Part I, Chapter 1: Fundamentals, Section 1.5

§1.5) Definition of Panel as an Operator

In order to develop a comprehensive theory of the mechanics of
sequential art, we must necessarily come to a functionally robust,
yet adequately general definition for what it means for an object
to be "inside" or "outside" of a panel. This section develops the idea
of a panel as a binary relation that maps objects from the space of
elements E to the spaces defined by a set of borders G. We develop
notation for the operator as Pm{E,G,k}.

1.5.1) Logical Definition of Panel:

Let Pm(x) be a predicate consisting of the statement "is contained within a border G", and x be any element of E or any n-tuple of elements selected therefrom. This predicate is true for all x, and is false for no x, e.g. for things not included in the universe of discourse.

(∀x∈E).Pm(x)...............Df.
(∀~x∈E).Pm(x) ⊃ f........Df.

1.5.2) Functional Definition of Panel:

With the logical description in hand, we now expand the definition into its full functional form.

Define: Pm{E,G,k}: E→G under k.

Where E is the tuple consisting of all elements in the layout L, and G is the set of all finite, simply-connected borders in L, such that E=(e1, e2, ..., et) and G={G1, G2, ...Gm}. Here, k is an index containing entries {σ1, σ2,..., σm}, where σ is some positive integer satisfying Σσm = t.

Then Pm is defined as the mapping which associates a certain sequence of elements out of E, specified by k, to a specific G.

Thus:

Pm{E,G,k} :=

G1 = (e1, e2, ..., e(σ1)) : k= σ1
G2 = (e(σ1) +1,e(σ1) +2, ..., e(σ1)+(σ2)) : k= σ2
...
Gm = (e(σt-m), ..., et) : k= σm

1.5.3) Some Properties of the Panel Operator:

Pm{E,G,k} is useful insofar as several interesting properties can be derived from this definition. These properties are analogous to the basic operations of arithmetic. They are: Union (Adding panels to each other) and Disjunction (Breaking panels up).

Union of adjacent Borders: ... Gn-1 ∪ Gn ∪ Gn+1 ...

Which can be represented by the operator as the following, given u, v, w ∈ E; Gn, Gn-1, Gn+1 ∈ G, σn, σn-1, σn+1 ∈ k:

1. Gn-1 ∪ Gn: Pm{E, Gn-1 + Gn, k}= Pm{u+v, Gn-1+Gn, σn-1 + σn}
2. Gn ∪ Gn+1: Pm{E, Gn + Gn+1, k}= Pm{v+w, Gn+Gn+1, σn + σn+1}
3. Gn-1 ∪ Gn∪ Gn+1: Pm{E, Gn-1 + Gn + Gn+1, k}= Pm{u+v+w, Gn-1+Gn+Gn+1, σn-1 + σn + σn+1}

Since E is well-ordered, we can write u+v+w=x, Gn-1+Gn+Gn+1 = G', σn-1 + σn + σn+1 = σ'. Thus:

Pm{E, Gn-1 ∪ Gn ∪ Gn+1, k} = Pm{x, G', σ'} .

Corollary: L = Σ G. ∎

However, this only makes sense for consecutive panels. We can run this oppositely to get disjunctions.

Part I, Chapter 1: Fundamentals, Section 1.4

§ 1.4) Elements as Atomic Propositions

So far our discussion has built up several of the necessary truths
of sequential art that have heretofore been taken for granted, or
left undefined. However, a necessary concept - that of the "element"-
has not yet been defined. This is because ASPM treats all sequential
art layouts as being comprised of these fundamental entities, whose
nature cannot be further defined. They are, therefore, "atomic".

1.4.1) Sequential Art As Logic:

Russell and Whitehead write in Principia Mathematica:

"Atomic propositions may be defined negatively as propositions containing no parts that are propositions, and not containing the notions 'all' or 'some'. Thus 'this is red', 'this is earlier than that,' are atomic propositions." [1]

We apply this definition in our treatment of sequential art. Just as in statements of logic there are fundamental propositions which cannot be split up into smaller propositions, there are objects in sequential art that cannot be split up into further objects. For example, the image of a man, as a symbol, is a single element. If one imagines a stick figure standing in an empty white box, the stick figure would be an element, and one could imagine equating the figure in the language of sequential art, with an equally valid statement in writing: "There is a figure".

