underlying structural difference of the corresponding graphematical systems. In fact, the underlying graph-models, tree and commutative graph, are hinting to the very difference of the structures of both calculi.

It seems that the strictly complementary behavior of both calculi is clearly established. This give the chance to disseminate them in a polycontextural framework, where they hold simultaneously, and allowing to study the complementarity of the indicational and distinctional aspects of events.

A similar comparison to the recursive arithmetic setting is available with the modeling of indicational and differentiational features in a cellular automata framework.

Both calculi are extracting interesting features out of the graphematic systems but are, as far as they are defined up to now, not yet covering the full range of its operative and formal properties. Both calculi, the CI and the CD, are “state"-oriented, i.e. the results of their demonstrations are, in fact, the states “*mark*” and “*unmark*”. Both states are atomic, there are no patterns, i.e. morphograms involved. Like it would be suggested by the constellations “(tt), (tf), (ff)" for the CI and “(tt), (tf), (ft)" for the CD. Hence, the variables are defined over tuples of states and not on atomic states.

The asymmetry of both graphematic systems is not yet mirrored in the calculi. It seems, that a further analysis, based on the recursive behavior of both graphematic systems might give some additional insight into the developed structure of the calculi.

In contrast to semiotic and numeric recursivity, i.e. recursivity in the mode of identity, Mersenne and Brown recursivity has to introduce a *normal* form (standard notation) selection from the possible semiotic representations of Mersenne and Brown “strings” or “numbers”. Similar to the trito-normal form (tnf) for trito-kenogrammatic operations.

**Examples for Mersenne calculi**

**Addition Sum**

Sum(a, 0) = a

Sum(a, Succ 0) = Succ(Sum(0, a))

= Succ(a) = {aa, ab, ba}. : R2.x

Sum(a, Succ a) = Succ(Sum(a, a))

= Succ(aa, ab, ba) = {aaa, aab, bba; aba, abb; baa, bab}.

Sum(a, Succ aa) = Succ(Sum(a, aa))

= Succ(aaa, aab, bba),

= Succ(aaa) = {aaaa, aaab, bbba}, : R2.x

= Succ(aab) = {aaba, aabb}, : R2.1, R2.2

= Succ(bba) = {bbaa, bbab}. : R2.1. R2.2

Sum(a, Succ aaa) = Sum(a,(aaaa, aaab, bbba)

= {aaaaa, aaaab, bbbba; aaaba, aaabb; bbbaa, bbbab}.

**Multiplication Prod**Prod(a, 0) = 0

Prod(a, Succ 0) = Sum(a, Prod(a, 0)) = Sum(a, 0)) = a

= Prod(a, a) = a

Prod(a, Succ a) = Sum(a, Prod(a; aa, ab, ba)) = Sum(a, (aa, ab, ba))

= {aaa, aab, bba; aba, abb; baa, bab}.

**Examples for Brown calculi**

**Addition Sum**

Sum(a, 0) = a

Sum(a, Succ a) = Succ(Sum(a, a))

= Succ(aa, ab, bb) = {aaa, aab; abb; bbb} : R2.x

with {aba, bba} ∉ bnf

Sum(a, Succ aa) = Succ(Sum(a, aa))

= Succ(aaa, aab, bba, bbb)

= {aaaa, aaab, bbba; aaba, aabb; bbaa, bbab; bbbb}.

with {aaba, bbaa, bbab} ∉ bnf

Sum(a, Succ ab) = Succ(Sum(a, aa))

= Succ(aa, ab, bb) = {aaa, aab; abb; bbb}.

Sum(a, Succ bb) = Succ(Sum(a, aa))

= Succ(aa, ab, bb) = {aaa, aab; abb; bbb}.

**Multiplication Prod**Prod(a, 0) = 0

Prod(a, Succ 0) = Sum(a, Prod(a, 0)) = Sum(a, 0)) = a

= Prod(a, a) = a

Prod(a, Succ a) = Sum(a, Prod(a; aa, ab, bb)) = Sum(a, (aa, ab, bb))

= {aaa, aab; abb; bbb}.

**Comparision**

Prod(a, Succ a)

Brown: Sum(a, Prod(a; aa, ab, bb)) =

Sum(a, (aa, ab, bb)) =

{aaa, aab; abb; bbb}.

Mersenne: Sum(a, Prod(a; aa, ab, ba)) =

Sum(a, (aa, ab, ba)) =

{aaa, aab, bba; aba, abb; baa, bab}.

