Formula of metaphor

This equation, (\(f\left(\frac{S^{\prime }}{S}\right)S\equiv S(+)s\)), represents the formula of metaphor as developed by the psychoanalyst Jacques Lacan. It is a linguistic and psychoanalytic concept, not a mathematical one, that uses algebraic notation to illustrate the function of metaphor in the unconscious. Explanation of the Formula The formula is derived from Roman Jakobson's work on aphasia and his opposition between metaphor and metonymy, which Lacan applied to his rereading of Sigmund Freud's work. \(f\): Represents the signifying function, or the effect of signification.\(\frac{S^{\prime }}{S}\): Signifies "the substitution of one signifier for another" (S' for S).\(\equiv \): This symbol means "is congruent with".\(S(+)\) s: Represents the resulting signification. The \((+)\) symbol represents the crossing of the bar (\(-\)) from the Saussurean algorithm (the division between signifier and signified), which represents "the emergence of signification".\(S\): The original signifier.\(s\): The newly created signified. The overall formula reads: the signifying function of the substitution of one signifier for another is congruent with the crossing of the bar, leading to the creation of a new signified. Lacan argued that the production of meaning (signification) is only made possible by this operation of metaphor, where the signifier passes into the signified. 

$f\left(\frac{S^{\prime }}{S}\right)S\equiv S(+)s$ Template:Short description

This page is about the psychoanalytic concept. For the linguistic trope, see Metaphor. For the related concept, see Metonymy (Lacan).

Formula of metaphor

The formula of metaphor is a conceptual expression developed by the psychoanalyst Jacques Lacan to formalize the operation of metaphor in the unconscious. Written in algebraic form as $f\left(\frac{S^{\prime }}{S}\right)S\equiv S(+)s$, the formula does not describe a mathematical relation but a linguistic and psychoanalytic process through which meaning (*signification*) is produced. Drawing on structural linguistics—particularly Roman Jakobson’s distinction between metaphor and metonymy—Lacan uses symbolic notation to articulate how a Signifier substitutes for another signifier, enabling the emergence of a new Signified.

The formula occupies a central place in Lacan’s reworking of Freudian theory and underpins his claim that the unconscious is structured like a language.

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Formal definition

In Lacanian theory, the formula $f\left(\frac{S^{\prime }}{S}\right)S\equiv S(+)s$ defines metaphor as a signifying operation in which the substitution of one Signifier ($S^{\prime }$) for another ($S$) produces a new Signified ($s$) through the crossing of the signifying bar.

• $f$ denotes the signifying function, that is, the effect of signification as such.
• $\frac{S^{\prime }}{S}$ represents the substitution of one signifier for another.
• $\equiv$ signifies congruence, indicating a structural equivalence rather than a causal or quantitative relation.
• $S(+)s$ designates the resulting signification, where the symbol $(+)$ marks the crossing of the bar separating signifier and signified in the Saussurean algorithm.
• $S$ is the original signifier occupying a position in the symbolic chain.
• $s$ is the newly produced signified that emerges as an effect of the metaphoric operation.

Read as a whole, the formula states that the signifying function of substituting one signifier for another is structurally equivalent to the production of new meaning through the crossing of the bar.

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Theoretical background

Lacan’s formula of metaphor emerges at the intersection of structural linguistics, Freudian psychoanalysis, and his own reformulation of the unconscious as a symbolic structure. Three figures are especially decisive: Roman Jakobson, Ferdinand de Saussure, and Sigmund Freud.

Roman Jakobson: metaphor and metonymy

Lacan derives the opposition between metaphor and metonymy from Jakobson’s analysis of aphasia, which distinguishes two fundamental axes of language:

• the axis of selection (similarity and substitution), corresponding to metaphor;
• the axis of combination (contiguity and displacement), corresponding to metonymy.

Lacan radicalizes this linguistic distinction by situating it within the unconscious. Metaphor and metonymy become fundamental mechanisms of signification rather than merely rhetorical figures. The formula of metaphor formalizes the metaphoric axis as an operation of substitution, standing in strict opposition to the metonymic process elaborated in Metonymy (Lacan).