Given the Panel Sequence Theorem (§1.2), if we draw two panels, we have set up a sequence. And this is the same as saying "If A (panel 1) then B (panel 2)". The act of demarcating some arbitrary number of elements in sequence equates to creating a linear chain of inference.

Hence, an element is defined as any object depicted in a layout that cannot be broken down into further qualifications. In practice, of course, it may be difficult or impossible to precisely point to every single element in a layout, but since the object of our task is to achieve a perfectly idealized mathematical model that can serve as a Platonic ideal for all sequential art works, for our purposes we shall treat elements as discrete objects.

1.4.2) Sequential Art As Language:

As McCloud writes in Making Comics:

"Comics is a secret language all its own, and mastering it poses challenges unlike any faced by prose writers, illustrators or any other creative professionals." [2]

This statement is true. As mentioned in the introduction, sequential art is a framework built upon writing and art, and exists at the unification of the two disciplines. From this union, and from the special demands imposed on art and writing by the demands of presentation of a visual story on the page, comes a unique grammatical structure built to convey meaning effectively. To that end, insofar as it is, in fact, an orderly system of meaning, sequential art yields useful insights when subjected to analysis by reductionistic logic.

This is to say, in general, that, as a language unto itself, sequential art is by definition an ordered system. One can imagine that, just as the statements of formal language are ordered by the structure of grammar, so too are the statements of sequential art are ordered by a unique form "grammar". Whereas in language, statements such as "If given X, P(x) implies Y; X is true, therefore Y" exist and can be written down, so too does the language of sequential art possess a similar if-then structure, conveyed visually by elements on the page. The fact that there are elements, and that they are encoded by a formal logic, implies a sequentiality similar to that possessed by elements of a language. One can imagine a set of elements in a given layout is in many ways "meant" to be ordered in a given sequence - otherwise, insofar as it as a work of sequential art, the layout would be meaningless. Similarly, a sentence in written language without the structure of grammar, is also meaningless.

Thus, by definition, sequential art, in order to be considered sequential art at all, must needs possess an underlying orderliness. It is from orderly, sequential placement of the elements, structured by the grammatical logic of sequential art, that it derives meaning.

1.4.3) Logic and Language Imply Atomic Truths

Given 1.41 and 1.42, we come to the conclusion that, to gain meaning as a language, sequential art employs elements that are fundamental, and place them in sequence. Therefore, it is meaningful to write statements such as e1 ⊃ e2 ⊃ ... ⊃em-1. ⊢ em, or molecular propositions em ⊃ ~em+1 because the elements can be treated as ordered sets of atomic propositions. Since sequential art possesses a unique grammar and structure, many aspects are therefore amenable to the same forms of mathematical treatment as logic. Concepts such as design, emotional affect, aesthetics, symbolism, metaphor and so on, are, however, philosophical in nature, and not within the scope of logical sequential art as treated by ASPM. ▮

1.4.4) References:

• [1] Whitehead, A. N. & Russell, B., Principia Mathematica to *56, Second Edition, Cambridge at the University Press, 1964, p. xv
• [2] McCloud, S., Making Comics, Harper-Collins, 2006, p. 2
  

Part I, Chapter 1: Fundamentals, Section 1.3

§ 1.3) Uniqueness Theorem

The Uniqueness Theorem states that any given element et must absolutely
choose to belong to one panel and no others. Therefore, the set
0f elements et belonging to Pn is the set of those objects with the property
of not belonging to any prior or subsequent panels.

1.3.1) Plain language proof:

Given the Axiom of Panels and its corollary, let us define a panel ϕ(x) with the property of being the set of all elements contained in both Pn and Pn+1. Now every x that is an element of ϕ(x) will have the following properties:

1. It belongs to neither Pn-1, nor Pn+1 .
   
2. It belongs to neither Pn, nor Pn+2.

In terms of predicate logic for a two-valued function ϕ(x,y) where x=y.