*"By adding reentry as a third term, Varela took Spencer-Brown’s work a step further and left behind the tame, two-valued world of Aristotelian logic. Varela made the radical assertion that reentry, along with paradoxical dynamics it entails, is built right into the very structure of the form. Varela upheld reentry as the cornerstone to autonomous functioning in nature.” *Terry Marks-Tarlow, Mario E. Martinez, *Francisco Varela (1946-2001)*, in: Society for Chaos Theory in Psychology & Life Sciences 2001, 9 (2), 3-5.

http://www.biocognitive.com/images/pdf/Francisco_Varela.PDF

http://memristors.memristics.com/MorphoReflection/Morphogrammatics%20of%20Reflection.html

Brownian and Mersennian calculi are accessible to category-theoretic considerations. And might therefore been distributed over different contextures of a polycontextural category. Interactions between different contextures containing different calculi are well ruled by the construction of bifunctoriality between discontextural contextures. Such a polycontextural dissemination is just a first step to emphazise with the means of category theory the complementary aspects and their interplay of Mersennian and Spencer-Brownian calculi.

With such a bifunctorial framework it is possible to distribute complementary formulae over the polycontextural grid.

From the point of view of the third contexture, both calculi are accessible to analysis of their semiotic properties and the embedment into the general framework.

Albeit the insistence of a “two-dimensional” notation and conception of the indicational Laws of Form, any dimensionality is reduced to one-dimensional linearity by the primary arithmetic law J2 and the law of reflexion. The same holds complementarily for the CD. The law of double differentiation is reduced to a single differentiation. Their might be some topological detours but the results of such journeys is always something simple and never involved in any non-reducible complexity. Such a *indicational* and *differentiational* simplicity is, together with the *identificational* simplicity of semiotics (logic), a strong criterion for mono-contexturality.

As it was pointed out with the paper “*Diamond Calculus of Formation of Forms”* there is a natural way to disseminate the calculus of indication over different loci: the double meaning of the “zero'-indication as a nullity in its calculus and at once as an indication in a neighboring other indicational calculus. This is formalized by the law of *enaction*. Obviously, the same holds complementarily for the calculus of differentiation too.

The “*Diamond Calculus*” paper is not yet dealing with the complementary calculus of differentiation, CD. The elaborations for the CI of *“interactional and reflectional*” indications, *enactions* and *retro-grade recursivity* have to be mirrored in the complementary setting of the CD.

With the concept of distributed and mediated enaction, the limitations drawn by the mono-contexturality of the CI and the CD are well overcome. The price to pay is an acceptance of other graphematic systems as grid for dissemination, keno- and morphogrammatics, of the trito-structure of graphematics. This polycontextural approach is not considered in this study.

Indicational CAs are identical to the indCAs introduced in previous papers.

Calculus of indication: a=a, a!=b, ab=ba

For 1D indCA:

Indicational normal form (inf):

inf([ ]) = inf([])= [])

inf([]) = inf([]) = [)]

inf([ ]) != inf([])

inf([]) !=inf([])

If two formal systems have a very close familiarity as a duality or even a complementarity, and are therefore to some degree nearly indistinguishable, but you nevertheless discovered in a strange situation of an insight a decisive difference between them. Then it might easily be possible, as in my case, that you get nightmares of endless oscillations and manifestations of something you don’t yet have access to, and what, as far as you guess, what it could be, you anyway wouldn’t like at all.

That’s what happens with the discovery of the complementary calculus of indication, a calculus I call a Mersenne calculus of *differentiation *and* separation*, in contrast to the Spencer-Brown calculus of indication and distinction. I have never been a friend of this calculus of *The Laws of Form*, therefore to get involved with its complementary calculus is no pleasure at all.

Obviously, to get rid of the headache with the CI and its ambitious and annoying celebrations, especially in German humanities, the best is to show, or even to prove, that there is a complementary calculus to the calculus of indication, too. And furthermore, both calculi are based on the general graphematic system, where they are located as the two only non-kenomic systems together with the graphematic system for semiotics as the sole pure identity system.

In fact, this is also a very late contribution to clear a situation that has not only produced unnecessary nightmares, I couldn’t avoid with my early papers, but also a late confirmation for the importance of Gotthard Gunther’s approach in this context that was demolished by the propaganda of the so-called Second-Order Cyberneticians, following Heinz von Foerster’s enthusiasm for self-referentiality, fixed points and the famous *circulus creativus* based on a non-existing chapter about the self-referentiality of reentry in George Spencer-Brown’s *Laws of Form*.

It turns out that the calculus of indication is the smallest possible calculus of the graphematic indicational system. Because Mersenne and Brownian calculi are conceived as complementary, this restriction holds for the original Mersenne calculus too.

With that, the sectarian propaganda for the CI boils down to a strictly one-sided and utterly restricted endeavour.

