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When the expression "the name of the father" first appeared in Lacan’s work, in the early 1950s, it is without capital letters and refers generally to the legislative and prohibitive function of the "symbolic father" as the one who lays down the taboo on incest in the Oedipus complex.
"It is in the 'name of the father' that we must recognize the support of the symbolic function which, from the dawn of history, has identified his person with the figure of the law."
Legislative and Prohibitive Function
The rexpression is at once a semi-humorous religious allusion and a play on the near-homonyms non and nom: the name-of-the-father (le nom du père) is also the father's "no" (le "non" du père) to the child's incestuous desire for its mother. (the legislative and prohibitive function of the symbolic father)
In Lacan's 1955-6 seminar, The Psychoses, the expression becomes capitalized and hyphenated and takes on a more precise meaning; the Name-of-the-Father is described as the fundamental signifier which permits signification to proceed normally.