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Porter Institute for Poetics and Semiotics

A Structural Model of Phenomena with Embedding in Literature and Other ArtsAuthor(s): Viveca FüredyReviewed work(s):Source: Poetics Today, Vol. 10, No. 4 (Winter, 1989), pp. 745-769Published by: Duke University PressStable URL: http://www.jstor.org/stable/1772809 .

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A StructuralModelof Phenomenawith

Embeddingn Literature nd OtherArts

Viveca FuredyHebrew University

Speaking of the play with a play-within-the-play, Dieter Mehl (1965:

42) once said that there had "hardly [been] any attempt to treat the

subject comprehensively," and that he was "not even sure whether itwould be possible at all, because such a bewildering variety would haveto be included [and because the subject] is ... much more complexand less easily defined than many other conventions." The same holdstrue of the whole of the structural category to which the play with a

play-within-the-play belongs, namely, what may be called "embedding-embedded objects,"which also includes phenomena such as novels with

novels-within-the-novel,paintings

withpaintings-within-the-painting,and films with films-within-the-film (or with paintings or plays-within-

the-film), Chinese boxes and Russian dolls, paradoxes, quotations andfree indirect speech, mise en abyme(in some usages of the term), and

many other things.' Nevertheless, the present paper, taking Mehl'swords as a challenge, constitutes an attempt to treat "comprehen-sively" or theoretically not only the play with a play-within-the-play

1. Studies of embedding include Voigt 1954; Nelson 1971 [1958]; Mehl 1961,1965; Genette 1972, 1983; Dallenbach 1977; Bal 1981; Ron 1987. A book which,

among other things, also deals with embedding and which acted as a catalyst formy own model is Hofstadter 1980. I regret very much that McHale 1987, which

-although his approach differs from mine-partly overlaps with my discussion,arrived too late for me to be able to relate to it within the body of the article. Icould only add a few references in footnotes, which, I am afraid, do not do this

interesting book justice.

Poetics Today 10:4 (Winter 1989). Copyright ? 1989 by The Porter Institute forPoetics and Semiotics. ccc 0333-5372/89/$2.50.

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746 PoeticsToday 10:4

(see Furedy 1983) but the structure of embedding as such, and thusto account for all possible forms of such phenomena, literary as well

as nonliterary, and for their relation to each other. As I hope to show,a radical decision to focus on the boundary between the embeddinglevel and the embedded level enables one to demonstrate not only that

(in their "pure" state) there are only three kinds of such phenomena,each of which has its own defining characteristics and effects, but alsothat they are actually three facets of the same underlying structure.This "Ur-phenomenon" of embedding is brought into being when a

boundary of the type which creates discontinuous hierarchical levels,which I shall call "logical levels," is inserted into a continuum by means

of an act of "punctuation." Such a phenomenon, which for the sake ofsimplicity will from now on be referred to as "object,"is radically dif-

ferent from a corresponding "simple" object without embedding (a

play, novel, painting, and so on); furthermore, its two parts are equally

important and interdependent for their character as embedding or

embedded-hence the somewhat unwieldy name used to refer to the

entity as a whole.The concepts used in the model presented here were borrowed

and adapted from various disciplines, psychology and communication,

logic, and computer programming, among others. The most impor-tant of these concepts are continuum, punctuation, boundary, and

logical levels.

Punctuationnd Continuum

The term punctuationis taken from the field of communication, into

which it was introduced by Bateson and Jackson (1964), followingWhorf (1956). They regarded conversation as a continuous sequenceinto which punctuation, a sort of usually nonverbal metamessage, is

introduced by the participants. There is normally, but by no meansalways, tacit agreement between them about this punctuation; how-

ever, it is not inherent in the language but arbitrarily imposed on the

sequence of verbal interaction. When there is no agreement, conflict-

ing punctuations may cause a serious communicational impasse. On

the other hand, the very arbitrariness of the punctuation of a con-

tinuum may, when foregrounded, be exploited in art or in jokes, as

in the example of the rat which said, "I have got my experimentertrained. Every time I press the lever he gives me food" (Watzlawick,

Beavin, and Jackson 1967: 55). Not only communicational continuaare subject to punctuation; it is ubiquitous and unavoidable. Any act

of focusing or classification, and thus any act of perception, which in-

volves both, is an act of punctuation. Alice's hesitation as to whether

what Humpty Dumpty is wearing should be called a cravat or a belt

is a case in point, but so is the usually unconscious decision to call

something a flower or a rose.

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Furedy * Phenomena with Embeddingin Literature 747

There are basically two modes of punctuation: one which divides acontinuum into units on the same logical level and one which divides

it into units on different logical levels (see below). Examples of thefirst mode is the division of a column of mercury into degrees (thearbitrariness of which is easy to perceive when considering that thereare four different such punctuations in use: Celsius, Fahrenheit, Kel-vin, and Reaumur) and the division of a verbal text into chapters,paragraphs, beginnings and endings, and so on. The second mode of

punctuation is employed, for instance, in the case of wallpaper with anabstract pattern on which is hanging a painting with the same abstract

pattern, to distinguish between the "real" pattern on the wallpaperand the "art"pattern on the canvas or paper. Similarly, a painting ofan artist painting another painting or a novel narrating the story of anovelist writing a novel is punctuated in the second mode at the edgeof the canvas and the beginning of the text, and again at the edgeof the painted canvas and the novel-within-the-novel. Since it is thismode of punctuation which creates embedding-embedded objects, itis this mode that I shall be referring to from now on, although muchof what I shall say about punctuation and the nature of the boundaryprobably holds true for the first mode as well.

The application of either mode of the concept of punctuation to anextended object such as a literary text necessitates the further distinc-tion between two kinds of punctuation, namely, the kind undertakenad hoc, during the process of reading the text as it unfolds in time and

space, which might be termed sequentialpunctuation,and the kind pos-sible only post hoc, at the end of the reading process, when all infor-mation is available (although permanent gaps may, of course, remain)and when one is in a position to make the final revision of the sequen-tial

punctuation (whichof course also involves continuous

revision).This latter punctuation might be called retrospective. he distinction iscalled for not only because of the nature of the process of reading, withits gradual accumulation of data, but also because of the possibility of

encoding in the text punctuational clues, or boundary markers (seebelow), which are intentionally misleading and which therefore make

subsequent repunctuation necessary. Both literary characters and thereal reader are engaged in punctuational activities, and there are often

gaps between the two. In the case of the present model of embedding-embedded

objects, it is the reader, offstage spectator, or the like who isthe agent defining the presence of embedding, that is, whose punctua-tion (or perception of the punctuation in the text?) decides whetherthe object in question is simple or embedding-embedded.