In-between I have written some papers dealing with the complementarity and applications of the concepts of the CI and the CD in the framework of a general graphematics.

There might be still too much undeliberated obfuscation involved, at least, some clear aspects of the new calculus of differentiation, CD, and its complementarity to the calculus of indication and distinction, positioned in the system of graphematics, are now elaborated as far as it takes to get a primary understanding of the new situation.

**Specification**

This study *“Interplay of Elementary Graphematic Calculi”* is a direct continuation of the previous paper *“Graphematic System of Cellular Automata” *which is studying 9 levels of graphematical inscription.

*"Freilich sind die primäre Algebra und die Aussagenlogik nur bedingt isomorph (strukturgleich). Neben der ungewöhnlichen Verwendung des leeren Ausdrucks, der einer beliebigen Folge von adjunktiv zugefügten Konstanten für >>das Falsche<< entspricht und der >>topologisch invarianten Notation<<, die die Reihenfolge von Adjunktions- und Konjunktionsgliedern irrelevant macht, und damit die entsprechenden Kommutativ- und Assoziativgesetze im Rahmen der primären Algebra überflüssig werden läßt, besteht ein entscheidender Unterschied zu nahezu allen gängigen formalen Systemen in der anadischen Verwendung (das heißt Verwendung ohne Stellenzahl für die Argumente) des Operators **.Während in der gewöhnlichen Aussagenlogik die Negation einstellig ist und eine Adjunktion als zweistellige Verknüpfung >>a<< und >>b<< zu >>a oder b<< verknüpft, hat der Operator *

**Recalling Varela:**

CI: Calculus of Indication,

PC: Propositional Calculus,

Variables: A, B, ... ∈ CI, PC.

Procedure: Π.*Definition* B.1

If A is ¬ B, write for A in CI;

If A is B C, write BC for A in CI;

If |- A in PC, write Π(A) = in CI;

If |- ¬A in PC, write Π() = in CI.

*Lemma* B.2

To every expression in PC there corresponds an indicational form.

*Lemma* B.3

Every demonstrable expression in PC is equivalent to the cross, , in CI.

(Varela, Principles of Biological Autonomy, 1979, p.285)

**DeMorgan**

= A

= corresponds to: ¬ (¬ A ¬ B) = A B.

**contra***"I want to conclued by emphazising once again, that the calculi of indication are not a subtle form of logic. They really intent something quite different ..." *(Varela, 1979)

Nevertheless, there is a strong isomorphism between the CI and Boolean algebra (Kaehr, Schwartz).

Such a kind of isomorphism is not (easily) to establish between the CD and a Boolean algebra.

*Lemma* D.2

To every expression in PC there corresponds a *differential* form.

*Lemma* D.3

Every demonstrable expression in PC is equivalent to the mark, , in CD.

**In contrast: Logical complementary systems**Johannes Oetsch and Hans Tompits, Gentzen-type Refutation Systems for Three-Valued Logics

While the traditional method to show that a formula is not valid is exhaustive search for counter models, refutation systems establish invalidity by deduction and thus in a purely syntactic way.

In fact, already the forefather of modern logic, Aristotle, studied rules that allow to reject assertions based on already rejected ones."

publik.tuwien.ac.at/files/PubDat_187985.pdf

Combinatorial studies are determining the number and structure of indicational and differentiational constellations. Constellations are similar to logical functions for Boolean algebras and propositional logics.

The hidden dynamics of such constellations are becoming manifest and productive if the constellations are interpreted as rules of cellular automata or finite state machines.

The classical presentation of the *Laws of Form*, and its calculus of indication, are not giving any hint to dynamize the very structure of the calculus. Dynamics are studied inside the calculus on the base of so called second-order formulas and reentry functions based on speculations about recursion in the framework of the CI (Kauffman, Varela, et al).

The new turn, presented in previous papers, is changing the static constellations into dynamic CA rules. Of special interest is the functional change of morphograms into morphic CA rules. But the same mechanism works for indicational and differentiational calculi too.

Where are the constellations from? Constellations are automatically introduced with the use of *variables* for *values* and to build *expressions* by concatenation of terms. Therefore, the study of constellations instead of singular values (states) as results or cases of demonstrations (proofs) is well defined.

Following the classification of the valuation of constellations by Spencer-Brown, there are, as for propositional logic too, 3 different classes: *truth, untruth, or contingency*.

The first we learn from the constellations of the values ∈ CI is that the correspondence between the CI and Boolean logic, with a distribution of for the CI and m, m=2, for Booleans, is not as close as it seems. Neither for the CD, with .

Hence, the story of the *isomorphism* between the CI and Boolean algebra has to be reconsidered again.

