Tuesday, July 19, 2011

Outlines 2

Although it is somewhat of a blunt instrument, mathematical logic is the only tool, I feel, that is capable of giving a field which hitherto has not had adequately rigorous theoretical grounding.


The thesis of this book boils down to this:

(1) "Ordered sequences of images and text consist of a series of assertions about a conceptual world."

(1.1) "In order to make sense of the assertions given, there must be a grammar around which a visual language is formed."

(1.1.1) "The grammar of this language can be adequately described by first order logic (F.O.L.)."

Descriptive language consists of assertions of the form, "It is the case such that p is true", or:
  1. ⊢p

Wednesday, July 13, 2011

Let ">" be an asymmetrical two-place predicate such that a is the "predecessor" of b, and b>a is false for ">", if {a, b, c, ...} form a well-ordered set.

Let P1 and P2 be any arbitrary sets on L.

⊦⌜ a>b⌝ & [a∈P1 & b∈P2 ] ⊃ ⌜P1>P2

Tuesday, February 22, 2011

Part I, Chapter 1: Fundamentals, Section 1.7

§1.7) Comics as Language, Separate from Art

Is it possible for pictures to denote logical statements in language? Ordinarily, no. For example, an image of a cat on a mat can certainly be understood, insofar as it can be seen by an entity with visual apparatus compatible with our own, but without the referential concepts of "cat", "mat" and some conceptual understanding of "on", it will appear to that entity as a bundle of shapes and colours. It requires, therefore, the existence of a surrounding linguistic context, which I call "concept-space", to establish the necessary semantic apparatus for pictures to denote statements. This is why abstract art seems confusing to some people, upon first viewing, as they have not entered the concept-space necessary to provide the viewer with a ready supply of concepts to understand the ideas denoted by the visual objects.

In the example given earlier, one must already have an idea of what a cat and a mat is, and be able to draw the abstract relation from this concept space to understand what the complex of visual objects in the picture means. It is then understood that the picture contains two certain elements, let us call them "c" for "cat" and "m" for "mat". In addition, the concept-space must also contain a universe of predicate relations, if the viewer is to be able to establish syntactic relations between elements in any given picture. Therefore, the concept of "On", in the example, is a dyadic relation "R" that also exists within the concept-space of the picture.

It is only when the concept-space is understood, that we can truly say that the picture of a cat on a mat, denotes the logical statement "cRm".

This has definite applications to the theory of comics. For one, all comics, I think, draw upon the same concept-space -- that all comics are commonly understood to be attempting to convey a meaningful series of sentences, in visual form. I...t is generally understood that comics will have some distinct sequence of images, and in most cases, text. This distinguishes them from art alone, because while individual artistic pieces draw upon their own individual concept-space, all comics, by virtue of being comics in themselves, must draw from one general concept-space; all comics must in some way show some sequence of events in a world, and must do so in a certain order if the sequence is to be understood.

Hence, it is always possible to draw linguistic inferences from objects denoted in a comic layout -- they are, in a sense, always pictures of a state of affairs, a representation of a world of facts, in the Wittgensteinian[1] sense, and whose facts, being the individual elements of a comic layout, are connected by definite syntactic relations. That is to say, while a painting of a cat on a mat may not be equivalent of the sentence "the cat is on the mat", as in the concept space of the painting in question what appears on the painting may be quite different from consensual conceptions of what a cat and a mat are in reality -- that a painting may denote a cat and a mat without being a one-for-one representation of a cat and a mat -- a comic showing a cat on a mat, by virtue of it being a comic, will always have a one-for-one relation to the sentence "the cat is on the mat".

This is because, as a sentential representation of some given sequence of events, a comic must establish the semantic correspondence between its images and the desired sentence in language, if there is to be any communication at all.

What, then, is the precise difference between an individual piece of art, and a comic?