BoundaryThe boundary, albeit essential to the model proposed here, has noexistence in the real world; it exists only in the system of percep-

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tion imposed on reality by the punctuator-perceiver. The boundaryis the discontinuity itself between the two discrete levels or units be-

tween which it distinguishes or, from a different perspective, creates.The boundary thus functions within a system without being part of it,

wholly in line with the view held in the discipline of communication,cited above, that punctuation, the insertion of a boundary, constitutesa metamessage. Being the mediating absence or "leap" between the

levels, the well-functioning boundary is a gap that bridges, the modus

operandi of the smooth movement of transition from one to the other

(Muken 1979). It has several interesting characteristics: It is a con-

tinuous "line" which divides another continuum, something like itself,

into discrete units, thus turning it into something unlike itself. By

intervening in the continuum, it turns difference (more/less) into dis-

tinction (either/or), the difference thus singled out by the boundary

being only one of a very great number, approaching infinitude, of

possible candidates. No distinctions can be made without boundaries;

conversely, the act of punctuation necessarily creates distinction.

Since the boundary itself is invisible, its presence must be indicated

by boundary markers. In literature and painting, the main marker

(not to be confused with the boundary itself) is a change in ontologi-cal status2: there are, as it were, two different "worlds"on either sideof the boundary. Thus Hamlet cannot intervene and stop the mur-

der of Gonzago; it happens in a different place, at a different time,and involves a different species of "people" from his own.3 A com-

mon indicator of the presence of such different worlds, and thus of a

boundary, in verbal and visual arts is the presence in the embedding

part of a "maker" of some kind who is responsible for the embedded

object (a narrator, painter, stage director, sponsor, or the like) or of a

certain kind of witness(an

audience within thetext,

as distinct from

an observer or an eavesdropper). In drama, another boundary marker

is the switch from outer play actor to inner play character, the bound-

ary in this case passing within a biological entity. In the verbal arts,there may be a change in dialect or register; in the visual or verbal-

visual arts, a change of style or period of costume, color-in films,color versus black-and-white-setting, lighting, and the like. In three-

dimensional objects and in paintings, as opposed to the verbal arts,there is also a boundary marking difference in size, the embedded ob-

ject of necessity beingsmaller than the

embeddingone. A

transgressedboundary too is marked, first, by there having existed, at some point,

2. McHale's (1987: 116) focus, on the other hand, is precisely on the "intensifica-

tion of ontological instability" n postmodernist film and fiction.3. Peter Handke's (1969) play Offendinghe Audience laborates on what happenswhen the boundary between play and audience is erased, thus by negation illumi-

nating the markers and normal functions of that boundary when it is intact.

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an intact boundary, and second, by the feeling of the strangeness ofthe interaction between the former levels (see below).

It should be noted that since some of the phenomena in this byno means exhaustive list do not necessarily indicate the presence ofa boundary (thus there may be different, parallel story lines set indifferent places; two characters on the same level may speak in dif-ferent dialects or registers; and so on), a number of such signs are

usually needed to mark the certain presence of a boundary. Even so,as Douglas Hofstadter (1980: 701) has shown in his interpretation of

Magritte's painting Ceci n'estpas unepipe, the punctuation of a paintinginto discrete logical levels, one of which represents "reality,"involves

not only complex translations, for example, of two-dimensional spa-tial clues into three-dimensional objects, but also a liberal dose of

"willing suspension of disbelief," without which not only does the real-istic pipe become a mere blot of color, but even the words "ceci n'est

pas une pipe," written on the canvas precisely in order to counter-act the "realistic pull" of that pipe, dissolve into smudges of paint.4The same, though involving other factors, holds true for embedding-embedded objects of different kinds as well.

As we shall see, the logical level-creating boundary can assume three

different forms: It can remain intact, become reified, or cease to func-tion and become transgressed (leaving, however, certain "traces"). Ineach of these cases, the relation between the two units distinguishedby the boundary will be different, and in each of these cases the effectcreated will also be different.

Logical Levels

The term logical levels is used here to avoid the interpretation ofthe preposition within in concepts such as novel-within-the-novel or

painting-within-the-painting as "physicallysurrounded by the embed-ding part of the object," as is often the case at least in literary criti-cism. The term as used here refers to hierarchically iscontinuous evels.The concept of hierarchical discontinuity is abstracted from Russell's

(1922, 1937 [1903], 1956 [1908]) distinction between logical types andTarski's (1956) between metalanguage and object language (distinc-tions with problems of their own, which, however, exceed the scopeof this paper. See Kneale and Kneale's [1962] analysis of the theo-ries and account of the problems which they, in their turn, create for

logic). It is in principle a logical concept; nevertheless, it has ontologi-

4. Thus a special Einstellung-a suspension of disbelief-is required of the viewerin order to perceive the distinct, and deceptive, logical levels (see below), andanother Einstellung-a suspension of belief-to blur them in order not to be

gulled by Magritte. (As may be gathered from this account, there is no such thingas neutral,purposeless, perception.)

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cal implications which, in practice, often serve as a kind of "discovery

procedure" (or marker) which helps one decide whether or not there

are two (or more) different logical levels in the object of analysis. Chiefamong these ontological implications is the impossibility of interactionbetween different logical levels. When such interaction nevertheless

does take place, as when the sun in an embedded painting casts a

shadow in the embedding one, or when a character in a book murders

the reader, as in a story by Julio Cortazar (1970), it does so preciselybecause the boundary between the logical levels is no longer intact,thus collapsing the distinction between them. The sense of oddness

which arises is a sign that the object in which this effect is created is an

embedding-embedded object with a transgressed boundary, and nota simple object in which there is nothing to prevent such interaction

to begin with. (See also the section on boundary markers, above, for

additional areas in which transgressions may manifest themselves.)After this brief introduction to the concepts of which the model of

embedding-embedded objects is constructed, we can now proceed to

an examination of the three subtypes of such objects created by the

insertion of the three different forms of boundary between the logicallevels of these objects.