**Logical representation of indicational constellations**Following the example of G.Spencer-Brown, a CI-expression is representing a set of logical functions (Laws of Form, 1979, p. 115).

Hence, the CI-expression "” represents 6 binary logical functions:

AB, BA

¬ (¬ A B), ¬ (¬ B A),

¬ (A B), ¬ (B A).

But that’s not really the point. Because of the property of permutation-invariance of the basic elements, the constellations “op2” and “op3” are representing 4 different logical realization of the CI-expression. And the constellation “op4” is representing 5 logical functions. The constellations “op1” and “op5” are invariant. All together, the 5 CI-constellations are representing the full range of binary two-valued logical functions: 1+4+4+6+1 = 16.

This kind of modeling is not taking into account the commutativity of conjunction respectively disjunction as in GSB’s model but is focusing on the primary structure of indicational forms. The commutativity might be added secondarily as a property of the logical connectives.

Because the DeMorgan laws and even Nicod’s function or the Sheffer stroke are representable by the CI on the base of concatenation (in contrast to superposition), all the PC functions are representable too.

The listed constellations op1, op5, and op2 and op3, correspond to Kauffman's classification. The constellations op4 and op6 are not represented in Kauffman’s graphs.

The second graph entails all constellations build by “binary” applications only, i.e. and are not considered.

The Brownian algebra is a system of 5 basic patterns, op1 - op5, and a concatenation/superposition operation. The calculus of indication is abstracted from this algebra and reduced to an *element*-oriented calculus of distinction, with the operator and element with its supplement ⌀, supported by the traditional concept of *variables* and equality, operated by concatenation on the base of linearized strings.

**CI-Algebra** = (Brown; op1, ..., op5, concatenation/superposition, , , =).

**Classification** of the constellations

op1: "truth”, tautology

op5: "untruth”, contradiction

op2-op4: "contingency".

The CI is still a calculus of the dichotomy of "truth" and "untruth", with an emphasis on indicational "*truth*".

Again, the relationship between CI-representations and CI-expressions has to be re-established.

[op1]: {op1, op5 } ∈ CI-rep ==> op1 =_{CI} op5

[op2]: {op2.i, i=1, ..,5} ∈ CI-rep ==> op2.i =_{CI} op2.j, ∀i, j=1, ...,5

[op3]: {op3.i, i=1, ..,4} ∈ CI-rep ==> op3.i =_{CI} op3.j, ∀i, j=1, ...,4|

[op4]: {op4.i, i=1, ..,6} ∈ CI-rep ==> op4.i =_{CI} op4.j, ∀i, j=1, ...,6

**Kauffman’s graphs**

**The same again**

(Michael Schneider)

Here, there are just two constellations that are coinciding:"op1" as logical tautology,“taut", and as logical contradiction, “contra”. The concatenation is logically defined as a disjunction. The other constellations are accessible to a direct one-to-one logical interpretations.

Definition of logical functions by the interpretation of a set of Mersenne constellations.

Because the necessary symmetry for the logical functions is not achievable with the Mersenne calculus, the corresponding generation of the PC functions out of a primary function on the base of concatenation, like Nicod or Sheffer, that is working for the CI, is not working for the CD.

**Classification** of the constellations

op1: "truth" = "untruth",

op2- op15: "contingency".

The CD is a calculus not so much of "truth" but of "*contingency*". Nevertheless it is still mainly focused on differentiational “truth".

The *pattern-oriented* approach is taking the fact of the permutative abstraction for the CI directly into account. Therefore the initials or axioms have to be adjusted with “Het: ” to the new situation. (cf. *Appendix* 8.3.2)

This approach and result is obviously NOT in correspondence with the classical definition of the LoF.

It might be speculated that with this new understanding of the *Laws of Form*, the specific novelty of the CI is getting much more accessible than with the propagated classical approach.

The proposed CA-modeling of the behavior of the CI and the CD is adequately implemented with the rules of their cellular automata.

As demonstrated in this paper, the differences between the calculus of indication and the calculus of differentiation becomes obvious with a complexity of just n=2 (and elements m=2). In the same sense logical basics like semantics are well defined. This is not anymore the case for Stirling calculi that are based on the trito-structure of graphematics. Here a complexity of at least n>=3 is necessary to demonstrate significant calculable differences between the Stirling constellations.

As it is obvious, the operators of distinctions and differentiations are very close to the combinatorial operations of permutation. The logical negation is clearly a permutation of logical values. Indication is changing the states of configurations from the marked to the unmarked and back. Differentiation has similar properties. This fact offers two different readings: an element- and a pattern-oriented understanding.

