Concept lattice formalisms of Hébert’s “semic analysis” and “analysis by classification”

1.1 Nomenclature

1.1.1 Semes

Greimas and Courtés (1979: 278) define a seme as the “‘minimal unit’ of meaning” that is located on the “content plane” and corresponds to the pheme, the unit on the “expression plane.” Like the constitution of phonemes as comprised of phemes, so too, semes (which we denote /seme/) are the components of sememes. But a seme itself is not autonomous or “atomistic,” instead it “exists only because of the differential gap that opposes it to other semes” (Greimas and Courtés 1979: 278). This theoretical position is as much indebted to Saussure (1983), as to Greimas’ recognition of how Levi-Strauss uses constitutive units for identifying signifieds in myths in the following form:

(1) A n o n − A ≅ B n o n − B .

Here, Greimas mirrors the notion of Levi-Strauss (1955) that “greater constitutive units” can be framed in terms of “distinctive features.” That is to say that semes themselves follow this type of structure of opposition and form what Greimas calls semic categories (Greimas 1987: 16). These semic categories constitute the content plane, and therefore are considered anterior to the individual semes themselves. But there is also structure here too, which becomes fully formed and visualized through the semiotic square (Greimas and Rastier 1968: 88) via the utility of the relations of contradiction (), contrariety (⟷) and implication (⤏) between semes:

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What is fundamental then to the semiotic square is homologation, or what Greimas and Courtés (1979: 144) consider “a rigorous formulation of reasoning by analogy” that they denote as A: B: A′: B′, which is a generalization of the case of the semiotic square as: s 1 : s 2 : : s ̄ 1 : s ̄ 2 .

In terms of classifying semic categories themselves, Greimas and Courtés offer the following:

An examination of different semic categories allows us to distinguish several classes: (a) figurative semes (or exteroceptive semes) are entities on the content plane of natural languages, corresponding to elements of the expression plane of the semiotics of the natural world, i.e., corresponding to the articulations of the sensory classes, to the perceptible qualities of the world; (b) abstract semes (or interoceptive semes) are content entities that refer to no exteriority, but which, on the contrary, are used to categorize the world and to give it meaning: for example, the categories relation/term, object/process; (c) thymic semes (or proprioceptive semes) connote semic microsystems according to the category euphoria/dysphoria, thus setting them up as axiological systems. (Greimas and Courtés 1979: 279)

Hébert (2020: 145–146) also specifies that semes can be thought of as generic, for which the corresponding sememe belongs to a semantic class as a semantic paradigm, made up of sememes. There are three types of generic semes, those that are microgeneric, mesogeneric and macrogeneric, for which all three correlate to the three semantic classes that Rastier (1997) contextualizes within the domain of his interpretive semantics. These semantic classes are taxemes, or the minimal classes by which sememes are distinguished, domains, which are connected to social contexts and human endeavors (e.g., dictionaries, scientific disciplines, etc.), and dimensions, or those most general classes of oppositions (e.g., animate vs. inanimate). Hébert give the example of the taxeme //tableware// and its three sememes, each of which

contains the microgeneric seme /tableware/ and is distinguished from the other sememes of the same taxeme by a specific seme: /for piercing/ in ‘fork,’ /for cutting/ in ‘knife’ and /for containing/ in ‘spoon.’ Since this taxeme comes under the domain //food//, the three sememes also contain the mesogeneric seme /food/. (Hébert 2020: 146)

Unlike a generic seme, the specific semes highlighted by Hébert (2020: 145) are ones that distinguish a sememe from all other sememes of the same class. Hébert also distinguishes between inherent semes and afferent semes. In the case of the former, an inherent seme is one that belongs to a sememe’s type and is actualized by default unless it is part of a virtualized structure. For example, ‘albino crow’ contains the inherent seme /black/ in the type of the sememe ‘crow’ that is virtualized because of ‘albino.’ Thus the seme /white/ here is actualized. Hébert notes that afferent semes are “present only in the sememe’s token, that is, only by contextual indication” (Hébert 2020: 146). Thus if a seme is present in a context it is normally associated to, it is actualized, and if it is missing it is virtualized.

