Lévi-Strauss's most ambitious project was the analysis of myth, culminating in his four-volume Mythologiques (1964-1971), which is among the most extraordinary works of scholarship produced in the 20th century. He analyzed hundreds of myths from indigenous North and South American cultures and argued that they all, despite their surface variety, belong to a single interconnected system organized by the same structural principles.
His basic analytical move was to identify what he called mythemes: the minimal structural units of a myth, analogous to phonemes (the minimal units of a language). The meaning of mythemes, like the meaning of phonemes, is not found in them individually but in their relations: in the oppositions and mediations they form with other mythemes.
His famous analysis of the Oedipus myth demonstrates the method. He arranges the mythemes of the story in a grid. Reading horizontally across, you get the narrative sequence (the events of the story in order). Reading vertically down, you get the structural relations (the patterns and oppositions that recur across the story). The horizontal reading gives you the story; the vertical reading gives you its unconscious logic.
In the Oedipus myth, Lévi-Strauss identifies two fundamental oppositions: the overrating vs. underrating of blood relations (Oedipus kills his father and marries his mother on one axis; the Spartoi who kill each other on another axis), and the denial vs. affirmation of the autochthonous origin of humans (the monsters the heroes kill vs. the physical difficulties the heroes have in walking, which in Greek tradition signals birth from the earth). The myth, on his reading, is a logical instrument for thinking through the contradiction between the cultural belief in human descent from the earth (autochthony) and the biological reality of sexual reproduction. The myth does not solve the contradiction. It mediates it: by setting the two poles in structural relation, it allows the culture to hold both without explicit contradiction.
This mediating function is what Lévi-Strauss thinks myths are fundamentally for. They are not proto-science (primitive attempts to explain natural phenomena), not allegories of moral truths, not expressions of individual psychological complexes. They are instruments of logical thinking: ways of working through fundamental contradictions in human experience by setting oppositions in structural relation and providing mediating figures and narratives that hold the tension without requiring resolution.
The most celebrated of Lévi-Strauss's oppositions is the raw and the cooked, which gives the first volume of Mythologiques its title. Cooking is a transformation of raw nature into cultural product, and he argues that the raw/cooked opposition structures an enormous range of human thinking about nature/culture, life/death, human/animal, and the relationship between them. The specific transformation (boiling, roasting, smoking) encodes further distinctions: boiling preserves the food in its own juice (endocuisine, associated with the domestic) while roasting exposes it to fire without mediation (exocuisine, associated with the social and the festive). These are not mere cooking techniques. They are a grammar for thinking about social relationships.