On the level of trito-structures, i.e. with the introduction of a Stirling calculus, the pattern property becomes dominant. Therefore, such trito-patterns are called *morphograms*, and their ‘elements’ *kenograms*. The position of a kenogram in a morphogram becomes crucial for the definition of Stirling patterns, i.e. Stirling morphograms. The identification of kenograms are depending on the position of the kenogram in a morphogrammatic pattern.

The operations of distinction and differentiation are generalized and applied as reflectors to morphograms. What played a relatively reduced role in the CI and the CD, *concatenation* and *superposition*, becomes a significant feature in the prominent roles as *coalitions* and *cooperations, *supported by other genuine morphogrammatic operations.

Like concatenations are “additions” and superpositions “multiplications” of strings, Stirling *coalitions* are replacing CI- and CD-concatenation, and Stirling *cooperations* are replacing CI- and CD-superpositions. Furthermore, the whole apparatus of deconstructed identity or equality relations has to be applied: *equality, equivalence, similarity, bisimilarity* and *metamorphosis*.

Superposition for the CI and the CD is not differentiated. It is realized by an unstructured monadic operator. In contrast, operators in Stirling systems are structured. A morphogram is not just an operand but an operator too. This fact is reflected by the structured coalitions and cooperations.

Because this figure of thought is definitively abandoning classical features dictated by identity thinking, the best exemplification, until now, is its interpretation in the framework of *morphic cellular automata*. A morphogram, then, is at once a *pattern* (morphé) and a *rule* (dynamis) of transclassical computation.

Despite the big ambitions declared by GSB with his calculus of indication as having surpassed the distinction of operator and operand, the manoeuvre works only for the highly trivial case of an unstructured operator, the cross. There is no chance, and obviously also no intention, to invent operators able of superposition of higher complexity, superpositions of structured distinctions. Obviously, something like “Q(P(X))" with Q and P defined as composed superposition terms is excluded from the world of forms. And certainly, such a case is not an example for a recursive form. The *Laws of Form *have a succession in the* Laws of Formations* (The Calculus of Idempositions, GSB, L. Kauffman, 1968) preserving the principal features of LoF.

What is possible for the CI is a highly restricted form of iteration as a superposition of the cross: From iteration:

to composed formulas with iterations: (Wolfram rule).

**Superposition as indicational cooperation**What is the superposition of the expression

But how could a complex operator be defined and applied for a more complex version of the CI? Concatenation might be generalized to addition (coalition) and superposition to multiplication (cooperation).

Quite different approaches to complex dimensionality are given by William Bricken, Jeff James and L. Kauffman.

*"From a boundary perspective, the notation of both arithmetic and algebra is dimensionally impoverished; a one-dimensional string imposes structural and computational rules that are not essential to the inherently parallel structure of numerics.”*

http://www.wbricken.com/pdfs/01bm/06number/bnums-complete/04bnums-sb.pdf

Given a = , and b = ( ⌀), then a superposition like a^{b} would have to be defiend as . But for a start it would be easier to define a superposition as a multiplication “a • b”. Hence, with ( ) •_{CI} ( ⌀) under the conditions of a Calculus of Indication, the result is a unary set of constellations.

By definition, indicational system (Brownian) are graphematically positioned between deutero- and trito-grammatic systems, and achored in identity systems (Leibniz). Their permutativity, (ab) = (ba) connects them to deutero-systems (Pascal) and the abstraction from the permutations to the trito-systems (Stirling). Because of the identity property of homogeneous constellations, (aa) != (bb), Brownian systems are additionally partly anchored in identity systems (Leibniz).

Considering the sketched approach, it is quite natural to generalize the *base* of the calculus of indication, and complementary, the *base* of the Mersenne calculus, to non-restricted calculi of arbitrary complexity. Therefore, a reduction to a two-element calculus is a restriction dictated not by formal but by philosophical or ideological reasons.

**Is the emptiness empty at all?**The aim of GSB's

May be, simplicity is not anymore what we are looking for.

The cultural belief into the metaphysical doctrine: *“the emptiness is empty”* and the *“void is void”*, etc., has become as shallow as Edinburgh’s famous Ghost House. Everybody who knows *Loch Ness* knows about the dynamics of emptiness, the Lochness of the Loch Ness.

An interesting consequence, still to study, is the application of the new situation to *reentry* forms.

Take 12.2: f = , then the usual construction of a contradiction, as shown by Varela, becomes questionable, again.

With the application of the initial = ⌀ to the formula "f = ”, we have, according to Varela’s construction:

f = , then

=

= , with substitution

=

= f , with substitution.