1.1.2 Isotopies and molecules

Greimas (1983: 81) describes an isotopy (denoted /isotopy/) as emerging from a syntagm, that is, a grouping of at least two semic figures. As Hébert (2020: 147) suggests, the sentence “I only use a knife for picking up peas” contains the (mesogeneric) isotopy /food/, which indexes the sememes ‘knife’ and ‘peas.’ In addition, it virtualizes the inherent specific seme /for cutting/ in ‘knife’ and actualizes the afferent seme /for picking up/. An isotopy can thus be extended over “two morphemes, two words, a paragraph, or a whole text” (Rastier 2016: 497). Thus given a pair of generic isotopies, that is, ones that correspond to one of the three semantic classes of taxeme, domain and dimension, we have what Rastier (1997: 35) identifies a semic molecule, or “the recurrence of groupings of relatively stable specific semes.” In Figure 1 we give Rastier’s diagram of the relation between these generic isotopies, and how their constituent sememes are covered by a semic molecule.

Figure 1:

Rastier’s (1997: 36) diagram of the relation between themes, generic isotopies and a semic molecule.

1.2 Motivation

Our motivation for this article is what we perceive as parallelisms between the analytic methods and conceptual frameworks of structural semiotics as espoused by Greimas (1983), Rastier (1997) and Hébert (2007, 2020, to the mathematics of extended lattice theory and “formal concept analysis” forwarded by Ganter and Wille (1999). In order to ground the connection between these fields, we examine the work of Hébert using his framework of “semic analysis” and “analysis by classification,” which directly emerges from Rastier’s interpretive semantics and Greimas’ notion of semic categories.

For Wille (2005: 9), the mathematization of real-world concepts to discover underlying structures and their accompanying semantics through lattice constructions is a promising inter-disciplinary field because it has the potential to “activate implicit conceptions and experiences concerning the underlying domains.” The mathematization of concepts thus points towards subjective theories which “contain implicit and explicit assumptions about objects and events, their conditions and causes, their characteristics, relations and functions” (Wille 2005: 8). This approach neatly mirrors what Rastier (1997: 5) asserts about objectivity in semiotic analysis, in which “meaning has no existence of its own beyond its utterance and its interpretation” Marrone (2022: 22). similarly notes that semiotic analysis ultimately seeks “to reconstruct the hierarchy of underlying layers” in subjective experiences. In our estimation, this is facilitated directly through imposed selector functions that seek to articulate, collate and render meaning in texts or images via associations between elements of a set.

To these ends, we firstly provide a mathematical frame to the generation of class taxonomies suggested by Hébert (2007: 167–169) in his analysis of the poem <<Quelle affaire!>> (“A Sorry Business!”) by Gilles Vigneault. We also provide a formalization of the structure of semic isotopies through his reading of The golden ship by Émile Nelligan (Hébert 2020: 157–160) as well as the characteristics of inter- and intra-semic molecules at work within Réne Magritte’s painting La clef des songes (Hébert 2020: 160–168). Through these examples, we introduce relative mathematical formalisms in order to provide a systematic description of the approach of structural semiotics to the conceptualization of a seme as the minimal unit of meaning, and how semes form the basis for larger emergent forms.

2.1 Formal concept analysis

Formal concept analysis (FCA) is a branch of mathematics pioneered by Rudolph Wille (1982) as a means in which to extend lattice theory. As Ganter and Wille (1999) contend, the basic notions of FCA center on a formal context and its formal concepts. These notions were developed in order to develop a system for formalizing “knowledge discovery” within any domain (Stumme et al. 1998: 451). FCA is a direct mathematization of ‘conventional’ concept through lexical fields such that concepts themselves are considered

cognitive acts and knowledge units potentially independent of language. Only if they are used to give meaning to linguistic expressions, they become so-called word concepts which are conventualized and incorporated. The meanings of words for an individuum presuppose conceptual knowledge of that individuum which turns linguistic expressions into signs for those concepts. (Wille 2005: 6)

Like Hébert’s recognition of the importance of considering how an intension of a term determines its extension, FCA provides a formalized method in which to study what Hébert calls classes and sub-classes, though in FCA these are known as superconcepts and subconcepts. Indeed the superconcept/subconcept relation in FCA was primarily influenced by how Frege (1952) identified the dual relationship between intension and extension and how each generates the other such that “the extension is the class of things in the world for which the intension is a true description” (Wilson and Keil 2001: 177). Below we introduce FCA through a number of standard definitions (Ganter and Wille 1999).

