Graphic Formalism / On the Bias / Alexander R. Galloway


What does it mean to think and act “on the bias”? Inherently formal and spatial, if not also graphical, the diagonal line has played any number of important roles: from the diagonal of the unit square (which nearly destroyed Pythagoreanism and, later, played an important role in Plato’s Meno), to the modern intervention of Georg Cantor’s diagonal argument (where in 1891 he demonstrated that the real numbers are uncountable), to the structuralism of A.J. Greimas and Jacques Lacan, to Gilles Deleuze and Félix Guattari’s postmodern machine, defined as a diagonal that cuts through an assemblage. More recently Sara Ahmed has used what she called “oblique or diagonal lines” to characterize queerness, and indeed the diagonal—whether alone or via synonyms like the oblique, the slanted or askew, the non-orthogonal—has come to indicate differences and deviations of all kinds.1

Here I will focus on two small snippets from what is otherwise a vast graphical domain: Greimas’ semiotic square, used to reveal the complicated dynamics generated from simple opposition; and Lacan’s L-Schema, one of the many diagrams that populate Lacan’s work, in this instance a drawing of how the subject is formed. Both of these graphical renderings entail four terms arranged in a square. Yet, as we will see, the square acts as a kind of foundation, from which it’s possible, necessary even, to derive diagonal lines. The square evokes its own diagonals, just as the diagonals imply a square to traverse. Each diagonal furnishes something different depending on context. It might be a sense of dynamism. It might be narrative. Or, as we will see first with Greimas, the diagonal defines, animates, and motivates analogy, as it crosscuts binary opposition.

Let us begin then where Greimas does, deep within deep structuralism—Lévi-Strauss, Hjelmslev, Dumézil, and all the rest—where the structure of binary opposition plays such an important role in the derivation of the diagonal. Like Gottlob Frege grounding arithmetic in a logical invariant (the law of identity), Greimas grounds semiotics not in identity but in something like the reverse, in the invariant of presence and absence.2 According to Greimas, all things, no matter how different, share at least one fundamental characteristic with everything else, namely their own susceptibility for internal contradiction. Greimas, always the technician, presented this universal susceptibility as an algebraic formalism (Figure 1).

Figure 1: Equivalence of Ratios (A.J. Greimas).

Which might be glossed in normal language as one thing is in relation with its own negation, just as another thing is in relation with its own negation.3 Greimas describes the two pairs as “homologized contradictions” forming a “correlation.” I have no hesitation in also labeling his algebraic equation a definition of analogy, given that analogos, since Euclid at least, has been defined as the equation of two ratios.4

Greimas’ famous semiotic square, then, stemming from the definition of analogy and from his structure of correlation, is an explicit redrawing of the equation itself only now translated into a higher graphical form. Except with a torsion! The analogy equation is not plopped down over the square, one for one, but rather the analogy equation is twisted along its midsection, with the two lower terms changing place (Figure 2). And out of this torsion, miracle of miracles, two diagonal lines appear, themselves ultimately upstaging the square itself, or at least tracing its most seductive contours.

Figure 2: Semiotic Square (A.J. Greimas).

The excesses and deficiencies of structuralism are well known, and my aim here is not to defend these escapades, merely to luxuriate, as Fredric Jameson does, in the surprise and wonder of the square itself. For this binary device, this two birthed from a one, contains, after all the terms and mutual relations are accounted for, no less than ten positions, at least according to Jameson’s count.5 The one divides into ten. And so the seemingly sterile square, all logic and line, bursts forth with fresh bounty like a plant in spring. (Deleuze: “structure constitutes the principle of a genesis.”6) After all, this is the advantage of tools like the semiotic square, why we use them and need them, because they open access into places we can’t get to on our own, whether due to physical or psychic limitations. In other words, if concepts like “self” or “white” or “man” seem constricting, rest assured that they have already induced a multiplicity of other meanings and nonmeanings. As Jameson narrates it, “white” and “black” inherently already contain, or indeed are contained by, a host of other terms including “mestizo,” “nonblack,” and even “colorless”; and if these latter terms still appear ideologically subordinate, the mechanism itself undercuts the logic of domination simply by proliferating the many alternatives and confirming their logical necessity.7 Or as Rosalind Krauss once described the semiotic square, “a set of binaries is transformed into a quaternary field which both mirrors the original opposition and at the same time opens it. It becomes a logically expanded field.”8 In this sense, the square’s diagonals are a potent tonic against all manner of philosophical skepticism: we can know real features of the world; we can discover unknown experiences based on what we already know; and these discoveries can and will expand if not also destabilize the primary signifiers that previously set them in motion.