"*But this leads to a contradiction, because substituting in (12.2) we have = *

which would render the calculus of indication inconsistent, and thus useless, by confusing every form

This result is obviously based on the “*element-oriented*” approach that is excluding any pattern-oriented considerations.

A pattern-oriented understanding of the calculus would clearly deliver a different result. Following the sketched introduction to the pattern-oriented CI we would not get “⌀ = ”, which is obviously a contradiction based on an atomic constellation, but the pattern “⌀⌀ = ”.

But the different approach is directly applicable to Varela’s start formula "f = “.

For f = p we get val(p) = (, ⌀) and val() = (⌀, ) and applying Het, we get p =, and therefore, by substitution, f = , which holds in the pattern interpretation of the CI as an *equivalence relation*. With that, the Brownian emphasis on the difference between the CI and negational logic gets some further justification.

With that, the still identity-logic inspired construction Varela’s is losing all its resonability.

It is also an open question if this direct modeling of the reentry form corresponds to the Brownian approach. To my knowledge, reentry occurs in the CI only in *context* with a form defined by other constituents, i.e. variables, and not in isolation.

A more profound analysis of the situation, considering polycontexturality, is given by Elena Esposito in *Ein zweiwertiger nicht-selbständiger Kalkül*:

*"Wollte man dies formalisieren, müßte man auf einen Kalkül von Wiedereintritten rekurrieren, das heißt auf einen Kalkül von Beziehungen nicht mehr zwischen Bezeichnungen, sondern zwischen Beobachtungsordnungen. Das jedoch ist nicht mehr die Aufgabe des Indikationenkalküls.” *Elena Esposito*, Ein zweiwertiger nicht-selbständiger Kalkül*, in:* *Kalkül der Form, (Dirk Baecker, hrsg.), suhrkamp taschenbuch, Frankfurt/M. 1993, p.96-111.

Apart of a logical interpretation of CI-constellations, there is a reflector-oriented analysis of direct interest. The merits of a reflector-approach becomes evident in a polycontextural framework of complexions of CIs (Kaehr, 1981). The reflector-analysis was successfully applied for the study of the behavior of composed morphograms in morphogrammatics.

The calculus of indication intents to strip down the logical “corset” of variables and functions down to the structural bones of its mechanism. Therefore, a reflectional analysis is achieving more directly the attempts of a skeleton-analysis than the common approaches of logical comparisons. A sketch of a reflector-analysis is given for Brown, Mersenne, Stirling and Deutero-structures.

**Reflector analysis for CI**

**Reflector analysis for CD**

refl(1) = (1) refl(4) = (7)

refl(2) = (12) refl(5) = (15)

refl(3) = (11) refl(6) = (9).

(1) : self-reflective

(i), i = 2,...,15 : reflective

**Reflector analysis for Stirling**

refl(1) = (1), refl(5) = 5, refl(6) = (6), refl(7) = (7) : self-reflective

refl(2) = (8) : reflective,

refl(3) = (4) : reflective.

**Reflector analysis of deutero-structures**

refl(i) = (i), i = 1, 2, 3.

This short analysis of different reflector-systems makes the position of Mersenne and Brownian calculi inside the graphematic system more clear.

Calculi shall be developed that are dealing not just with atomic elements (states) but with *patterns* of behaviors, as far as this is possible within the framework of calculi like the CI and the CD.

With the introduction of the CI- and the CD-**equivalence** relation over CI-expressions *e* and CD-expressions *d* the connection between the *intentions* of the calculi CI and CD as founded in their graphematical systems is re-established.

It seems, that there is a *discrepancy* in the formalization of the calculus of indication and its intention. This intention seems to be properly represented by the graphematics of the CI-intention.

Similar holds for the formalization of the calculus of differentiation CD.

The gap in the formalizations is produced by the mapping of expressions of sets of atomic “states” to sets of atomic “states”, in the CI the states Cross and NoCross, i.e. from {, } to { , } and in the CD from {, } to the “states" {, }.

Hence, the use of a *functional* representation, i.e. F:{x, y}^{n} --> {x, y}, without an **abstraction** from the permutation of states for the CI, i.e. F, and the abstraction from the equality of homogeneous states for the CD, i.e. CD, seems to mislead the intentions of the interventions. The transition from the “*primary arithmetic*” to the “*primary* *algebra*” with its use of variables is based on the assumption of the correctness of the step-wise atomic “*substitution*”, e.g. “Let p = . Then”. This shows the unresolved complicity with “identity” assumptions, well in contrast with the claims of the *Laws of Form.*

Albeit differently motivated, and not touching the fundaments of the CI formalization at all, a similarity to the above remarks could be constructed with Varela’s reflection axiom “a = ” (12.27) and “p() = a” for his *“generalized Brownian algebra”* of “*periodic* *sequences*” and “*wave forms*"(Varela, p. 150/51). But Varela’s “transposition algebra” of “wave forms” is not considered as a proper Brownian calculus for forms.