Let us first consider the formal context K σ ≔ K 1 | K 2 given in Figure 4. Here we use Hébert’s elements, or leaves from his finite rooted trees of << Quelle affaire!>> in Figure 3 to construct G, a set of objects for which a “×” in the binary table of K σ indicates a incidence relation to a set of attributes M that correspond to what Hébert calls a collection of classes. In order to formalize the incidence relation of gIm in K σ , we relabel the classes so that they reflect in a similar manner how Greimas and Courtés (1979: 29–30) discuss classemes as “those semes which are recurrent in the discourse and which guarantee its isotopy.”

Figure 4:

The formal context K σ as the apposition of formal contexts K 1 and K 2 , whose objects and attributes are sourced from Hébert’s (2007: 168) two disconnected finite rooted trees that visualize the “Thematized classification in << Quelle affaire!>>” (c.f. Figure 3). The dashed line indicates the articulation of M ₁ and M ₂ such that M ₁\coloneq {is an actant, is from culture, is a predator, is a person, preys on seals, preys on fish, is an anti-predator}, and M ₂\coloneq {is from nature, is an animal, is terrestrial, is aquatic, is a fresh water habitat, lives in fresh water, lives in salt water, is prey, is a seal, is a fish}.

We construct K σ through two contexts K 1 and K 2 that share the object set G, though split Hébert’s classes into M ₁ and M ₂ such that M ₁ ∩ M ₂ = ∅. This follows Hébert’s two original disjunct finite rooted trees as classifications of the oppositional classes: //culture// and //nature//. From K σ and its subcontexts K 1 and K 2 we can see that subsets of objects may share attributes and thus be considered formalized structures that form “cognitive acts and knowledge units potentially independent of language” (Wille 2005: 6). These units are called formal concepts and given some units may be contained in other units, leads to the notion of a partial order on a context and a hierarchy of concepts.

From our formal context K σ , consider the following two concepts, c 1 and c 2 found in the set of all formal concepts B ( K 1 ) :

We can see that given c 1 contains objects that are a subset of the objects of c 2 , we say that c 1 ≤ c 2 , or that c 1 is a subconcept of c 2 . This is more readily seen through a Hasse diagram found in Figure 5, and Figure 6. Here we can visualize the order (≤) on B ( K n ) that generates B ̲ ( K n ) . We find that the edges of a diagram that connect objects (whose labels are shaded and sit below vertices) to other vertices, some of which contain attributes (whose labels sit above vertices). The concept c 1 has an intent (upwards connecting edges) that we can denote using the derivation operator as:

while for c 2 this is:

given that the vertices where all objects of c 2 meet through upwards paths are the attribute vertices {is a predator} and {is an actant}. Indeed, given that B ̲ ( K σ ) is isomorphic to a complete lattice L (Definition 2.2) through Ganter and Wille’s (1999: 20–21) proof of their basic theorem on concept lattices via the mappings γ ͞ : G → L and μ ͞ : M → L , we see there exists two types of concepts: object concepts and attribute concepts.

Figure 5:

Hasse diagram of B ̲ ( K 1 ) .

Figure 6:

Hasse diagram of B ̲ ( K 2 ) .

There is then some relation between what Hébert (2007: 166) calls monoclassification and polyclassification within his finite rooted trees to object concepts and attribute concepts within a complete concept lattice. Consider the following concepts from B ̲ ( K 2 ) (Figure 6):

What Hébert calls monoclassifications are concepts in the form μ{m}≔(m ^↓ , m ^↓↑ ) that have the smallest (empty) intents and largest extents: e.g., μ{is an actant} in B ̲ ( K 1 ) and μ { is from nature } , μ { is terrestrial } ∈ B ̲ ( K 2 ) . Duly, polyclassifications are any concepts that are subconcepts of these superconcepts such that | A ^↑ | > 1 for a concept in the form (A, B).