Ferdinand de Saussure: signifier and signified

The formula presupposes Saussure’s distinction between Signifier and Signified but reconfigures it. Lacan emphasizes the bar separating the two terms and asserts the primacy of the signifier. Meaning is not inherent in the signified but produced through operations within the signifying chain.

The $(+)$ symbol in the formula represents the crossing of this bar. Metaphor is thus the operation by which a signifier produces an effect in the register of the signified.

Sigmund Freud: condensation

Lacan aligns metaphor with Freud’s concept of condensation (*Verdichtung*) in dream-work. Condensation involves the substitution and overdetermination of signifiers, a mechanism Lacan formalizes linguistically as metaphor. In dreams, symptoms, and slips of the tongue, multiple unconscious signifying chains converge into a single formation.

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Lacan’s seminars

Seminar III: The Psychoses (1955–1956)

In *Seminar III*, Lacan links metaphor to the integrity of the symbolic order. Psychosis is characterized by the foreclosure of a fundamental signifier (notably the Name-of-the-Father), resulting in the failure of metaphoric substitution. The formula of metaphor thus provides a structural criterion distinguishing neurosis from psychosis.

Seminar V: The Formations of the Unconscious (1957–1958)

The formula is articulated most explicitly in *Seminar V*. Lacan presents metaphor and metonymy as the two primary mechanisms of the unconscious, aligning them respectively with condensation and displacement. Here, metaphor is defined as the operation through which meaning is produced via signifier substitution.

Seminar XI: The Four Fundamental Concepts of Psychoanalysis (1964)

In *Seminar XI*, metaphor underlies Lacan’s mature conception of the unconscious and interpretation. Meaning remains contingent and unstable, directly linking metaphor to the Sliding of the Signified. Interpretation intervenes at the level of the signifier rather than supplying meaning.

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Relation to metonymy

The formula of metaphor must be read in direct contrast to Metonymy (Lacan). Whereas metaphor operates through substitution, metonymy operates through displacement along a chain of signifiers.

• Metaphor: substitution producing meaning.
• Metonymy: displacement deferring meaning and sustaining desire.

Together, these mechanisms structure unconscious formations, speech, and desire.

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Diagrammatic explanation

Schematic representation

$\begin{array}{c}{S}^{\prime }\\ S\end{array}⟹S(+)s$

$\begin{array}{ccc}{S}^{\prime }& \to & \text{substitution}\\ \downarrow & & \\ S& \stackrel{(+)}{\to }& s\end{array}$

Metonymy, by contrast, may be schematized as horizontal displacement:

mediawiki Copy code $S_1\to S_2\to S_3\to \dots$

Clinical implications

Symptom formation

In clinical psychoanalysis, the symptom functions as a metaphoric formation in which unconscious signifiers are condensed into a single signifying expression. The symptom is not merely pathological but a structured production of meaning.

Interpretation

Lacanian interpretation intervenes at the level of the signifier. Rather than explaining meaning, interpretation restructures signification, allowing new metaphoric articulations or disrupting existing ones.

Subjectivity

The formula of metaphor articulates Lacan’s claim that subjectivity itself is constituted through language. Meaning emerges only through metaphoric substitution within a structure otherwise governed by displacement and slippage.

Historical reception

Following Lacan’s death, the formula of metaphor became central to post-Lacanian debates. Lacanian schools emphasized its diagnostic and structural value, particularly in differentiating neurosis and psychosis. Structuralist readings praised its formal rigor, while post-structuralist interpretations highlighted the instability of meaning emphasized by the Sliding of the Signified.

Clinically, debates persist over the extent to which analytic interpretation should explicitly target metaphoric formations. Despite differing emphases, the formula remains one of Lacan’s most influential theoretical contributions.

Comparison of metaphor, metonymy, and sliding of the signified

Concept	Primary operation	Direction	Relation to meaning	Freudian correlate	Clinical function [[Metaphor (Lacan) Substitution of signifiers Vertical Produces signification Condensation Symptom formation - [[Metonymy (Lacan) Displacement along chain Horizontal Defers meaning Displacement Sustains desire - Sliding of the Signified Structural instability Continuous drift Prevents final fixation — Explains interpretive openness } See also Metaphor (Lacan) Metonymy (Lacan) Sliding of the Signified Signifier Signified Name-of-the-Father Unconscious (psychoanalysis)