(x,y)ϕ(x,y): ~x ^ x
(x,y)ϕ(x,y): y ^ ~y

Using a truth table we can compute the value for these statements.

~x ^ x
F F T
F F T
T F F
T F T

y ^ ~y
T F F
T F F
F F T
F F T

And, since both statements are equally false, the statement ~x^x = y^~y is true for all cases. Substituting y=x for y, we get ~x^x = x^~x. Being true in all cases, then, means that we can say without a doubt that the statement (x, ...)ϕ(x,...):{x NOT in Pn-1, x NOT in Pn, x NOT in Pn+1, x NOT in Pn+2, ...} is always false for any given set of x's, since by mathematical induction, we can see that the proof works for the n+1 case as well. Therefore, by induction, we conclude that because ϕ(x) contains all the elements x that do not belong to any union of spaces Pn U Pn+1, it is therefore equal to the empty set. Therefore, any element et must belong to one and only one Pn. ▮.

1.3.2) Proof in Symbolic Logic.

A proof in symbolic logic will follow.

NB: Bear in mind that these assertions make no statement on the movement of the eye over the design. At this stage, we are merely trying to build up precise definitions for concepts such as "belonging to" or "exists".

1.3.3) Implications of the Uniqueness Theorem.

The Uniqueness Theorem implies that, whatever their position in space related to the layout, an element will belong to that panel where it was first read. Imagine if each panel were on its own unique layer, and whatever arrangement of elements was free to exist either within or outside the panel boundaries. By the Uniqueness Theorem, all elements first encountered within those panels will belong strictly to those panels and those alone, since each dwells within its own unique space.

Part I, Chapter 1: Fundamentals, Section 1.2

§ 1.2) Panel Sequence Theorem

It is now possible to define our first theorem. Panel P1 containing a finite
subset of elements e1...et, 1≤t≤m, implies a final panel Pm.

1.2.1) Proof:

By the Axiom of Extension (§ 1.1.4) we have a set of elements defined by the property that each en+1 follows sequentially from en, where 1≤n≤t, on the set of well-formed formulae e1 to et.

Lemma 1:

1. Given e1, and e1 ⊃ e2, therefore e2. (modus ponens)
2. en-1 ⊃ en
3. et ⊃ et+1
4. e1, e2, ..., et, et+1,...,em-1 ⊢ em ▮.

This is true by the Deduction Theorem of the propositional calculus.

Now, given Pn = {e1, e2, ..., et}, let us define Pn+1 = {et+1, et+2, ..., es}, s ≤ m-1.

Let us write x as a variable that takes on values e1,..., et and satisfies the conditions for Pn. Then the definition of Pn becomes a predicate Pn(x) and all et in Pn can be written as the statement: ∀xPn(x).

Now, given Lemma 1:

1. Pn(et) ⊃ Pn+1(et+1)
   

So:

1. ∀x1Pn(x1) ⊃ ∀x2Pn+1(x2); x2 = {et+1, et+2, ..., es}
2. ∴ Pn(x1) ⊃ Pn+1(x2)
3. ∴ Pn(x1) ⊃ Pn+1(x2) ⊃... Pm-1(xm-1) ⊢ Pm(xm) ▮.

1.2.2) Corollary to the Panel Sequence Theorem.

Given t=m, the Axiom of Completion implies P1 = Pm.
That is, the panel containing all elements e1...em contained within L is
the the only and final panel. Let us call this the Identity Panel PI.

1.2.2.1) Proof:

1. ∀x1P1(x1) ⊃ ∀x2Pn+1(x2)
2. x1= e1, e2..., em = xm
   
3. ∀xmP1(xm) ⊃ ∀xmP2(xm+1) . ⊃ f (Axiom of Completion, §1.1.5)
   
4. ∀xmP1(xm) ⊃ ∀xmP2(xm+1) . ⊃ f . ⊃ ∀xmP1(xm) = ∀xmPm(xm)
5. ∴ P1=Pm
6. ⊢P1(xm) : PI ▮.