In the case of the concepts γ{Newf’ndland}, γ{land}, γ{stone}, γ{sand}, we also have what Hébert (2007: 169) identifies as the isotopy /mineral/ given their intent has the attributes {is terrestrial, is from nature}. Similarly, what Hébert (2007: 169) identifies in << Quelle affaire!>> as the significance of the text “lady walking on the ice floe” as a mediating term to the oppositional semes /land/ and /sea/ is expressed in B ̲ ( K 2 ) through the superconcept/subconcept relation of μ{is from nature} ≤ (G, G ^↑ ), where {land, sea} ∈ γ{is from nature}, and {lady} is an element of the concept with the largest extent (G, G ^↑ ), or the supremum.

2.1.1 Nested lattices

Hébert also identifies in Vigneault’s poem the potential of classification to compare human predation and seal predation, and the parallel “between the recursivity of predations (man hunts the seal who hunts the fish) and the recursivity in the transmission of information (fish A tells B who tells C, ‘and so on down the line’)” (Hébert 2007: 169). In order to examine this aspect within the context of FCA we can firstly look more closely at the relation between the sub-lattices B ̲ ( K 1 ) and B ̲ ( K 2 ) in terms of G, and in respect to the complete lattice formed from K σ . This is achieved this through a tensor product of B ̲ ( K 1 ) and B ̲ ( K 1 ) which Ganter and Wille (1999) proved is isomorphic to a direct product of formal contexts, which is to say more generally that:

(16) B ̲ ∏ t ∈ T K t ≅ ⊗ t ∈ T B ̲ ( K t ) .

Definition 2.9:

(Direct Product): The direct product of two formal contexts, K 1 ≔ ( G 1 , M 1 , I ) and K 2 ≔ ( G 2 , M 2 , I 2 ) produces the formal context K 1 × K 2 ≔ ( G 1 × G 2 , M 1 × M 2 , ▽ ) in which (g ₁, g ₂)▿(m ₁, m ₂): ⇔((g ₁, m ₁) ∈ I ₁ ∧ (g ₂, m ₂) ∈ I ₂).

Definition 2.10:

(Tensor Product): The tensor product of two complete lattices L 1 and L 2 is the formal concept lattice of the direct product of their contexts, that is, L 1 ⊗ L 2 ≔ B ̲ ( L 1 × L 2 , L 1 × L 2 , ▽ ) such that (x ₁, x ₂)▿(y ₁, y ₂): ⇔(x ₁, ≤ y ₁) ∧ (x ₂ ≤ y ₂).

In Figure 7 we give the Hasse diagram of the tensor product of the two complete sublattices of B ̲ ( K σ ) such that the diagram expresses B ̲ ( G , M 1 , I ∩ G × M 1 ) as the outer lattice, for which each vertex contains B ̲ ( G , M 2 , I ∩ G × M 2 ) . In order to show which objects and their associated intents in K 2 intersect with K 1 , we shade these nodes and omit labels (c.f. with Figure 6).

Figure 7:

B ̲ ( K 1 ) ⊗ B ̲ ( K 2 ) .

The resultant tensor product of sublattices gives a clear indication as to the nature of Hébert’s oppositional construct of finite rooted trees based on the nature/culture distinction and the recursivity of predations. We can see this firstly through the fact that the intent (colored vertices) of the object subset {baby seal, seal} (//nature//) in B ̲ ( K 1 ) is part of the extent of the concept μ{preys on fish} in B ̲ ( K 2 ) , and that this concept has the object {fisherman} (//culture//) in its extent. This gives the relation: γ{fisherman} ≤ μ{preys on fish}, which further shows that both γ{fisherman} and γ{baby seal, seal} have the attribute {is a predator} in their intents. We also see from B ̲ ( K 2 ) and the vertices in B ̲ ( K 1 ) ⊗ B ̲ ( K 2 ) described by the attributes {is an actant} and {preys on fish}, that both the objects {baby seal, seal} and the subset of objects that contains the species of fish, {smelt, sole, …, bullhead}, intersect in their intent at {is prey, is an animal, is aquatic, is from nature}. While in B ̲ ( K 2 ) both these object subsets are equal in terms of |{g} ^↑ | (the cardinality of their intents), within B ̲ ( K 1 ) ⊗ B ̲ ( K 2 ) because {smelt, sole, … ,bullhead} ^↑  = {is an actant}, and {baby seal, seal} ^↑  = {preys on fish, is a predator, is an actant}, we can think of γ{baby seal, seal} as more specific, and thus a key theme in << Quelle affaire!>>.