Consider a picture of a vase on a table. A vase on a table in reality is something different from the phrase "a vase on a table", is something different still from a... picture of a vase on a table. The vase on the table in reality is a pile of atoms, interacting via physical laws, with another pile of atoms. Beyond that, there is nothing we can say. Sam, my artist friend, said that "Of course the vase on the table! I see the vase, and somewhere near the table, and so it's obviously on the table. You'd have to be a very bad artist for it not to have the property of 'on' the table." Herein is a distinction. I separate the linguistic property "on" from that collection of colours on a flat 2-d sheet, which reflect light into my eyes in such a way that my brain connects these shapes to some memory of vases and tables. I also separate the linguistic property "on" from the world of atoms interacting with each other through the physical forces of nature.

There is no "on" in nature. You cannot run the Large Hadron Collider high enough to produce evidence for a particle that means the word "on". No matter how deep you dig, how high you fly into the heavens, you will never be able to reach out your hand and isolate so intangible an idea as "on". To propose otherwise is Platonism.

Here is a painting. It is a collection of shapes and textures, on canvas. The paints have colours, and are arranged in all sorts of ways. Human beings with eyes can see this painting and connect the shapes it makes in certain ways to things in their experience. But what meaning exists in the paint itself? If there were no humans to see it, would the artist's concept of whatever he painted still be inscribed in the painting, embedded into the world as if he had plucked the concept out of his brain and put it into a jar of formaldehyde? This is patently ludicrous.

Draw back from the world of just paintings, to the world of "art" in general. Consider a Ming vase, a 20,000 year old Aboriginal cliff drawing, the pattern on a Zulu shield, a Navajo blanket, "Fountain" by Marcel Duchamp, and Da Vinci's fresco of the Last Supper. What do these all have in common, if art is to be some kind of "language"? They express a thing, so far as they are made to evoke some kind of memory, feeling, or image in the minds of humans. But what is this thing? Does the structure of the "thing" evoked by any of these items bear any resemblance to the structure of the "thing" evoked by any other? Is the African mask really saying something in the same way as my saying "Hi there everyone, I'm Anh-vu and I am arguing with you!" Any individual work of art can mean something different, evoke something different, in each person who sees it. Certainly the sense-data are similar insofar as we are all "seeing" it, but how do these sense-data assemble to make sense in our minds? Regularly or irregularly? If it is regular, then it is linguistic; if it is irregular then it is not.

Why is it that we can have disputes about what a painting "means", about what an artist was trying to "express", when we do not have disputes over what I mean when I say "two plus two equals four"? One is a statement of language, the other is an argument over differing opinions. Certainly one can quibble over the semantics of what Samuel Taylor Coleridge was really "symbolizing" when he wrote Kubla-Khan, but we would not quibble over the way in which he delivered those words.

When Coleridge writes:

"In Xanadu did Kubla-Khan,
A stately pleasure-dome decree".

We can ask "What did he mean to symbolize by this?" But two people won't read this same sentence, and the one says:

Anh-vu: I read it and what he wrote was "Goobla moobla boopity boo"

While the other says:

Sam: You're mistaken. What he wrote was "Paffle faffle naffle waffle".

When we are confronted with Duchamp's "Fountain", what kind of statement comes into mind? Do all people come to the same statement, or is each statement different depending on the viewer? One man says "Duchamp was saying that all of art is now in the toilet." Another says "Duchamp needed to go to the bathroom." Another says "Everything in the world is now art." Another says "Nothing is art." Just by looking at the same painting, multiple people may conclude, if asked what the artist was trying to "say", an infinite multitude of different statements. When one is confronted with a statement of language, there is only the one statement. What it is is clear. Coleridge said: "In Xanadu..." and not anything else. What he was symbolizing by it is a question of a higher order. But it's obvious that when a sentence of language says a thing, it says only that thing, nothing else.

Consider the African mask. What does it "say"? "I am strong"? "I am wise"? "I am fearsome"? "God is dead"? What? We can read a multitude of different statements just from the same object.

Return to the vase upon the table.