In any case, the diagonals are forged as a direct consequence of the aforementioned torsion. With one diagonal crossing the other, in front or behind no one knows, this double helix springs forth as the very essence of the diagram, its own little surprise, the thing that makes it so distinctive. The diagonals offer a genuine sense of motion and development; we’re really heading somewhere along those diagonal paths, like Dickens’ Pip, all anxiety and anticipation, boarding his four-horse stage-coach, destination London.

The secret of the square is thus narrative, not simply structure itself, Jameson has argued. Or rather some kind of shimmering between the two, where the square disgorges a complex of relations, while simultaneously indicating a path through those relations. So the semiotic square reanimates temporal diachrony, particularly in the form of narrative, even as it boasts and broadcasts a special kind of spatial thinking (which Jameson sees as emblematic of the shift from modernity to postmodernity).9 Diachrony offers a beginning, middle, and end; no different here, since this square is not hovering somewhere in outer space, its pitch and yaw determined only by computer, but rather stipulates a definite top and bottom, a definite beginning and end. The fourth and final term will be as important as the first, if not more so, for this ultimate terminus must be the result of a double negation, as it were, or at least the repository of a negation and an opposition summed together. In this way, the narrative of the square recounts two important truths. It says that the fourth term is the most interesting term, and that the proper path of reading is not clockwise or counter-clockwise, but rather to follow the eye’s path while reading (western) text, namely left-to-right and top-to-bottom, zigzagging like the letter Z. In other words, the diagonals, which traverse the square, set an internal narrative in motion. The diagonals are the narrative, or at least the capacities of narrative would be impoverished without such diagonality.

Yet even such a space of possibility is itself finite. This space of possibility, this “expanded field” as Krauss put it, still retains its outer limits, as if to insist in a menacing accent that these are the conditions of possibility, these and no others! Which only means that the semiotic square is an ideological square, a map of an ideological space, a map that lays out a set of boundaries while also providing a variety of real experiences and real forms of consciousness with which to populate it. Accordingly the vertical sides of the square begin to gain some attractiveness, even beyond that of the diagonals, for these complex and revelatory vertical vectors are the only relations not explicitly anticipated at the outset via the rules of opposition and negation. Greimas calls these two vectors the lines of “implication.”10 So “white” would implicate “non-black,” and “man” implicate “non-woman,” which, I suspect, might destabilize the original terms far more than opposition or even negation could do, those mechanisms already so well integrated into the infrastructure of metaphysics.

The key is that some kind of second-order process has taken place. This isn’t just language or speech, but some kind of affectation of speech, some gestural swirl applied in order to whip up language into something a little different. (Such an insistence on second-order language seems absolutely integral to structuralism as a method, and thus, in turn, has frequently become a primary target for those wishing to get beyond structuralism; Deleuze offers one variant of this, with his flat ontology, but there are many others.)

Which might be basis enough to revoke Lacan’s structuralist credentials, or at least to reclassify him as a very different sort of beast, for this is the man who stridently and consistently repeated his mantra that “there is no such thing as metalanguage.”11 To which Greimas, Barthes, Jameson, Krauss, et al. shoot back in response, oh yes there is, and it takes the shape of a square!

Would it be irresponsible, then, to label Lacan’s L-Schema a “simplified semiotic square” (Figure 3)?12 I think so, for the reason just given, but also because the four terms in Lacan’s graphical schema play very different roles than those in Greimas’ square. Although the temptation to reduce one to the other is already contained in both schemas, indeed in the very notion of the schema itself, which promises to write all scenarios, to right all wrongs, simply by sketching a toy model on the back of a sheet of paper and assuming it to have any sort of explanatory power whatsoever. The superficial resemblance between Greimas’ square and Lacan’s L schema—they both begin from four terms arranged in a square, they both carry a prominent x-shape at their heart (not unconnected to the literary technique known as chiasmus), they both promise to elucidate if not also resolve one or more fundamental antagonisms—this superficial resemblance loses its force once we give proper scrutiny to the four terms themselves and the relations between them. What were abstract placeholders in Greimas, logical constructions that could conceivably house any number of different things, appear now in the specific rectangular shape of the psyche, marked at its corners using the concepts of id, ego, object, and the unconscious borrowed from Freud.

