The Name-of-the-Father (French Nom du père) is a concept that Jacques Lacan fully developed starting in his Seminar The Psychoses (1955-1956). Lacan plays with the homophony of le nom du père (the name of the father) and le non du père (the no of the father), to emphasize the legislative and prohibitive function of the symbolic father.
The Name-of-the-Father is the fundamental signifier which permits signification to proceed normally. It both confers identity to the subject, naming and positioning the subject within The Symbolic Order and signifies the Oedipical prohibition (the "no'" of the incestal taboo). If this signifier is foreclosed, that is not included in the Symbolic Order, the result is Psychosis. See Foreclusion.
In On a Question Preliminary to Any Possible Treatment of Psychosis in 1957 (Écrits), Lacan represents the Oedipus complex as a metaphor in which one signifier (the Name-of-the-Father) substitutes another (the desire of the mother), meaning that all paternity involves metaphoric substitution. Lacan presents the 'paternal metaphor' in his Seminar ''La relation d'objet'' (1956-1957): it is the fundamental metaphor on which all signification depends (all signification is phallic). As previously stated, if the Name-of-the-Father is foreclosed, as in psychosis, there can be no paternal metaphor and hence no phallic signification.
Lacan distinguishes between the Symbolic, the Imaginary and the Real father. The Symbolic Father is not a real being but a position, a function. This paternal function imposes the law and regulates desire in the Oedipus complex, intervening in the imaginary dual relationship between mother and child to introduce a necessary symbolic distance between them (Dylan Evans). 'The true function of the Father is fundamentally to unite (and not to set in opposition) a desire and the Law' (Écrits). A subject may come to occupy the position of the symbolic father by virtue of exercizing the paternal function. Although the Symbolic father is not an actual subject but a position in the Symbolic order, a subject may actually occupy this position.
The Imaginary Father is an imago, the composite of all the imaginary constructs that the subject builds up in fantasy around the figure of the father, This imaginary construction bears little relation to the father as he is in reality (Seminar ''La relation d'objet). The Imaginary father can be construed as an ideal father or as the opposite, that is the bad father. As to the Real Father Lacan states that he is the agent of castration, the one who performs the operation of symbolic castration (Seminar The Other Side of Psychoanalysis). If the real father is the biological father of the subject, there is however a certain degree of uncertainty surrounding the question of who the biological father really is - Freud in Family Romances (1909) states that pater semper incertus est while the mother is certissima'' - therefore the real father is the man who is said to be the subject's biological father. The Real Father is an effect of language, thus the real here is the real of language (The Other Side of Psychoanalysis).
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