1.2.3) Summary

Having proved that Pn+1 necessarily follows from Pn, we arrive at the logical conclusion that if we are to assume that the set of et is orderly, then having exhausted all e1...et within Pn, one must continue on to Pn+1 until one has reached Pm. Essentially, if the elements are finite and ordered, then the panels must also be finite and ordered.

Part I, Chapter 1: Fundamentals, Section 1.1

1) Axiomatization of Fundamental Truths of Sequential Art

By developing a sufficiently rich language to describe sequential art,
it is possible to make powerful, general statements that are true of all
works of sequential art, irregardless of their individual differences.

§ 1.1) Formalization of First-Order Mathematical Sequential Art

To adequately ground mathematical Sequential Art in firmly logical
foundations necessitates the creation of a precise, logical language by
which all theorems of Sequential Art can be derived. What follows are
the axioms and formation rules necessary for first-order mathematical
sequential art.

1.1.1) Definitions

1.1.1.1) The Layout. The layout L ⊆ ℝ2 bounded by a finite, simply-connected border B. All objects so bounded are considered part of the Universe of Discourse. Any element or subset of L has a positive truth value, any element or subset not in the Universe of Discourse has a negative truth value.

1.1.1.2) Elements. E is the t-tuple consisting of all elements et in L. An element is an object in the Universe of Discourse.

1.1.1.3) Relations. Define the dyadic relation > where en-1 > en for 1≤n≤t. Let en-1 be defined as the predecessor and en be defined as the successor.

1.1.2) Formation Rules

1.1.2.1) Modus Ponens. From en-1 > en and given en-1 to imply en . In logical notation:

[[en-1 > en] & [ en-1 ]] ⊃ en. Df.

1.1.3) Axioms

1.1.3.1) Axiom of Enclosure. Within L, we may draw any number of finite, simply-connected borders enclosing some grouping of elements, subset of E. Every border is a member of the set of borders G.

1.1.3.2) Axiom of Sequence. The relation > will apply to every pair of elements in E.

Table of Contents

Part I: Diagnostics

Chapter 1: Axioms of Mathematical Sequential Art

Chapter 2: Formal Semantics

...

Part II: Prognostics

PART I



Diagnostics







Mathematics is the dream,
Physics is the reality.

Outline of the Book

The agenda of this book will be to provide precise, mathematical descriptions for every aspect of sequential art, and to derive theorems based on these axioms proving commonly-held notions about sequential art. The goal of this exercise is essentially to prove that these commonly-held notions consist of an independent, consistent logical framework that obeys certain properties; and, having a fundamental basis in logical operations, to be able to make broad, general statements that are true for all sequential art. To that end, we will develop a calculus for determining an abstract, general system of sequential art that exists entirely in the world of mathematics, but whose properties can be specified or modified to apply to any real work of sequential art.

The ability to make precise statements about any particular system governed by a certain set of axioms, is a fundamental step towards a comprehensive understanding of all systems governed by the same axioms. This is analogous to how a fundamental knowledge of physics is essential for making precise statements necessary for engineering and building specific machinery. A broad, fundamental understanding of the basic principles underlying all sequential art, then, is imperative for the growth and development of the field.

Thus, this book will be divided into two main parts: Part One will deal exclusively with the establishment of axioms, theorems, and precise definitions for the common tools and systems found within sequential art; while Part Two will deal with the material implications of these axioms and theorems. A substantial goal of Part One will be to develop the mathematical machinery necessary for describing the concept of flow through a sequential art layout, a precise, mathematical description of which will be the primary subject of Part Two.

Hence, one could broadly describe the two parts as largely being divided into, for part one, a treatment of "statics", of building up an analytic model for the base, logical machinery that govern the operations of sequential art; and for part two, a development of "dynamics", of creating a prognostic model using that machinery to create a deterministic system for governing flow. Necessarily, then, the first part will rely heavily on mathematical logic, set theory, and abstract algebra; while the second part will rely heavily on systems analysis, differential equations, and multivariable calculus.