1. The Intactand MultiplyingBoundary

The effect par excellence which most people associate with an

embedding-embedded object, such as a play with a play-within-the-

play, is that of infinite regression or recursion. The effect is especially

interesting, it seems to me, since it is actually created by a finite num-

ber of embeddings, often only one or two, somewhat in the nature

of an afterimage on the retina (cf. McHale 1987: 114, 124, passim).Leo

Spitzer(1957: 205) remarks on the same

potential

of the number

two in language, as in expressions like "It rains and rains," that "lan-

guage has chosen only two links in the chain, which are called uponto represent the infinite expansion (rains and rains and rains etc.)."The sense of infinity created by cases of repetition,ike these, however,

seems to me considerably weaker than that brought about by recursion,

the potentially endless addition of discontinuous hierarchical levels.5

The reason for this difference may well be that there is no theoretical

5. The conceptof

infinityis

mathematicallytrickyand

may giverise to

logicalproblems. As a mathematical layman speaking, presumably, to other laymen, I

have, however, often employed the expressions "potentiallyinfinite,""infinite re-

cursion,"and the like, meaning that we get a (nonscientific)sense of infinite recur-

sion, but that within the confines of our finite world-even the universe is finite,

according to some-an infinite expansion cannot in fact take place. I should there-

fore, unless the context indicates otherwise,be taken to mean "avery high number

of recursions, approaching infinity." I am grateful to my father for this caution.)

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necessity for repetition on the same logical level to go on ad infinitum.On the other hand, since every metalevel, when discussed, in its turn

becomes an object level, the concept of infinity is intrinsic to Russell'sidea of logical typing and Tarski's distinction between object languageand metalanguage, from which the concept of logical levels derives.A rewriting of the Mobius strip called "Frame-Tale,"with which JohnBarth's (1969: 1-2) novel Lost in theFunhousebegins,6 strikingly illus-trates this strong effect: "Once upon a time there was a story that

began 'Once upon a time there was a story that began "Once upon atime there was a story that began ... ,"' and so on. The first "Once

upon a time .. ." is on a level which we may call a. The next time it is

repeated, it is embedded in the level a, creating a level we may call 3;the following level is y; and so on. The construction of the strip mayalso be represented by the use of logical bracketing in the followingway: a (a ((a (((a .. )))))) (Breuer 1976: 229).

The effect of infinite recursion is thus inherent in the intact logi-cal level-producing boundary, and anyone wishing to create such aneffect need do nothing but use such an intact boundary. However,he may augment the effect thus created by increasing the amount of

analogy between the logical levels. In the example from Barth, the

analogy is maximal and the effect therefore very strong: The sen-tence of one level is repeated verbatim on the next. The greater thenumber of elements repeated on the different levels, in addition tothe multiplication of the boundary itself, the stronger the effect of in-finite recursion. Though not immediately evident in the case of verbal

repetition on different levels, it is important to remember that whatis repeated are tokens of a type, not the type itself, and that what wehave is analogy, not identity. (Identity, as we shall see, exists only inthe third

subtype.)The difference is easier to

perceivewhen we turn

to a visual embedding-embedded object, such as the famous cocoa tinon which is painted a girl holding another such cocoa tin, on whichis painted a girl, and so on. Not even a row of such tins coming offthe production line would be "the same," except in the sense of beingtokens of the same type, but the tins painted on the real tin areeven more clearly different from it. The real tin is three-dimensional,made of metal, and perhaps full of cocoa. The other tins are two-dimensional, made of specks of paint, and different from the real tinand from each other in size. To be able to

regardthe

cocoa tins as "the6. In the book, the words "Once upon a time there" are written on a strip on the

right-hand side of the first page and the words "was a story that began" on theother side, and instructions are given to cut the strip and fold it in such a waythat it becomes an endlessly continuous sentence, a Mobius strip. By spelling thesentence out, as I have done, I have converted it from a self-engulfing structureinto an infinitely recursive one.

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same tin" even in the sense of being analogous to one another, we se-

lect certain features as more significant than others, foreground them,

and interpret them. In this case, this involves translating spatial cluesgiven in the flat, painted tins into three-dimensionality and ignoringthe differences in size, material, and the like mentioned before. That

is, we focus on the similarities and gloss over the differences.

The perception of recursion, as distinct from repetition on the same

logical level, seems to involve a step in addition to the abstraction

of similarities. Take, for instance, a painting which depicts a painter

painting a picture of himself painting and so on. Here, as with the

tins painted on the metal tin, by inserting boundaries we give different

logical status to what in reality are areas of similar lines and colors;we see, as it were, a "real" artist painting an object (an artist) which,on the next logical level, the first frankly artificial painted canvas, is

in turn a subject painting an object (an artist), and so on. Here, same-

ness (lines and colors) is translated into distinct logical levels, thereby

multiplying the boundary and creating the effect of infinite recursion.

In other words, in order to perceive not only repetition but recursion,what is required is both an abstraction of similarity and punctuationof the continuum in such a manner as to

emphasizethe difference

between the logical levels (the "willing suspension of disbelief" men-

tioned above); we need a combination of distinct logical levels (differ-

ence) separated by a potentially infinite series of boundaries (sameness)and some analogy between the levels (sameness in difference).

The recursion resulting from the multiplication of the intact bound-

ary differs from repetition not only in involving distinct logical levels;whereas in repetition the whole unit can be repeated, in recursion it

cannot. In the heraldic device which Lucien Dallenbach (1977: 143)

thinks suggested the idea of mise en abyme to Gide, we see a shieldA, in the middle of which is another shield, B, in the middle of which

is an imaginary shield C, the first of the potential shields expandingthe recursion. We have no way of knowing what is "behind" shield B,but we assume it to be the continuation of shield A. This continuation

is not, and cannot be, included in shield B, nor does shield B include

itself, although it too is part of shield A. In other words, when the

point is reached at which the object would have to repeat itself, there

is a switch to the next logical level. As in linguistic subordination, the

lower logical level fulfills a structural function on the higher level, thuscompleting it, but, since the lower level itself here is incomplete, this

completion is never achieved. At the heart of recursion there is thus

a hole, an absence. These structures, like the sentence from Lost in

the Funhouse, never "bottom out," to borrow a term from computer

programming. The reason they cannot include themselves in them-

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selves and thus bottom out is the same reason that eliminated para-doxes from Russell's theory of logical types: Self in the different oc-

