This psychic dilemma (of the hero) or ideological operation (of the narrative) can be described also in terms of formal logic or rhetorical strategy. Rather than presenting it as a simple opposition (surface structure vs deep structure), of binary pairs (male/female; white/non-white; American/foreign), we could see John McClane's rational action/ unconscious desire in terms of a more three-dimensional, dynamic process, of which the narrative is the linear, sequential articulation, at the same time as it is layered, differently textured, and contradictory. Lévi-Strauss assumed that a society starts telling itself myths when, as a culture, it is faced with contradictory experiences which it cannot make conscious to itself. The myth preserves the contradiction at the same time as it 'resolves' it in another medium and through another modality - e.g. that of 'art' or 'narrative' -which is why he argued that myths are 'the imaginary resolution to real contradictions'. Using the Oedipus myth as his example, Lévi-Strauss established a set of successive equations with multiple variables (A:B = C:D), in order to show how a family of mythic narratives transforms initially contradictory statements into apparently unproblematic equivalences (see Lévi-Strauss 1972). In our example, it would be 'terrorists taking women hostages' = 'working-class male identity taken hostage by middle-class female values', with the result that 'male identity' is allowed to rescue 'women hostages'. A.J. Greimas (1983) recognized in such representations of unconscious societal contradiction a traditional tool of logic, similar to Aristotle's syllogism which establishes the rules of deductive reasoning (see Corbett and Connors, 1999: 38-9). In order to deduce valid conclusions from a given premise, logicians distinguish between universal and particular propositions (quantity) and between affirmative and negative assertions (quality). Once these premises are fitted into the so-called 'square of oppositions', propositions can be tested as to their validity by diagrammatically placing the terms involved into the different corners of the square, according to whether they are positive or negative terms (e.g. human/not human), and whether they make universal or particular claims (all humans/some humans). The diagonals, top and bottom, left and right will then be readable as establishing pertinent relationships. From this example, Greimas developed his so-called semiotic square, in order to provide a structural model for the conditions of possibility of narratives. Made up of contradictions (mutually exclusive terms), contraries (a double negative generating a positive term), and implications (terms generated by complementary relations), the semiotic square is intended to identify the 'elementary structure of signification' by specifying the 'semiotic constraints' underlying culturally meaningful narratives.
For instance, seeing the repetitions, redundancies and relationships of mutual implication in Propp's morphology of the Russian folk tale, Greimas sought to reduce Propp's linear succession of narrative events to a closed system, where the semiotic constraints of the syllogistic square could account for the permutation of functions, after Propp had already successfully demonstrated how the multiplicity of characters in folk tales can be reduced to a relatively limited repertoire of functions, and particular actions can be understood as general 'moves'. If in a tale someone pronounces a prohibition, this will inevitably be followed by a transgression, so that prohibition and transgression imply each other and only amount to a single move. Greimas
put forward the notion of semes (minimal semantic units) and arranged them in a four-term homology:
These four semes establish logical relations to one another. Seme SI is the positive term. S2 establishes a relation of opposition to SI; -Si is the contrary of Si; and -S2 is implied by Si. For example, if we invest SI with the semantic content /human/, S2 = /anti-human/, -SI = /not-human/, and -S2 = /not-antihuman/. James Kavanaugh finds these semes to be the dominant values in Ridley Scott's Alien (Kavanaugh 1980: 98):
Anti-human (the Alien)
(Jones the cat) (Ash the robot)
Greimas therefore shows how a narrative generates a sense of coherence and closure by a structure which contains negation, equivalence, opposition, and contradiction, as the logical types of relation elaborated to accommodate ideologically or culturally contradictory material. As Fredric Jameson sums up the advantages of the semiotic square:
The first merit of Greimas' mechanism is to enjoin upon us the obligation to articulate any apparently static free-standing concept or term into the binary opposition ... which forms the very basis for its intelligibility. [This] might take the form of the invention of some mediatory concept which bridged the gap [of the contradiction.] Or [the mechanism may function as] a value-system, in which raw materials coming in from the outside are at once given their place in the rectangular structure and transformed into symbolically signifying elements within the system.
(Jameson 1972: 164)
Jameson has himself drawn a number of such semiotic squares for Hollywood feature films, including Dog Day Afternoon and The Shining (in Jameson 1990: 35-54,82-98), in each case gathering up the socio-political 'material' that the film in question is said to transform.
Treating Die Hard as such an accumulation of culturally problematic 'material' from the outside, we can see how it is converted into binary pairs, and what sort of functions and relations are highlighted when we devise for these pairs the 'rectangular structure' of a semiotic square: male and female, working class and middle class, foreign and domestic, the police and the criminals could all be the binary pairs necessary for supplying the contradictory, contrary, and complementary relations, across which the imaginary resolution can be generated. There is even a 'mediatory concept bridging the gap', in the figure of the black patrolman Al Powell: we shall see how crucial not only he but 'race' is for the logic of the film's overall action and the plot's successive moves.
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