It is assumed by the author that the reader is at least generally familiar with these mathematical tools and can follow a cogent mathematical argument. Arguments will be done in the traditional theorem-proof structure, building conclusions based on logical deductions from the basic axioms, then using theorems to prove further theorems. Where necessary, simple diagrams will be provided.

Acknowledgments

Special thanks go to Eric Gerlach, my
philosophy professor at Berkeley City College,
for his excellent course in logic, that
turned me on to concepts such as syllogisms,
mathematical logic, and truth tables. I would
not have known about these interesting
systems had it not been for this course and
his clear and effective presentations.

Introduction

This is a declaration of war.

This is the response of the natural scientist, who knows the world empirically, based on reason and sound logic, to those who claim to know nothing. This is a rejection of negative, skeptical philosophies who reject the axioms of true, physical reality, in favor of unprovable, unknowable claims about the unreality of nature and the uncertainty of knowledge that are beyond proof and evidence; and an affirmation of empirical truths that appeal to common sense and logical deductions based on sound axioms.

The natural scientist embraces the future and values forward progress and knowledge; the skeptic claims that history is a circle and points to the past. The past is dead, a rotten framework built of driftwood and twigs, around a termite-hollowed core; it is in the future that promise and progress lie.

Thus spoken we turn to the aim of this book: the axiomatization and empiricization of a realm that which was once thought of as the domain of only the philosopher, the artist, and the writer. Here I speak of the world of sequential art.

Let me address the question of why we must axiomatize sequential art. The world of sequential art is a type of craft, hence it lies at the intersection of art and writing, the boundary inhabited by design and technical illustration. Art without writing is static, is frozen: a statue has no movement, a painting has form but does not change in time; writing without art is invulnerable to physical analysis, trapped within a world of symbols, metaphor, language and grammar. Taken alone, each are equally timeless and resistant to analytical techniques, but the demands of craft, when taken together, bring them down to a zone where they are amenable to logical treatment.

Yet in the history of analysis of sequential art, all treatises on the subject have been written exclusively by artists, not engineers nor scientists, and are necessarily comparative, inductive rather than deductive, without the technical vocabulary to precisely define the concepts of which they speak. McCloud comes closest to this, but even he fails at giving a totally empirical definition of the mechanisms which animate comics, and is forced to reason by means of analogies and vague pictures and symbols. The same has happened time and time again in every area of human knowledge.

Like DaVinci, Galileo, and Kepler of the past, the world of physics fully explored the myriad descriptive methods available to science of the time until it ran up against the boundaries of what was possible with their paradigm. With the coming of Newton's Philosophiae Naturalis Principia Mathematica the world of physics was transformed with a new paradigm; it was given new life and animation by the development of Newton's calculus, which axiomatized the principles of planetary motion and the movements of objects in space, generalizing and abstracting the inductive, empirical knowledge of before with deductive logic. Having the new paradigm of Newton's logic, science explored every avenue of research until it reached its limit in the early twentieth century, where again its boundaries were expanded and generalized further with the quantum theory and Einstein's general relativity.

Likewise, all theoretical developments in any area of human knowledge will explore every possibility, every method of analysis, until they reach the boundaries posed by their current methodological paradigm, at which point progress is impossible without forcing open those boundaries with a new concept, and establishing a new paradigm. And at the core of each new paradigm is a further abstraction, a further generalization, an expansion of the view to encompass larger principles and a more concrete description of the concepts of the previous.

Therefore the agenda of this book is revolutionary. We have currently reached the limit of where earlier theorists have explored and if we are to progress further we must discover new principles. Our aim is therefore to axiomatize the rules-of-thumb that describe the principles of sequential art, outlined by predecessors such as McCloud and Eisner, and create a paradigm where sequential art is enriched by precise definitions and refined by exact, deterministic analytical tools.

And at the core of this agenda, is logic. We will use logic as a hammer to smash down the hand-waving, flimsy architecture of those who came before; we will use our hammer of logic and the raw firmament of the past furnished by this demolition to build a new system on stronger foundations than ever before.

ATD, 2010