One person, when asked, "What does the painting say?" Replies, "It's a memoir of the artist's grandmother". Another: "The old vase is on the dirty table". Another: "The vase on the table bore telltale signs of years of neglect". Another: "I'm bored and there's nothing else in the room to paint." Another: "My vase is old, and I should clean my table".

These are all different statements, even if they can refer to the same sorts of objects. From the same painting bursts a multitude of different phrasings, interpretations, predicates. One can read the "statement" made by the painting of the vase on the table any number of ways. The form of the statement is nebulous, not specific.
Such sentential notions are unique to language, not works of art.

Yet sentential notions do exist in comics. We perceive art pieces as making statements any number of ways, but given a certain sequence of sequential art, one will always interpret that sequence in the same way; in a sense, if one is to understand the meaning of the events portrayed in the sequence at all, one must submit to the general concept-space which defines sequential art. There can be only one statement derivable from any given sequence of sequential art, and this statement, being a statement of logical facts, must be form-independent.

Tuesday, December 28, 2010

Part I, Chapter 1: Fundamentals, Section 1.6

§1.6) Elemental Semantics

From §1.1 we have a set of axioms for the further construction of the theorem of formal logic in sequential art. However, the practical meaning of the concept of "element" remains undefined. In this section, we shall define elements and provide intuitive examples of the way in which elements in a layout combine to create the formal semantics of sequential art as a language.

1.6.1) In axiomatic sequential art, as illustrated in previous sections, elements et in a layout L are considered to be atomic propositions. In formal logic, an atomic proposition is a statement of fact that can take either a value of "true" or "false", and which cannot be broken down further into smaller statements. For example, the statement "Socrates is a man" is an atomic proposition. In axiomatic sequential art, the illustrations of the objects themselves are considered atomic propositions. For example, if I were to draw a dog, it would be the equivalent of the written statement "There is a dog", or more simply - DOG.

Sequential art builds meaning out of assemblages of the elements in the same way that logic assembles atomic propositions with logical connectives such as AND, OR, NOT, and IF-THEN. These are called "molecular propositions". For example, the statement "Socrates is a man AND he is mortal". The logic of sequential art, similarly, composes elements into a larger semantic context. Imagine if I drew a dog in a running pose and drew movement-lines behind it. This would be equivalent to the molecular proposition: "There is a dog AND it is running", or more simply [DOG & RUNNING]. Adding on successive numbers of elements increases the complexity of each statement in the layout. For example, if one added a third element to the scene of the running dog - that of a sky in the background with no visible ground underneath, for example, a bank of clouds or a bird - then the context would increase in complexity and the statement would become [DOG & ( RUNNING & SKY ) ], and therefore "There is a dog and it is running in the sky". This is why, for brevity and simplicity of interpretation, the practice of dividing scenes into panels was born, which reduces the number of statements the reader must deal with in any given scene.

Fundamental to the nature of sequential art is the concept of sequentiality - that of a passage of time. This is, essentially, equivalent to the IF-THEN statement of logic. In static art, scenes depicted are of a "snapshot" of a single image. In the context of sequential art there is the implication of a time-interval between individual elements. Here is the fundamental difference between static art and sequential art. The contextualization of a sequence of events relies necessarily on the viewer filling in the essential logical step of "if there is this set of elements, then there will be this next set of elements."

Imagine if I drew the running dog in the previous example and then followed it by an image of the dog licking a man. The semantics of sequential art means that the following statement was produced: "If there is a dog and it was running, then it ran to a man and licked him," or rather: "A dog ran to a man and licked him." Symbolically: [ DOG & RUNNING ] -> [ DOG & (LICKING & MAN) ].

Therefore, one can build statements equivalent to written propositions visually, through the formal logic of sequential art.

Sunday, November 28, 2010

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) ⊃ 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.


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+1G, 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 GnGn+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.

Sunday, November 21, 2010

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

Friday, November 19, 2010

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

y ^ ~y

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.