currences of the word itselfrefers to objects on different logical levels.(But see Watzlawick, Beavin, and Jackson [1967] and Fuiredy [1983:chs. 7, 9] on the return of paradox.) Therefore, as long as the bound-aries of the recursive structures remain intact, the absence at theirheart will also continue to exist. Each time the edge of the abyss is

reached, another boundary is quickly inserted, another level created,as a futile-but perhaps necessary?-attempt to fill in the hole. Onlywhen the boundaries are erased, as in the third subtype, does thestructure finally bottom out and the "sameness" of analogy become

transformed into the sameness of identity.As pointed out above, the increasing of analogy between the logical

levels will only enhance an effect inherent in the intact logical level-

producing boundary as such. In contradistinction, if one wishes todiminish or stop the recursion, active measures have to be taken. Ishall mention only a few, taken from embedding-embedded objectsin the field of drama. Thus, in TinyAlice, Albee (1971) has a charac-ter deliberately and explicitly stop the recursions. In a room in MissAlice's castle, in which much of the action takes place, there is a model

of the castle which even includes a "model of the model of the castle"in the room of the model corresponding to the room of the castle inwhich the model stands.

BUTLER(a shy smile):Youdon'tsupposethat withinthat tinymodel there,there is ... another room like this,withyeta tiniermodel withinit, andwithin ...

JULIAN (laughs): ... and within and within and within and .. . ? No, I ...rather doubt it. It is a remarkablecraftmanship, hough. Remarkable.(Ibid.:26)7

A series of embedded objects on the same logical level (e.g., a seriesof 3s), that is, parallel rather than embedded one within the other,also reduces the effect of infinite recursion. Thus in Massinger's (1912[1626]) TheRomanActor, the potential of the first play-within-the-playto suggest infinite regression is reduced the moment the next play is

put on, and reduced even further when the third one begins. Rever-sal of the downward or inward movement of the intensional recursion

("popping" instead of "pushing," to use computer terminology again)

also undermines the feeling of potentially infinite recursion (see, e.g.,the dialogue "Little Harmonic Labyrinth" in Hofstadter 1980). Simi-

larly, if the outermost embedding level of a literary text includes God,

7. The play itself constantly subverts the notion that the castle is "real" and themodel only a replica.

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extensional multiplication of the boundary is rendered improbable,

though not, theoretically, impossible (see, e.g., McHale 1987: 115). A

diminishing of the feeling of infinite recursion also occurs when theembedded text fulfills a plot function within the outer, especially if

it is a play acted by lay characters from the outer play, as in RichardBrome's (1968 [1641]) A Jovial Crew. In it, the runaway children of

the outer play put on a play for their elders in order to obtain their

pardon. The inner play repeats some of the action of the outer, ex-

plaining why Oldrents' daughters have run off to live with beggarsand at the same time providing the solution to a prophecy that theywould be reduced to begging. When the play has achieved its purpose,Oldrents interrupts it, thus also decreasing the elements analogous tothe outer play. Although logically there is a clear distinction between

the characters of the outer play and the roles they are playing in the

inner, the boundary seems to be less clearly marked in this case and

therefore not as suggestive of recursion. The fact that they are actingout a representation of their own lives, and that the play has a practicalfunction within the plot of the outer play, further narrows the gap be-tween the discontinuous levels and thus weakens the boundary which

is thatgap,

albeit notsufficiently

to cancel itsfunction, only enoughto make it more difficult for it to multiply. It is as if, instead of the

dizzying multiplications of the well-functioning boundary, we have a

more sluggish process, one which demands an effort of imaginationrather than happening, as it were, by itself.

What would happen if an object were truly, not only hypothetically,

infinitely recursive? Each time it came to the point at which it had

to repeat itself, it would push to a lower, hierarchically discontinu-ous level (in computer language, it would "call on" another object like

itself). Although the hole of each level would be closed by the next levelthat came to stop that gap, that level in turn would have a similar hole,and so on forever. Such an object would never bottom out; it would

literally never end. Each embedded level would be surrounded by an

intact and well-functioning boundary, the sine qua non for its infinite

multiplication, but the embedding-embedded object as a whole would

not be closed. In other words, although paradox proprementditbelongsto the second subtype, there seems to be something paradoxical about

the first and, as we shall see, the third types as well.

2. The Intact but ReifiedBoundaryThe first of the two types of boundary which may be described as

"malfunctioning" (in a descriptive rather than evaluative sense) still

preserves the logical levels created by it as separate, though they be-

come incomplete and undecidable entities. This kind of boundary istransformed from a mediating absence into an obstructing presence,

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a boundary drawing attention to itself rather than facilitating pas-sage across it. Free passage from level to level is thus prevented, even

prohibited. The boundary allows one to cross it on condition thatone immediately recross it, and one is permitted to recross it onlyif one immediately re-recrosses it and so on, in perpetual oscillationbetween the two logical levels. When the passage is disturbed in this

way, a "repetition-compulsion" is created-perhaps a re-petition to beallowed to cross the boundary in the proper one-directional and un-conditional way-and oscillation results, a perpetual series of encoun-ters with the reified, obstructing boundary. If the main characteristicof the first group of objects is the multiplication of the boundary with

its concomitant effect of a sense of infinite recursion, what character-izes this category with the reification of the boundary is its concomitanteffect of undecidability and infinite oscillation.

Undecidability (ambiguity as to, e.g., status or reference or logicallevel for which no process of decision which would resolve it is known)actually occurs twice in these objects. At the first encounter with the

phenomenon in question, for example a paradox, what is undecidableis which level we are confronted with, a or p. Once the boundary be-tween the two levels has been crossed and we are obliged to recross it,

what is undecidable is the direction in which we are moving: whetherwe have returned to the original level of entry or gone on in the samedirection as that of the first crossing. Though the logical levels re-main clearly separate, the distinction between higher and lower levels,as will be discussed in more detail below, thus loses much of its normal

significance.When the levels distinguished by this type of boundary are incom-

plete, constantly referring to the other level for completion, and whenthere is simultaneously a conflict between them, we have the structure

of paradox.8 In paradox, an additional factor is the denial of accessto an inviolate, extrasystem metalevel; there is no safe and oscillation-free place outside the system or object with the reified boundary andthe contradictory, incomplete levels from which the system can be

regarded "objectively."Consider, for example, M. C. Escher's (1971:drawing 69) lithograph DrawingHands. It shows a sheet of paper nailed

8. My way of anatomizing paradoxes is based on Watzlawick's (1967) account of

pathologicaland

therapeutic paradoxes.Different

typesof

paradox which are notaccounted for with this method may exist. It should be noted, however, that theaim of philosophy is to resolve paradoxes and to rid logic and mathematics ofthem, whereas my concern is to describe the structure of the unresolved paradox,for which the distinction between logical levels and the concept of an inviolatemetalevel (see below) are highly illuminating. I was pleased later to find a similar

approach and a delightful anthology of mind-boggling paradoxes in Hughes andBrecht 1978 [1975].