Figure 3: L Schema (Jacques Lacan).

Jameson showed how narrative was still legible in the Greimas square despite its staid and static veneer; no such divination is necessary with the L Schema, as an explicit narrative springs from the diagram unassisted, instructing the viewer exactly where to begin, where to go next, and where to end. The L Schema starts on the forward slash, the right-leaning diagonal labeled “imaginary relation,” this being the well-known Mirror Stage, in which the subject sees itself in the form of an object, and likewise the object reflects back at it. “In the veil of the narcissistic mirage,” was how Lacan described the aa′ diagonal axis.13 Or, as he more plainly put it, the first diagonal is the plane of the mirror.14 A beginning, but merely a beginning, the axis of the ego reveals two additional zones within the diagram, first what is “shy of” this imaginary objectification, namely the Subject, and second what is “beyond” it, namely the Other. Lacan calls this second slash the wall of language.15 The L Schema represents the plane of the mirror bisected by the wall of language, one technology colliding with another.

Figure 4. Mary Kelly, “Post-Partum Document: Introduction, 1973” (detail). Courtesy Generali Foundation Collection—Permanent Loan to the Museum der Moderne Salzburg, © Mary Kelly, Photograph: Werner Kaligofsky.

If Krauss or Jameson were perhaps a better entrance into the semiotic square than Greimas himself, the artist Mary Kelly is likewise perhaps a better chaperone for the L Schema than Lacan, who we know was more prone to playful suggestion than pedantic exposition, even for his more complex inventions. (Like a child at night spooked by scary shadows, I will tiptoe past those frightful R and I Schemas, which follow the L Schema by pushing confidently if not always comprehensibly into the outer fringes of Lacanian formalization.) In her work “Post-Partum Document: Introduction, 1973” (Figure 4), Kelly appropriated Lacan’s diagram, turning it into a kind of crest or sigil traced onto the baby clothes of her young son, over four successive developmental phases. Lacan apparently labeled this the “L” schema since it resembled the Greek letter lambda λ—to my eye an imprecise resemblance—but Kelly recomposed the master’s diagram slightly by advancing the alphabet from L to Z, suggesting that the diagram ought to be read from left-to-right, top-to-bottom, beginning and ending at the Subject.16

So Kelly appropriated the Lacan diagonals, but she also altered them. The four outer vertices are labeled in the normal fashion as Subject, little other, little other prime, and big Other. The word “INTERSUBJECTIVITY” appears four times—not unorthodox, as Lacan himself explicitly called his graphic a “dialectic of intersubjectivity.”17 But Kelly added some new text labels for the lines: AXIS I, AXIS II, AXIS III, AXIS IV. She also added dates to cover the span of four months: SEPT 1973, OCT 1973, NOV 1973, DEC 1973. In essence, Kelly was posing that important feminist question: who is this graphic for?; where is the woman in the diagram? Is this a diagram of male subjectivity—in this case, her son—which would make the baby the subject, and the mother the object? Or is this a diagram of the mother’s subjectivity, where Kelly herself is the subject, with baby as object. For Kelly it was the latter. “It is a picture of her psychic life,” the art theorist Margaret Iversen stated unambiguously.18 Subjectivity was always somehow “femme” for Lacan. But perhaps it required Mary Kelly to violate the synchronic integrity of the L Schema, to turn it into a relational document of four months of a mother’s life, both lived and traced for posterity.

Logic, reduction, structure, relation—these graphical formalisms are metaphysical technologies, that much is clear, for Greimas if not also for Lacan, beginning as they do from a ratio between presence and absence (that is, from a logos), adjoining opposites together, spinning out binary relations into a complex of stipulations and suspensions. But what did Lacan mean by  a “dialectic of intersubjectivity”? Does the dialectic have a place here? Or is the dialectic blocked by the logos, by logical relation? I think it does have a place, if only due to the role that contradiction plays within these formalisms. And while contradiction is a part of classical metaphysics, only with the dialectic does contradiction take on its role as a kind of constitutive fabric of being (or rather being-as-becoming). Hence a tentative conclusion, that the diagonal is the mark of the dialectic. And if the dialectic continues or extends the logic of the ratio (logos) it does so only by reinterpreting the putative harmony of the ratio instead as a kind of majestic dissonance: these elements are actually antagonists, fomenting positive feedback rather than relaxing into homeostasis.