The aim of pattern-oriented formalization is defined by the attempt to take directly into account the graphematic structure of the CI and CD calculi as it appears within its tree and graph productions.

The fact that the expressions e_{1}= (aaba) and e_{2}= (aaab) of the primary algebra are indicationally equivalent is not properly represented by the ordinary formalization(s) of the *Laws of Form*.

The same holds for the CD. The fact that the expressions d_{1}= (aaaa) and d_{2}= (bbbb) are differentially equivalent is not properly represented by the proposed formalization of the *Laws of Differentiation*. (cf. Appendix 8.2.4)

**CI:**

e_{1}= (abaa)∈graph with the interpretation () and = (aaab)∈graph with the interpretation () results in the conflictive equation: e_{1} != .

From the point of view of the graph-generation for the CI, both patterns, eand e_{2}, are CI-equal.

Hence, the pattern related fact of *permutative* equality has to be considered: X =_{CI} perm(X).

**CD:**

d_{1}= (aaaa)∈tree with the interpretation () and d_{2}= (bbbb)∈tree with the interpretation (∅∅∅∅) results in the conflictive equation: d_{1} != .

From the point of view of the tree-generation for the CD, both patterns, d_{1}and d_{2}, are CD-equal, d_{1} =_{CD} .

Hence, the pattern related fact of *homogeneous* equality has to be considered: X =_{CD} hom(X).

A different *“calculus taken out of the calculus".*

Following the standard definitions of a calculus, the simple tectonics of alphabet, syntax and interpretation, “semantics”, proof theory has to be introduced.

**Calculus of Indication***Alphabet*

constants = {, ⌀}

variables = {p, q, r, ...}

*Syntax*

concatenation of the constants {, ⌀} and superposition of {}.

R0.1: ==>

R0.2: ==> ⌀

R1.1 n ==> n

R1.2 n ==>

R1.3 n ==> n ⌀

Concatenation and superposition of variables

R3.1: p ==>

R3.2: p, q ==> p q

With the rules R1 - R3 a production of strings based on the alphabet is defined.

All the CI-strings produced are accepted as CI-expressions.

*Example*

e = ppq q ⌀ r ⌀ .

Hence, any expression build on the base of CI-constants and CI-variables with the operators of *concatenation* and *superposition *of the constant* * is a CI-expression.

*Semantics*, i.e. indicational rules

J1: ⇔

J2: ⇔ ⌀

I1: ⇔ ⌀

I2:

*Valuation*

V1: The valuation of variables, p, ..., p_{n} is defined by the mapping val(CI): {, ⌀}^{n} --> {, ⌀}.

Wording for the valuation of variables:

“Let p = . Then ..."

"Let p = ⌀. Then..."

This defines a valuation *table*, like for the semantics of propositional logic but without a semantic-ontological commitment.

P: *Rules for proofs*, based on substitution and equality.

With this setting, and its further elaboration, the crucial situations of the CI as they are determined by the graphematic graph-structure of the indicational events are *not* approached at all.

Therefore, the semantic mapping for the variables has to be specified according to the “*pattern*"-structure of the indicational graph. With that, the algebraic rules are not just depending on the atomic values (constants) but on the structure of the valuation too. There are just two classes of patterns: *homogenous* and *heterogenous* to consider at first.

*Homogeneus* patterns are treated like atomic constants, i.e. X ∈ Hom ==> X != . *Heterogeneous* patterns are permutation-invariant, therefore: X ∈ Het ==> X = .

Therefore, according to the proposed conception of indication, the calculus of indication has to be specified by the meta-theoretic distinction of *homogenous* and *heterogeneous* valuations of its variables, i.e. X ∈ Hom or X ∈ Het.

Heterogenuos patterns are *classified* according the number of their elements.

The number of all patterns for an indicational graph are given by the combinatirial formula:

Because there are only two homogeneous constellation in an indicational graph for m = 2, the number of heterogenous situations is:

Recalling the usual machinery for calculi, propositional logics and the sketch of a definition of the calculus of indication, the procedure to establish a calculus of differentiation follows the same strategies.

Here, this manoeuvre shall be reduced to the essentials. And that is the pattern aspect of the homogeneous constellations of “valuations".

Second-order formulas have different motivations in Spencer-Brown's Laws of Form. A new turn is achieved with the introduction of the complementary Mersenne calculus. The CI is studying distinction in or of a world, while the CD is studying the differentiations of inscriptions of calculi in a graphematic (scriptural) space. Hence, a new second-order type of analysis is opened up: the differentiational study of distinctional inscriptions, i.e. the CD-study of CI-notations.

