Invented by French semiotician A.J. Greimas, the Semiotic Square is one of the favorite intellectual toys of theory-minded art critics. In a legendary essay, "Sculpture in the Expanded Field," Rosalind Krauss used it to chart the different manifestations of postmodern sculpture as reactions to the opposition "landscape" and "architecture," though she uses a quirky version of the device; a much more satisfying and orthodox example is the way Hal Foster used the Square to illustrate the different positions within Russian Constructivism, as outgrowths of the opposition between "Art" and "Production."
Basically, the Semiotic Square is a way of visually representing a matrix of possible relationships generated by a given opposition. The idea is relatively simple: Any principal opposition between contrary terms -- between "a" and "b" -- can be expanded to include a secondary pair of "contradictory" terms, "non-a" and "non-b." These contradictory terms have a natural relation of affinity with the respective contrary terms of the original binary, thus allowing you to form a kind of map of potential relationships within a given presupposed opposition. (You get, in Krauss words, "a quaternary field which both mirrors the original opposition and at the same time opens it.")
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