3.1 Preconcept lattices

We can further consider the example of the generalization of formal concepts as preconcepts. These smaller structures are subconcepts, and allow for the identification of the smallest semantic units associated to sentences (line combinations). Burgmann and Wille (2006: 80) first described preconcepts as a means in which to “mathematize the notion of a ‘preconcept’ which is used in Piaget’s cognitive psychology to explain the developmental stage between the stage of senso-motoric intelligence and the stage of operational intelligence.”

Given that preconcepts are smaller units of meaning according to the condition A ⊆ B ^↓ (⇔ A ^↑  ⊇ B), we firstly apply an operator V on K 3 from The golden ship in order to generate all preconcepts. We call V ( K ) ) a derived formal context such that:

Wille (2004: 5) has previously proved that

That is, a complete protoconcept lattice is isomorphic to the lattice of a derived context.

Consider then the derived context V ( K 3 ) we provide in Figure 11. What we will call the atoms of its corresponding preconcept lattice B ̲ ( V ( K 3 ) ) given in Figure 10 are defined more generally as the preconcepts (∅, M \{m}) such that m ∈ M as well as the preconcepts ({g}, M) such that g ∈ G and {g} ^↑  = M. The coatoms are duly the preconcepts (G \{g}, ∅) such that g ∈ G as well as the preconcepts (G, {m}) such that m ∈ M and {m} ^↓  = G (Burgmann and Wille 2006: 81). We call the set of all atoms in a complete lattice J and the set of all coatoms M .

There exist preconcepts that are the images of subsets of J and M because atoms and coatoms are described through set complements. We indicate these images through the condition on a complete lattice L, where a is an element of A ⊆ J ( L ) , and b is an element of B ⊆ M ( L ) such that:

Here, a prototypic object that we denote as ↖ a, is the image of a coatom, or the set complement of a single attribute in M, while a characteristic attribute as ↙ b, is an image of a set complement of a single object in G. Semantically we can think of these elements as the Boolean operations of negation “¬” on a formal concept, or an opposition “⌙” (Wille 2004: 2–3):

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For example, the concept formed from the attribute ‘↙abyss’ in B ̲ ( V ( K 3 ) ) is equivalent to what Greimas and Courtés (1979: 60–61) call contradiction. Here, they describe ¬/abyss//abyss/ as an assignment of the binary present/absent. Regarding ↖-images, we use the example of ‘ ↖afferent seme,’ which is an opposition of the afferent seme type. Here an apposition is equivalent to what Hébert (2020: 146) describes as the virtualization of a seme: that is, an opposition to its actualization.

But from these conditions we also have subpreconcepts which we understand as representing the smallest units of meaning. Consider the first sentence across lines 1–2 of The golden ship in light of Hébert’s isotopy /navigation/:

There was a fine ship, carved from solid gold

With azure reaching masts, on seas unknown.

Let the set R≔coloneq {azure, ship, masts, seas} be a collection of signifieds in G of which we gave in K 3 (Figure 9a), and which we associate to lines 1–2 of the poem. In B ̲ ( V ( K 3 ) ) , there exists a subpreconcept that we label p 1 in Figure 10 such that

R is embedded in p 1 , such that its intent is the subset {actualized seme, ↙abyss, /navigation/}. As the smallest preconcept formed through R, p 1 is a coatom that semantically implies a type of negation of the signified ‘abyss’ in addition to pertaining to the features of an {actualized seme} and the signified as an element of the seme as isotopy /navigation/ (Figure 10). It is a subpreconcept to the preconcepts formed through the attributes {actualized seme}, { ↙abyss}, {/navigation/}. This shows the isotopy /navigation/ contains two large concepts μ{↙abyss} and μ{actualized seme} as features or preconceptual planes (Figure 11).