This is similar to GSB’s the study of Russell/Whitehead’s logic with the means of his CI. This analysis of logic by the CI offers a specific reduction of the logical material: *“this represents a reduction of the mathematical noise-level by a factor of more than 40000” *(LoF, 1972, p. 117)

Hence, a minimal graphematic *comparistics* of formal languages is in place: logic, distinctional and differentional calculi.

A self-application of CI formulae are a reentry of the form into the form.

A CD self-application is analyzing the form of the CI.

The interesting feature of memristivity that has been discovered for morphogrammatics in general, gets its special realization in the calculi of Spencer-Brown and Mersenne too. Both are founded in graphematic systems and have partially morphogrammatic properties.

Memristivity is closely related to the fact that a repetition of a term in a morphogrammatic system has to decide as what kind of a representation of the term the term has to be repeated. This question is obsolete for identity systems, like semiotics. It occurs at first on a low level with the CI and the CD. According to the relevant abstractions, a term in a CI constellation is involved into its permutations. Thus the choice of a representation is defined by the range of the number of permutations of the elements of the term.

Following the idea of complexity reduction of graphematic systems in respect to semiotic sign systems, the measure of reduction, the reduction quotient, is in fact representing the degree of memristivity of a system. As it is clear, semiotic systems are of zero memristivity, i.e. measured as - 1= 0.

**Example: Logic/Indication**

This approach gets a more precise elaboration with the combinatorial formulas of the specific representations of morphogrammatic situations in semiotic realizations.

As we learned from the papers “*Diamond calculus*” and “*Quadralectics*”, there is a precise architecture and mechanism of 4-fold disseminations of calculi, esp. the calculus of indication. What wasn't established in the previous papers was an exact specification of the laws of a “*complementary*” calculus of indication, i.e. the Mersenne calculus. Dual-forms to existing calculi had been well defined but not elaborated.

With the approach to Mersenne calculi the promise for a complementary calculus is fulfilled and the gap is finally closed.** Quadralectics of CI and CD**

Quadraletic (CI, CD, CI^{D}, CD^{D}) =

**Complementary and inverse forms**

Because of the principle of "*perfect continence*”, there are no dual forms in a logical sense in the calculi of forms. What is reflecting the formation of forms are parallax and complementary, i.e. diamond formations of form. This is mirrored firstly, by the systems of inverse forms. Hence, the basic, and not yet disseminated planar forms, are the forms of *complementarity *and* inversion *(here: duality).

Complementary calculi are based on complementary *graphematic* systems. Therefore, there is no mechanism given by a calculus to define its complementary calculus by the means of the calculus. Dual systems remain in their graphematic framework, i.e. the dual of a calculus is defined by the means of the calculus itself. Hence, the graphematics of the CI is complementary to the graphematics of the CD, while the graphematics of the dual systems are unchanged.

**Distinctions between distinction systems**

Beyond the systematics of planar distinctions, a polycontextural theory and calculus of distinctions, is demanding for distinctions between discontextural distinction systems. This might be realized by the introduction of topological and knot-theoretic constellations of distinction systems. A simple start could be a 3-dimensional distinction system with the set of planar distinctions and reentries at each contextural position and the transcontextural distinctions and reentry forms between distributed contextural distinction systems.

An interesting interplay between the calculi of distinction and differentiation might be introduced as a chiasm between the calculi and their gaps. In this sense, a gap of a calculus is the placeholder for its neighbor calculus.

One of the main terms used in this presentation of a complementarity between the Spencer-Brown calculus of indication and the Mersenne calculus of differentiation is the term “*complementarity*” itself.

As shown in other papers, the most fundamental and most direct approach to define the term “complementarity” is, as far as I know, accessible by the theory of diamond category. Diamond category is defined as an interplay of categories and saltatories. Such an interplay defines the complementarity of categories and saltatories in diamonds.

One crucial motivation to use diamond categorical constructions is the fact that the “concatenation” of marks in the CI is not defined by properties of the calculus but based on cultural intuition. This lack of definition is fixed with the strategy of the matching conditions for morphisms.

Complementarity = diamond (Brown, Mersenne).

diamond (Mersenne ∐ Brown) = Mersenne || Brown

Semiotics in this chiastic diamond diagram is reflecting the ‘common’ and *acceptional* features of Brown and Mersenne, while complementarity is the place where the *rejectional* difference between both are kept together.

Given this chiastic diamond modeling, iterative and accretive *combinations* to build networks, grids or meshes out of it are following naturally.