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diagonally with drawing pins onto a darker background. At the upperend of this sheet, there is a "three-dimensional" right hand holding a

pen and drawing a two-dimensional cuff on the left side of the paper.Out of this cuff emerges a "three-dimensional" left hand holding a

pen and drawing a two-dimensional cuff on the right side of the paper.Out of this cuff emerges the upper right hand, with which I beganthe description. The right hand thus draws the left and is drawn by it,and vice versa. Neither is complete without the other, because neither

would exist if it were not drawn by the other and because neitherwould be a "drawing" hand were it not drawing the other. (The am-

biguous title of the lithograph expresses the same paradox.) These

hands are, logically, on different levels: The hand drawing is on alogical level immediately above that which it draws. Although this dis-

tinction is preserved in the lithograph in the sense that the levels are

still separable, it is also and at the same time subverted. Not only is

each level by itself incomplete-without the other level, each hand is

simply a slightly odd, two-dimensional/three-dimensional hand-but

the assignation of one of the hands to one of the logical levels is im-

possible. It can be assigned to either level, thus being undecidable,and the two levels, "drawing" and "drawn,"conflict with each other,

thus creating a paradox. The paradox can be escaped only by 'jump-

ing out" of the system, that is, by moving to an inviolate metalevel

separated from the oscillating levels by a boundary of the first type.Hofstadter (1980: 690) suggests that this can be done by imaginingEscher'shand drawing both of the hands in the lithograph. (This level

may, of course, in turn become part of an oscillating structure with

a reified boundary, necessitating yet another inviolate metalevel, and

so on.)In a scene in Albee's play, the characters are assembled in the room

in the castle in which the model of the castle stands. The butler men-tions that the real castle once was in England. The following conver-

sation ensues:

Miss ALICEas if suddenly remembering): Yes, it was! Every stone, marked

and shipped.

JULIAN: Oh, I had thought it was a replica.LAWYER: Oh no; that would have been too simple. Though it is a replica ...

in its way.

JULIAN: Of?LAWYERpointing to the model): Of that. (JULIAN laughs a little; the LAWYER

shrugs.) Ah well.

JULIAN (to MIss ALICE): Did your father ... did your father have it ... put

up?....BUTLER (to JULIAN, pointing first to the model, then to the room): Do you

mean the model ... or the replica?

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Furedy* Phenomena with Embeddingin Literature 757

JULIAN: I mean the ... I mean . . . what we are in.

BUTLER:Ah-ha. And which is that? (Albee 1971: 58-59)

It is perfectly obvious which is a room in a real castle and which is a

model; what is undecidable is, which is a replica of which. Is the modelthe object-level model of which the real castle is a metalevel represen-tation, or is it a metalevel replica of the object-level real castle? The

concepts "replica" and "model" (in the sense of "real object to copy,"not "small-sized copy") are incomplete and refer us to the other as

part of their meaning, and they conflict with each other. The scenethus has all the ingredients of a paradox.

In plays with plays-within-the-play with a reified boundary, for in-stance, Pirandello's (1952 [1923]) Each in His Own Way, there is thesame undecidability and the same paradoxical oscillation between thelevels, in this case between the inner play and the outer play. The two

levels, though undoubtedly separate, keep changing places in a pro-cess of infinite oscillation, and because of this oscillation their statusas inner or outer plays is undecidable. In the case of extended texts,what is incomplete is not something in a particular word, as in the

previous example. Rather, something in the content of one of the

levels will send one to the other level for completion, and vice versa.Thus, in Pirandello's play, the inner play "imitates" an action whichhas "really" taken place on the logical level of the outer play, and thetwo "real" people in the outer play whose previous actions are actedout in the inner play in turn "imitate" an action performed by their

counterparts in the inner play which had not taken place in outer-playreality. Which play is "inner" and which "outer," which object leveland which metalevel? Whether we look at it in terms of content orstructure, within the limited "set" or "universe" of the play as a whole,there is a conflict between the two levels of the play, and thus paradoxarises.9

In the two dramatic examples, as in Escher's drawing, access to aninviolate metalevel has been barred. To break through to it requires adeliberate repunctuation, sometimes very difficult to imagine. Whenno such repunctuation is effected, all messages-assertions, actions,events, and so on-are "partof the game," or subject to undecidability.We are thus faced with an illusion of alternatives which we cannot

escape, except by refusing to watch theobject altogether.

The chal-

lenge to decide the undecidable and the inescapability of the challenge

9. In the set of embedding-embedded drama, a play can only be either inner orouter, and it must be one of them. The universe consisting of these two alterna-tives is exhaustive; tertiumnon datur.For a case involving "the interchangeabilityof narrative levels,"in this case whether the narratorcreates the character or viceversa, see Rimmon-Kenan 1982.

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produce the characteristic of the phenomena with two logical levels

separated by a reified boundary: infinite oscillation in a game without

end. One's knowledge that there aredistinct logical levels is of no avail.The boundary functions like a revolving door to which one is chained,with no hope of escape, rather than like the normal opening of the

"gap that bridges." Since each side or level is incomplete and thus un-able to exist without the other level to which it refers for completion,one is forced to shuttle back and forth in the vain, but eternal, hope of

arriving at some final, untangling, order-creating "full stop"-a pointwhich, as we have seen, can exist only outside the system to which one

is chained inside the revolving door.

3. TheTransgressedBoundary

In the case of the first two types of boundary, something always mul-

tiplies: either the boundary itself or our encounter with it. In the case

of the third type, however, the boundary can almost be said to dis-

appear. Almost, but not quite: Traces of the original, intact boundaryremain, and this is the reason I prefer to call it a transgressed bound-

ary, rather than an erased one. It should be noted that the "trans-

gressions"take

placebetween

logical levels,not

between,for

example,characters from different levels; the interaction of characters across a

previously intact boundary is only one of the most frequent markers

of this transgression. The new entity created when the boundary is

transgressed and the hitherto discrete logical levels collapse is com-

pletely different from entities not created in this way. (The feeling of

peculiarity, mentioned before, which it evokes plays an important partin the discovery procedures for boundaries of this type.) Within this

entity, the original distinction between logical types or levels is can-

celed, remaining only as the backdrop against which this "composite"or, to use Hofstadter's term referring to Russell's theory of logical

types, "typeless" level is perceived. As opposed to the first two kinds of

boundary, in the case of the transgressed boundary every notion of in-

finity within the structure itself (except in the case of self-engulfment;see below) is eliminated; it bottoms out and reaches an end-or a be-

ginning. In Breuer's (1976: 230) formulation, this typeless unit is a

"futile attempt at 'reconciling' the two levels." Instead, infinite oscilla-

tion is set up between the "composite level,"created when a boundary

is transgressed, and the "inviolate level" beyond it, belonging to thelogical system in which such composite levels cannot occur and no

rock bottom can be reached. The "infinity"here thus moves out of the

painting, novel, or play and seems to involve the concepts themselves

rather than their fictional manifestations.

For a boundary transgression to be perceived as such, an intact

boundary must also be perceived. Logical levels can be subverted only

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after being established or, at least, after retroactively being seen tohave existed. In other words, "all boundaries cannot be transgressed

all the time." Whether a boundary is to be regarded as transgressedor not usually depends on "instructions" for punctuation encoded inthe object. When confronted with the paradoxes created by the reified

boundary, we had to take deliberate steps to reach an inviolate meta-

level, that is, to insert an intact boundary around the two levels created

by the reified boundary, if we wished to escape the oscillations (an

equally deliberate decision to preserve the sense of paradox ratherthan resolve it can, of course, be taken by refusing this option of es-

cape). In the case of the transgressed boundary, on the other hand, we

just as deliberately have to unlearn the punctuation previously used.From the point of view of "normal"punctuation, the clues encoded inan object with a boundary of this type are instructions to "mispunctu-ate." When reading such a text or regarding such an object, then, we

paradoxically obey the injunction of the object to disobey the rules weare used to regarding as inherent in embedding-embedded objects.As a result of this mispunctuation, the analogy between the two logicallevels created by the intact boundary is transformed into identity.

Some cases ofmispunctuation

are reversible.They

are the result of

misleading clues or mistaken conclusions as to the place or nature ofthe boundary. Usually it is the intratextual characters who make such

mistakes, as in the case of the "real"deaths in the plays-within-the-playin Elizabethan and Jacobean revenge drama, such as Kyd's (1959 [c.1582-92]) The SpanishTragedy.Here, as in Middleton's (1965 [1605])comedy A Mad World,My Masters,the onstage spectators perceive anintact boundary where none exists. The opposite occurs in Massinger's(1912 [1626]) TheRomanActor,when Domitia, watching a play in which

the lover, acted by the outer-play Roman actor Paris, with whom sheis beginning to fall in love, is going to hang himself, rises from herseat, crying, "Restrain him, as you love your lives!" (3.2.281-82), thus

erasing a boundary which is, in fact, intact and thereby revealingsomething about herself rather than about the nature of the boundary.In these plays, the offstage spectator knows the correct punctuation,but there are plays where he knows as little as, or even, as in JosephHeller's (1967) WeBombedn New Haven, less than, some of the onstagecharacters, thus reversing the more frequently occurring type of gap

of knowledge.Within the subtype of embedding-embedded objects with a trans-

gressed boundary, further subdivisions can be made. Gerard Genette(1972: 245), in his discussion of narrative levels in Figures3, discussesthree such subdivisions, to which I shall add another two, namely,self-reference and self-engulfing. (I shall also add something I call

"pseudotransgression," a phenomenon which, though structurally dif-

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ferent from the others, has a similar effect.) Genette (1980 [1972]:236) defines boundary transgression in narrative, which he calls meta-

lepsis, as any form of transition between narrative levels other thanthat "achieved ... by the narrating, the act that consists precisely of

introducing into one situation, by means of a discourse, the knowledgeof another situation."10 His theory is admirably simple and elegant.

Although Genette does not make this subdivision explicit, all his nar-

rative metalepses, apart from the temporal ones, belong to one of two

categories: Either the transgressive transition takes place from the

inner level outwards, or the other way round.

Metalepsis rom heInnerLevelOutwardsGenette gives the example, mentioned above, of a story by Cortazarabout a man murdered by a character in the novel he is reading. Ex-

amples taken from plays include characters (but not actors) speakingto their audience, as when the Peasant in Calderon's (1976) The Great

Stage of the World,a character in the inner play, answers the comments

of the World, a member of the onstage audience.1

Metalepsis rom he OuterLevel nwards

Anexample

mentionedby

Genette(1972: 244)

is Sterne's narrator's

asking the reader to help him put Mr. Shandy to bed. This imaginaryact would be a metalepsis from the outer level inwards. The scene

from The RomanActor, mentioned above, is another example, as are

several of the interventions of the merchant and his wife in the inner

plays of Beaumont's (1969 [c. 1607]) TheKnightof theBurning Pestle.

TemporalMetalepsisGenette (1972: 245 n.3) quotes his school history teacher, who pro-duced the following metalepsis: "Nous allons etudier maintenant le

Second Empire depuis le Coup d'Etatjusqu'aux vacances de Paques."

(The technique of showing something which has happened or will

happen elsewhere, for instance, in the form of a "magic show," as in

Corneille's [1961 (1636)] L'Illusioncomique,seems to be a relative of

this particular form of metalepsis, but in reverse; it involves the trans-

position onto the embedded logical level of something which "really"

10. It is not clear from this formulationwho should be the possessorof "theknowl-

edge of another situation,"but presumablyit is the reader. In my model, the status

of an object as embedding-embedded is defined by the spectator, reader, or thelike outside that object.Metalepsismeans"takinghold of (telling)bychanging level"

(Genette 1980: 235 n.1). It should be noted that, although I add some metaleptic

phenomena to Genette's list, my own "taxonomy"may not be exhaustive, either.

11. In this play, the transgression serves a further, characterizingfunction: The

Peasant is churlish and insolent, and the boundary transgressions,which he is the

only character in the play to commit, constitute a structural counterpart to the

moral transgressionsof which he is guilty.

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belongs on the embedding one, in other words a mispunctuation con-

sisting of the insertion of an intact logical level-creating boundary

where in "reality" there is none.)

Self-Reference

Self-reference is what would occur "if the title of a book were includedin the book's own bibliography or if, in the index, one found refer-

ences not only to those pages where a name or a notion occurs in itsactual usage, but also to the page of the index entry" (Breuer 1976:

229). The index of a book has a metarelation to the book to which it

belongs; it refers to names and concepts occurring on the object level

of the text of the book itself. When the index refers not to the text butto itself, we do not have an intrusion of the index into the text or viceversa. Instead, we have a metalevel becoming its own object level, thus

collapsing the logical levels. It is as if a boundary had momentarilybeen inserted where it did not belong, dividing the index into two

levels, only to be immediately erased. The effect is clearly metaleptic,in the wider sense of "creating a unit with composite logical levels." 12

At this point, the question may arise whether the index referenceto its own entry, that is, the entry referring to itself, is not a case of

undecidability and oscillation between two levels, of a reified bound-ary rather than of boundary transgression. In spite of the apparentsimilarities, however, the two are quite different. In the case of un-

decidability, no procedure exists for finding out whether what we are

dealing with is a metalevel or an object level. Our index entry, however,is not undecidable: There is a procedure for deciding, for example,whether "p. 735" refers to a page in the text or a page of the index-one simply has to look at the page numbers of the text and the index.The problem is therefore not that it is undecidable but that such a ref-

erence is "improper," violating the rule that the item referred to mustbe on a logical level distinct from and immediately below the level ofthe referring item; in other words, self-reference is possible only whena boundary has been transgressed.

Self-EngulfingIn Escher's lithograph Print Gallery(Ernst 1976: Fig. 56),13we see a

boy standing in a picture gallery, looking at a print on the wall. This

print shows a harbor with a ship in the foreground and buildings on

the shore in the background. To the right, the buildings are enlargedand come closer to the spectator, becoming bigger and bigger. In the

12. I sometimes also use the term "boundarytransgression" n this wider sense.13. See Ernst 1976: 98 for a reproduction of Escher'sSpiralswhich beautifullyillustrates the structure of self-engulfing: They look like a snake swallowingitself,and, because of the gaps between the spiralingstrips, one can see the "tail" nsidethe "entrails."

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building closest to us, a woman is standing at a window looking out.

Below her flat is the entrance to a picture gallery-which turns out to

be the same gallery in which the print hangs of which the house formsa part! The gallery is "in"a picture, which is itself"in" the gallery, and

thus all logical levels collapse and intact boundaries are transgressed.(As already pointed out, this is the only kind of embedding-embedded

object with a transgressed boundary in which an effect of infinity is

created within the structure itself.) This is not a case of a lower level

invading a higher level or vice versa; here, one level is engulfing not

only the next but also itself (or perhaps both levels are invading and

swallowing each other?).There is one part of the print which does not participate in the en-

gulfing movement, however, namely, the blank space in the middle in

which Escher's signature appears. It is not part of the world of the con-

tent of the print at all; it belongs to an extrapictorial level, the level

of Escher's Print Gallery.Unlike us, the boy in the print cannot see it.

The space was not left blank by chance or because Escher needed a

place to sign his name; it is logically and mathematically impossible to

fill that space, to achieve total self-engulfing (ibid.: Fig. 59; p. 33).

Similarly,in Pirandello's

play TonightWe

Improvise,owards the end

of the first act we see the characters of the play on their way to the the-

ater. They are late and afraid that they have missed the first act. They

hurry offstage and after a moment appear in the real auditorium,

noisily entering a box. The first act, which has indeed just ended, is

the first act of the outer play in which these same "spectators" ap-

peared as characters. They have thus come to watch themselves go to

the theater to watch themselves go to the theater and so on, and the

play thus engulfs itself.14

Although the blank space in the self-engulfing structures mayseem

to resemble the "hole" in the recursive structures discussed above,

actually there are two essential differences between these phenomena.The first is that the hole in the recursive structures is created by the

"legal"transition from one logical level to the next-the multiplicationof the intact boundary-and, although the next level in the hierarchyis also incomplete and therefore unable to make the recursive struc-

ture bottom out, each hole is filled by the infinite series of logical levels.

In the self-engulfing structures, on the other hand, as we have just

seen, nothing can enter the "space"of the hole, and the logical level isa typeless one. The second difference is that recursive structures are

14. Pirandello 1932 [1930]: 82ff. In this play, there is self-engulfing at the bound-

ary around the play as a whole as well (see the interlude in the lobby). The self-

engulfing metalepsis seems to be marked not by the interaction of any two charac-

ters across a boundary, but by a spectator and a watched one or an author and a

character-in-his-work and the like.

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Figure 1. Self-engulfmentcorrectlypunctuated.

based on analogy, whereas self-engulfing, as shown by the word self,like all

boundary transgressions,translates

analogyinto

identity.This

is actually a "mistranslation," concomitant with the "mispunctuation"of the transgressed boundary. If we begin looking at Print Galleryatthe bottom, we may consider the gallery entrance as belonging to levela and everything depicted in the picture exhibited in the gallery as be-

longing to level P. The second time we get to the entrance, it is thus alevel 3entrance and the pictures exhibited in the gallery are on level y,and so forth. With "correct"punctuation, then, the movement is spiral(Figure 1). Since, however, the logical levels a, f3,y, are represented by

the same unrepeated objects-the gallery entrance and the picture inthe gallery, in this example-not by a series of recursive analogues likethe girls with the cocoa tins, the underlying "correct"spiral movementin effect collapses into a circular one (Figure 2).

The self-engulfing structures also remind us of the oscillating struc-tures created by the reified boundary both in that we keep returningto the same place, and in that the exclusion of the "inviolate" levelof Escher's signature from the structure, like the barring of accessto the inviolate metalevel in the oscillating structures, seems to cre-

ate undecidability. Although the undecidability of self-engulfing andoscillating structures may be the same (I am not certain it is), thereare two important differences between them, one of which I have justdiscussed. The other difference is that in the oscillating structures,the oscillation is two-directional, back and forth between the sametwo logical levels (Figure 3). In self-engulfing structures, however, themovement, as in recursive structures but not oscillating ones, is in one

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Figure 2. The effect of the self-engulfing structure.

boundary

x y

(metalevel) (object evel)

Figure 3. Oscillation at the reified boundary.

direction only; but unlike recursive structures and like oscillating ones,we keep coming back to the same place (with the proviso about the

underlying spiral structure).

PseudotransgressionA "metaleptic" effect without actual boundary transgression is created

in verbal texts when attention is drawn to thenormally

invisible in-

tact boundary, which is then foregrounded and "made strange" (but

apparently not actually reified, to judge by the effect). Fabian's well-

known lines in Twelfth Night are a case in point: "If this were played

upon a stage now, I could condemn it as an improbable fiction" (Shake-

speare 1975 [1601]: 3.4.128-29). Similarly, in The Roman Actor, Caesar,on seeing Philargus, Parthenius' miserly father, comments, before the

play-within-the-play showing the cure of a miser:

Can it be

This sordid thing, Parthenius, is thy father?No actor can express him. I had heldThe fiction for impossible in the scene,Had I not seen the substance.

(Massinger 1912 [1626]: 2.1.268-72)

Like those detectives and policemen in detective fiction who are fond

of saying, "If this were a detective story, such and such would hap-

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pen, but since it is not . .. ," these characters are speaking about a

hypothetical logical level embedded in their own; furthermore, they

are unaware of the analogy between the object referred to and theirown "reality."They are not stepping out of their own existence intoanother reality on a different logical level; on the contrary, they are

firmly entrenched in it. It is the reader's or the spectator's perceptionof the analogy between the fictional nature of the objects referred toand the fictionality of the characters' own existence which draws atten-tion to the normally invisible boundary between the two levels, thus, in

opposition to the characters' intention to establish the reality of their

existence, giving rise to a metaleptic effect emphasizing its artificiality.To return to embedding-embedded objects with a structural meta-

lepsis, not only a metaleptic effect, when this object is a work of artthe composite level is further circumscribed by an untransgressableboundary, which, because we are so used to it, is not always clearlyperceived as such. If what is transgressed is the boundary betweenlevel a and level t3,the boundary between level a and the reality of thereader or spectator of course remains intact. But even if the boundarybetween level a and extradramatic reality seems to be transgressed,the one around the work of art as a whole remains intact. The momentsuch a transgression takes place, for example, when spectators inter-

rupt a play or interact with characters in a play, as in Beaumont's The

Knight of the Burning Pestle or Pirandello's TonightWeImprovise,whatwas or seemed like reality becomes part of the a level of the play; itis part of the script of the play and as such is distinguishable fromthe reality beyond the boundary of the script. This boundary betweenthe work of art and reality can therefore be transgressed by analogyonly, by a boundary transgression within the "script."The essence ofa work of art

as such depends on the existence of a boundary betweenit and reality; if, hypothetically, it would really succeed in erasing this

boundary, it would self-destruct (Aspelin 1977). Everything that canbe said about transgression of the boundary between a work of artand reality must therefore always be understood in the context of this

proviso.However, although the composite and typeless unit with the trans-

gressed boundary is surrounded by an intact boundary, the nature oflatter is affected by the transgression of the boundary within the unit.

Unlike the intact boundaries discussed in section 1, this is a threatenedboundary which seems peculiarly temporary. Evidently, what is sug-gested when a boundary is transgressed is that at any moment the next

boundary may in turn be transgressed. In other words, although at no

point can all boundaries be transgressed simultaneously, because atall times there will be an intact boundary around the composite unit,a recursive series of transgressable boundaries is created whenever

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one boundary is transgressed! At this point, the model of embedding-embedded objects with different types of boundaries seems to become

self-engulfing.Nevertheless, although the insertion of an intact, well-functioning

boundary into a continuum already implies its possible reification or

transgression, there is still an important difference between the multi-

plication of intact boundaries and the multiplication of transgressableboundaries. The first, however hypothetical, is within the system of

the embedding-embedded object, which can become recursively ex-

panded. The boundaries which multiply when a boundary is trans-

gressed, on the other hand, are the boundaries separating the level-

less system from the system with logical levels. Although as soon as a

boundary is transgressed it becomes part of the typeless unit within

the object, the boundary which is added whenever one is transgressedis not yet transgressed and thus does not yet belong to the level-less

system (this is the logical reason why the transgressed boundary will

always be fictional, "part of the script").15The oscillation at the intact but threatened boundary around the

level-less unit is thus between the whole system based on logical levels,which

permeatesour

thinkingbut leads to infinite recursion, and

another system, in which at least two logical levels have collapsed into a

typeless unit, thus escaping infinite recursion by bottoming out. Inter-

estingly, although the typeless unit is outside the logical level system, it

may in turn become the basis for a system which may be either typelessor not; in the latter case, it forms, as it were, an axiom of the systemwith logical levels. Not only religious metaphysical systems have been

based on such a typeless level; Rolf Breuer (1976: 234 nn. 25, 26),

referring to Wittgenstein, makes the intriguing suggestion that logic

itself may bottom out in this way: Wittgenstein'sthesishad been: there are no verifiable i.e. meaningful)statementsexceptthose in the field of the naturalsciences.However, hiscontentionitself is,of course,nota statementof thiskind and thereforemeaningless, f correct.

AlthoughWittgensteinknewthat a statementcannotreferback to itself, heneverthelesswrote:"Logicmusttake careof itself."

Which system, then, is theoretically stronger, perhaps the "ultimate"

system, the rock-bottom typeless system or the system with logicallevels which never bottoms out, never reaches the "first cause"? Does

15. It is also interesting to note that the "infinity"of the system itself is here lim-

ited in a way which is not the case with the other two types of boundary. If there

were n intact boundaries in recursivestructures,or n meetings with the boundaryin the oscillating ones, there are only n-1 transgressed boundaries here. (Whatwe have n of here is transgressableboundaries, one of which, as we have seen, is

always outside the system.)

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the very notion that every boundary is potentially transgressable implythe collapse of Russell's and Tarski's distinction between logical levels

on which, as we have in part seen, so much of Western thinking isfounded? Or does the fact that there is always a part of the systemwith distinct logical levels which is not swallowed up by the typelesssystem indicate that it is, after all, the "correct"way of perceiving and

thinking? Whether one ultimately decides in favor of subversion or

preservation of the system of logical levels is largely a matter of tasteand temperament. Both decisions are equally "right"or "wrong," be-cause both are made by an arbitrary act of punctuating an essentiallydialectical continuum, in which both systems continuously and simul-

taneously subvert and reinforce each other.

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