Figure 9:

Formal context K 3 and corresponding Hasse diagram B ̲ ( K 3 ) of Hébert’s semic analysis of the isotopy /navigation/ from Émile Nelligan’s The golden ship.

Figure 10:

Hasse diagram of the preconcept lattice B ̲ ( V ( K 3 ) ) .

Figure 11:

The derived formal context of ( V ( K 3 ) ).

3.2 Inter- and intra-semiotic isotopies

Hébert (2020: 163) also presents semic analysis through reduced analytical semic tables for tracking isotopies that occur within Réne Magritte’s 1930 oil painting La clef des songes (Key of Dreams). Magritte’s painting (Figure 12) is set out in a 6 × 2 grid in which various commonly encountered objects are depicted with labels that describe other objects that are seemingly unconnected. Hébert utilizes the example as a means to distinguish between the notion of intra-semiotic repetition, that is, when a molecule or semic iteration is for example limited to a text, to that of inter-semiotic repetition, or when a molecule or isotopy can be found activated both within a text and an accompanying image.

Figure 12:

René Magritte La clef des songes (Key of Dreams), oil on canvas, 1930. The work comprises of 6 images (tokens) each accompanied by a word (signifier). Image 1 (top left): egg; word 1: l’Acacia (Acacia). Image 2 (top right): shoe; word 2: la Lune (moon). Image 3 (middle left): hat; word 3: la Neige (snow). Image 4 (middle right): candle; word 4: le Plafond (ceiling). Image 5 (bottom left): glass; word 5: l’Orage (storm). Image 6 (bottom right): hammer; word 6: le Désert (desert).

In order to explore these notions in regards to FCA, we provide a map of Hébert’s semic table to a pair of complete contexts visualized in Figures 13 and 14. These ordered concept lattices demarcate Magritte’s La clef des songes into formal concepts after Hébert’s table for which B ̲ ( K 4 ) maps images (tokens), and B ̲ ( K 5 ) maps words (signifiers).

Figure 13:

Hasse diagram of B ̲ ( K 4 ) .

Figure 14:

Hasse diagram of B ̲ ( K 5 ) .

The Hasse diagrams provide a way in which to quickly locate intra-semiotic isotopies as semic molecules given that according to the definition by Rastier (2016: 499–500), a molecule is simply “a stable grouping of semantic features” that is structured in some way through a collection of semes. As an example consider the following attribute concept in the form (B ^↓ , B) from B ̲ ( K 4 ) (Figure 13):

which we equate to the semic molecule /water protection/ + /black/ + /clothing/ through its tokens, the images of a hat and a shoe in La clef des songes. The intent of these two tokens yields:

Similarly in B ̲ ( K 5 ) (Figure 14) we have:

with the following intent:

as the semic molecule /container/ + /hot/ with signifiers “ceiling” (la Plafond) and “desert” (le Désert). But given we also have object concepts in the form (A, A ^↑ ) such as found in (34) and (36) means that for large formal contexts with large intents, we would ideally like to identify subsets of M that are informed through a collection of primary attributes that are dependent on a secondary attribute(s). This would allow for a stricter definition of Rastier’s notion of the stability of a semic molecule through a reduction in the attribute set via a distinction between μ{m} concepts that contain equivalences (i.e., m≅m′ | (m, m′) ∈ M), and γ{g} concepts whose intents may not readily describe complete semic molecules – i.e., there may be other tokens or signifiers that satisfy parts of the attribute subset.

3.3 Attribute dependency formulas

In order to explore inter-semiotic isotopies between B ̲ ( K 4 ) and B ̲ ( K 5 ) we introduce Bĕlohlévek and Sklenár̆’s (2005) attribute-dependency formulas as a method for rule-based attribute reduction and concept exploration within each lattice.