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This is part of the cluster Graphic Formalism. Read the other posts here.

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  1. Sara Ahmed, Queer Phenomenology: Orientations, Objects, Others (Durham, NC: Duke University Press, 2006), 61.
  2. For Frege, the count of all things disobeying the law of identity is zero, precisely and universally. See Gottlob Frege, The Foundations of Arithmetic: A Logico-mathematical Enquiry into the Concept of Number, trans. J.L. Austin (Oxford: Basil Blackwell, 1950).
  3. A.J. Greimas, On Meaning: Selected Writings in Semiotic Theory, trans. Paul J. Perron and Frank H. Collins (Minneapolis: University of Minnesota Press, 1987), 5.
  4. Greimas, On Meaning, 50. For Euclid’s definition of analogos as a/b = c/d, see Euclid, Elements, vol. 2, trans. Thomas Heath (New York: Dover, 1956), 114.
  5. Fredric Jameson, “Foreword,” in Greimas, On Meaning, xiv; this essay was also reprinted in Fredric Jameson, The Ideologies of Theory (New York: Verso, 2008), 516-533.
  6. Gilles Deleuze, “How Do We Recognize Structuralism?” in Desert Islands: and Other Texts, 1953-1974, trans. David Lapoujade (Los Angeles: Semiotext(e), 2004), 172.
  7. Jameson, “Foreword,” xiv.
  8. Rosalind Krauss, “Sculpture in the Expanded Field,” October 8 (Spring, 1979): 30-44, 37. The concept of the “expanded field” carries multiple meanings for Krauss: a field of grass or earth; art making as a field of practice; the area of the Greimas square on the page as a spatio-logical field.
  9. The semiotic square thus serves as a graphical marker for “a period intellectually given over to space in a way radically different from the preceding generation of the modernists, in thrall to temporality” (Jameson, “Foreword,” xxi-xxii).
  10. Greimas, On Meaning, 49.
  11. Although Lacan repeats this formulation a number of times, one place to look is Jacques Lacan, The Seminar of Jacques Lacan: Book XIX, …or Worse 1971-1972, trans. A. R. Price (Cambridge: Polity, 2018), 4.
  12. The historical lineage here is complicated and would require a study of its own. Lacan’s L Schema dates to the mid 1950s; Greimas published his square in 1966. Greimas himself references three sources, comparing his square “to Robert Blanché’s logical hexagon…as well as to the structures called the Klein group in mathematics and the Piaget group in psychology” (Greimas, On Meaning, 50). Jean Piaget published a similar kind of logical square in his 1949 book, Traité de Logique (Paris: Armand Colin, 1949). And historians have noted similar kinds of logical squares going back to Aristotle. I thank William Lockett for his insight into Piaget in particular.
  13. Jacques Lacan, Écrits, trans. Bruce Fink (New York: Norton, 2005), 460. He also references the mirror stage on page 40, as well as in his second and third seminars. See Jacques Lacan, The Seminar of Jacques Lacan: Book II, The Ego in Freud’s Theory and in the Technique of Psychoanalysis 1954-1955, trans. Sylvana Tomaselli (New York: Norton, 1988), 243, and Jacques Lacan, The Seminar of Jacques Lacan: Book III, The Psychoses 1955-1956, trans. Russell Grigg (New York: Norton, 1993), 161.
  14. “…the plane of the mirror, the symmetrical world of the egos and of the homogeneous others” (Lacan, Seminar II, 244).
  15. Lacan, Seminar II, 244.
  16. In fact Lacan himself also suggested a Z shaped narrative for the L Schema. See Lacan, Écrits, 459.
  17. Lacan, Écrits, 40.
  18. Margaret Iversen, “Visualizing the Unconscious: Mary Kelly’s Installations,” in Douglas Crimp, et al., Mary Kelly (London: Phaidon, 1997), 41.