Consider the set C ≔ { φ 1 , φ 2 } as a subset of all AFDs, in which we associate φ ₁ to B ̲ ( K 4 ) , and φ ₂ to B ̲ ( K 5 ) such that:

Here, we will to focus on the seme /white/ and the nominated dependent semes as they associate to the tokens (objects) {Image 1 (egg)} and {Image 4 (candle)} in B ̲ ( K 4 ) as well as signifiers {Word 3 ‘snow’ la Neige} and {Word 2 ‘moon’ la Lune} in B ̲ ( K 5 ) . As Bĕlohlévek and Sklenár̆ (2005: 180) note, in general, we can consider expressions within ADFs as terms over B in the form y ⊑ t(y ₁, …y _n ). Here, we designate each attribute m ∈ M as a term, in addition to identifying terms through Booleans such as (t ₁ ⊔ t ₂). For example, we see in (38) that (/solid/ ⊓/familiar/) can be expressed as/concrete/(/solid/ ⊓/familiar/) given {/solid/, /familiar/} ^↑  = {/concrete/}.

Because the seme (attribute) /white/ is common to both concept lattices, we use the ADFs above in order to construct a dependency, and thus seek out concepts in both lattices for which given the seme /white/, we also expect in φ ₁ the molecule /concrete/(/liquid/ ⊓/damageable/) as well as the molecule /concrete/(/solid/ ⊓/familiar/) in φ ₂. We can then express the ADFs as the finite rooted trees T(φ ₁):and T(φ ₂):

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We denote the formal contexts built from these trees as K φ 1 and K φ 2 whose attributes are subsets of K 4 + K 5 . Here we mean that for each term t in C there exists a corresponding attribute m in (M ₄ ∪ M ₅). In order to model concepts that are associated to both tokens and signifiers in Key to Dreams, we glue these two subcontexts through a subposition.

In Figure 15 we give the clarified context K θ C of the subposition of contexts K φ 1 and  K φ 2 as well as its corresponding concept lattice. The lattice shows both semic molecules that are intra-semiotic and inter-semiotic in Key to Dreams. Consider the following intra-semiotic concept within the set of signifiers in the form (A, B):

and its corresponding intra-semiotic concept within the set of tokens:

Figure 15:

Formal context K θ C and corresponding Hasse diagram B ̲ ( K θ C ) .

These two intra-semiotic isotopies are themselves subconcepts to formal concepts formed at the nodes labelled with the attributes {/inanimate/} and {/damageable/, /liquid/} in B ̲ ( K θ C ) :

The formal concepts c 3 and c 4 are thus two inter-semiotic molecules as superconcepts ( c 1 ≤ c 3 , and c 2 ≤ c 4 ) that cover a subset of signifiers and tokens in Key to Dreams. These two molecules have a greatest lower bound as the subconcept c 0 (Figure 15):

which we contend is the smallest and most generalized expression of the inter-semiotic molecule in Key to Dreams obtained through an ADF restriction on the seme {/white/}.

3.4 Attribute implications

The use of ADFs to assist in constructing subconcept/superconcept relations between molecules also points towards how we can parse implications from a concept lattice in order to establish /seme/ → /seme/ structures that define a semic molecule’s structure.

Although Hébert does not discuss implications directly within Rastier’s framework of interpretive semiotics, Greimas and Courtés (1979: 152) equate implications to presuppositions. That is, they are envisaged as “logically prior to implication: the ‘if’ would not find its ‘then,’ if the latter did not already exist as the presupposed.” Within the framework of FCA, implications are element/subset relations that emerge from the intent of the formal contexts.

For example, in the concept lattice B ̲ ( K θ C ) (Figure 15), the two inter-semiotic molecules we identified as c 3 and c 4 contain the following attribute (seme) implications respectively:

such that both sets of implications also imply {/white/}. But we can further cross-reference these implications as they pertain more generally to the lattices B ̲ ( K 4 ) (Figure 13) and B ̲ ( K 5 ) (Figure 14) to find further implications such as:

We also find from our earlier example in Equation (33) of the semic molecule /water protection/ + /black/ + /clothing/ from the tokens {Image 3 (hat), Image 2 (shoe)} that: