2.1 General features

As a structural-functional model of language, FDG aims at meeting the three standards of psychological, pragmatic and typological adequacy set by Dik (1997: 12–14). In keeping with these requirements, FDG presents itself as “a typologically-based theory of language structure” and, more specifically, as the Grammatical Component of a wider theory of verbal interaction, “systematically link[ing] up with a Conceptual, a Contextual, and an Output Component” (Hengeveld & Mackenzie 2008: 1). The overall model is thus designed to be compatible with psycholinguistic evidence to the extent that, following Levelt (1989), language production is understood as “a top-down process, which starts with intentions and ends with the articulation of the actual linguistic expression” (Hengeveld & Mackenzie 2008: 2). In addition, the concern with pragmatic adequacy is embodied in the close interaction between the Grammatical and the Contextual Component, the latter being viewed “as a companion to the Grammatical Component, collaborating with it to achieve contextually appropriate outputs” (Hengeveld & Mackenzie 2014: 204). The general architecture of FDG is represented in Figure 1, where the model is presented from the viewpoint of language production. It should be stressed, however, that this is a merely conventional choice: as pointed out by Hengeveld & Mackenzie (2008: 2), “the model could in principle be turned on its head to account for the parsing of utterances. It is clear that listeners analyse phonetic input into phonological representations, which are subsequently grouped into morphosyntactic constituents, from which meaningful representations are then constructed”. For an actual implementation of this idea see Giomi (2014).

Figure 1

General layout of Functional Discourse Grammar (Hengeveld & Mackenzie 2008: 13).

As shown in Figure 1, the communicative intention developed in the Conceptual Component triggers the grammatical operation of Formulation, which is responsible for structuring the interactional and ideational aspects of that communicative intention into two separate levels of grammatical analysis: the Interpersonal and the Representational Level. These two levels of analysis, which deal with pragmatics and semantics respectively, “describe language in terms of its functions and meanings, but only in so far as these functions and meanings are encoded in the grammar of a language” (Hengeveld & Mackenzie 2008: 5). This dualist “form-oriented, function-to-form” approach is reflected in the organization of the Grammatical Component in the fact that the pragmatic and semantic structures created by Formulation are in turn translated into morphosyntactic and phonological structures through the operations of Morphosyntactic and Phonological Encoding, each of which organizes the input received from all of the higher levels into a separate level of formal expression: the Morphosyntactic and the Phonological Level. Finally, the phonological structure elaborated by Phonological Encoding on the basis of the information passed on from all of the higher levels serves as the input for the post-grammatical operation of Articulation, which takes place in the Output Component: the role of Articulation is thus to translate the phonological representation passed on from the Grammatical Component into acoustically perceivable signals.

Each level of the Grammatical Component makes use of a separate set of language-specific primitives such as (e.g.) lexemes for the Formulation levels and morphosyntactic and phonological templates for the Encoding Levels. These primitives are the “building blocks” of the grammar (Hengeveld & Mackenzie 2008: 19) and are retrieved from a dedicated storehouse referred to as the Fund. The Fund is organized in such a way that, for each language, each contentful element of the Formulation levels is stored in connection with the corresponding expression format at the Encoding levels. For instance, illocutionary values such as Declarative and Interrogative will be stored as abstract placeholders DECL(arative) and INTER(rogative) in the subsection of the Fund which hosts the primitives used at the Interpersonal Level: in English, these two abstract interpersonal primitives are connected to two separate templates at the Morphosyntactic Level (characterized by the occurrence of the subject phrase before or after the finite verb) and to two distinct prosodic patterns at the Phonological Level (characterized by falling and rising intonation, respectively). Other languages, however, only distinguish declaratives and yes/no questions by means of different intonational contours (e.g. most Romance languages), and not by morphosyntactic means as in English or Japanese (where a specific particle is used in interrogative utterances). This means that in the former type of languages only the Interpersonal and the Phonological Level are called upon for the expression of Declarative and Interrogative illocution, whereas in English and Japanese Morphosyntactic Encoding is also involved.

2.2 Layered structure

The common feature of all four levels of grammatical analysis is that they share “a hierarchically ordered layered organization and are displayed as a layered structure” (Hengeveld & Mackenzie 2008: 14). This is illustrated in (1)–(2) for the Interpersonal and the Representational Level, respectively – that is, the two levels of the grammar which deal with the content of linguistic utterances:

(1)	General structure of the Interpersonal Level (adapted from Hengeveld & Mackenzie 2008: 49)

(2)	General structure of the Representational Level (adapted from Hengeveld & Mackenzie 2008: 142)

Each layer of each level is distinguished by its own variable and is labelled with reference to its specific functional or formal properties (for the Formulation and the Encoding levels, respectively). In general terms, the maximal form of each layer/variable can be represented as in (3) (Hengeveld & Mackenzie 2006: 671), where v is the general symbol for “any variable” (cf. also Smit and van Staden 2007: 144; Hengeveld & Mackenzie 2008: 14; Keizer 2015: 32):

Each variable must bear an index so as to distinguish it from other occurrences of the same type of variable: this index is given in numerical form in general representations and as a letter of the Latin alphabet (starting with i) in the analysis of concrete expressions. Further elements introduced in (1)–(3) are the colon (:), indicating a relation of restriction, and the general symbols ♦ for lexemes, π for operators, σ for modifiers and φ for functions. On the whole, the general format in (3) is to be read as follows:

v₁ represents the variable of the relevant layer, which is restricted by a (possibly complex) head that takes the variable as its argument, and may be further restricted by a modifier σ that takes the variable as its argument.¹ The layer may be specified by an operator π and carry a function ϕ. Heads and modifiers represent lexical strategies, while operators and functions represent grammatical strategies. The difference between operators and functions is that the latter are relational, holding between the entire unit and other units at the same layer, while the former are not, applying only to the unit itself. (Hengeveld & Mackenzie 2008: 14)

As an example of a fully developed layer, making use of all the elements in (3), consider the Representational Level analysis of the NP the young girl in (4):

In this representation, the young girl is analyzed as denoting a concrete, “first-order” entity of the class Individual (variable symbol x), whose head is restricted by a Lexical Property girl (variable symbol f). Lexical Properties denote “zero-order” entities, that is, intangible properties that only exist in relation to specific referents: these referents may be concrete, as in this case, but also eventive “second-order” entities such as States-of-Affairs or Episodes (described as “thematically coherent combinations of States-of-Affairs that are characterized by unity or continuity of Time (t), Location (l), and Individuals (x)”, Hengeveld & Mackenzie 2008: 133) or more abstract “third-order” entities such as Propositional Contents.² The (x_i) variable in (4) is thus to be read as “an Individual (x_i) such that (x_i) has the Property (f_i) of being a girl”; this Individual is further restricted by a modifier young, which is again analyzed as a Lexical Property providing further information on the entity designated by the variable. All lexemes inserted at the Representational Level are analyzed as Lexical Properties,³ the specific uses to which a given lexeme can be put being restricted by the inventory of representational frames available for the level in question in a given language (see again Figure 1).

Returning to the representation of the Individual (x_i) in (4), note that this variable is further specified by an operator 1, i.e. Singular, as expressed in English by the lack of plural inflection on the head-noun girl. Finally, the whole unit is assigned a semantic function Actor (represented by the subscript A on the closing variable): this is because the State-of-Affairs described in (4) is a dynamic one, that is, one in which the action designated by the predicate requires an input of energy on the part of one of the participants. Given the general layout of the Representational Level presented in (2), a fully developed analysis of the utterance in (4) can be formalized as shown in (5), where the predicate dance (f_j) and its argument the young girl (x_i) form a Configurational Property (f_i^c) – that is, a nuclear predication frame divested of any kind of “situatedness” in relation to the real world and of any characterization in terms of the speaker’s attitudes. Note further that the whole Episode (ep_i) is assigned an absolute tense operator pres(ent).

3.1 Types of head

In (4) I gave an example of a lexical head, that is, of a Lexical Property filling the head of another variable at the Representational Level. This, however, is not the only possible type of head. To start with, the head of a certain variable may be occupied by one or more abstract features, which will trigger a specific form at the lower levels. I have already mentioned the case of abstract features such as DECL(arative) or INTER(rogative) for the Illocution layer of the Interpersonal Level; another example is that of the features ±S(peaker) and ±A(ddressee) for the interpersonal layer of Subacts of Reference. A Subact of Reference (variable symbol R) is “an attempt by the Speaker to evoke a referent, i.e. a null, singleton, or multiple set of entities or qualities” (Hengeveld & Mackenzie 2008: 88). By specifying the head of a Subact of Reference as [+S, –A] the speaker indicates that the representational unit corresponding to that Subact denotes (a set of entities including or deictically related to) the speaker herself or himself: this will trigger first person marking at the Morphosyntactic Level, if the referent in question is an Individual, or a deictic pro-adverb such as here if the referent is an entity of the type Location. Conversely, by specifying a Subact of Reference as [–S, +A], the speaker indicates that the representational unit corresponding to that Subact denotes (a set of entities including or deictically related to) the addressee, which will trigger second person marking (or a pro-adverb such as there, to be interpreted as ‘there where you are’). Referential Subacts may also be specified as [+S, +A], in which case a first person plural (or dual) inclusive form will be used; finally, a Subact of Reference which does not coincide with or include either of the speech participants may be specified as [–S, –A], triggering third person marking (or, again, a pro-adverb such as there, this time in the sense ‘there where neither of us is’). Number distinctions, as we saw above, are not a matter for the Interpersonal Level but will be captured at the Representational Level. Thus, whenever a language distinguishes between singular and plural marking in its pronominal and/or agreement system, we need to make reference to both levels of Formulation in analyzing the relevant forms. Some of the possible combinations are illustrated in (6) for first, second and (deictic)⁴ third person pronouns (adapted from Hengeveld & Mackenzie 2008: 118), where IL = Interpersonal Level and RL = Representational Level; in the right column, the operators 1 and m distinguish singular from plural number. Note that with first and second person forms the semantic category designated at the Representational Level is represented as an Individual (x₁) (since speech participants are by necessity humans, except of course in fantasy fiction or when the speaker rhetorically addresses a non-human entity); third person pronouns, by contrast, may denote entities of various types, whence the use of the general variable symbol (v₁).

As can be deduced from the Interpersonal Level analysis of personal pronouns in (6), the general frame underlying any variable headed by an abstract feature will take the following form:

The above example also serves to illustrate a further possible type of head – or, more accurately, the possibility of absent heads, as exemplified in the rightmost column of (6). Abstracting away from the possible, but by no means obligatory presence of operators, the general frame in this case is the simplest possible:

As we will see in Section 4.4, this configuration represents the default scenario for certain layers of the Interpersonal Level. At the Representational Level, however, it only obtains when the variable in question is expressed (i) as a proper name (since proper names are analyzed as heads of Referential Subacts at the Interpersonal Level) or (ii) as a proform, for instance a deictic personal pronoun as in (6), a phoric pronoun of whatever type or a pro-adverb like here or there. Clearly, in all these cases the range of possible referents of the proform is never restricted by lexical means: accordingly, the head of the variable corresponding to the proform is not occupied by a Lexical Property, as in the case of the Individual girl in (4), but is simply left unspecified.

Distinct from the absent head is the possibility of a variable having an empty head. This only happens with a specific type of proform, namely, one that anaphorically retrieves a Lexical Property that has been evoked in the preceding discourse: an example is English one, as used in (9):

At the Representational Level, such anaphoras can be analyzed as follows:

While the two Individuals (x_i) and (x_j) bear different indexes, reflecting the fact that they designate two separate entities, the Lexical Property (f_i) in the head of (x_j) is co-indexed with that in the head of (x_i) but differs from it in not being lexically specified: it is precisely this circumstance that triggers the anaphoric proform one. If the lexeme man had been selected from the Fund a second time and inserted into the head of (x_j), the proform would not have been used and we would have had a lexically headed NP an honest man. In general terms, the difference between lexically headed and empty-headed variables can be formulated as in (11)–(12):

Last but not least, the head of a variable may be configurational. This accounts for the possibility of complex heads consisting of a configuration of equipollent units. The prime example of an equipollence relation, at the Representational Level, is the relation between a predicate and its argument(s), each of which bears a different semantic function. As an example, consider the complex NP the brother of the landlord, as analyzed in (13):

In FDG, relational nouns like brother are analyzed as one-place predicates taking the genitive-marked complement as their argument: following Mackenzie (1983), this argument is assigned the semantic function Reference (cf. the paraphrase “(x_i) is brother with reference to (x_j)”, see Hengeveld & Mackenzie 2008: 203). Note that, like dance and the young girl in (5), the predicate brother and its argument the landlord are enclosed within square brackets, indicating that their relation is one of equipollence (as opposed to round brackets, which indicate hierarchically asymmetric relations, such as those between a variable and its head or modifier(s)). These two units together constitute the head of a Configurational Property (f_i^c), which in turn is the head of the Individual (x_i) (the brother of the landlord). This Individual can therefore also be said to have a configurational head, since the variable (f_i^c) that restricts its denotation is by definition one that consists of a configuration of equipollent units. This means that a configurationally headed variable will be built on one of the two general frames in (14a–b):

Summing up, the following types of heads are distinguished in FDG:

3.2 Modification in Functional Discourse Grammar

As mentioned in Section 2.2, in FDG modifiers are characterized as a lexical strategy for specifying additional information about a certain unit. In this sense, modifiers may be regarded as the lexical counterpart of operators. This is reflected in the fact that each layer is provided with its own, distinct set of operators and modifiers. For instance, the layer of Configurational Properties has phasal aspect and participant-oriented modality among its operators: the former category relates to the internal temporal structure of a State-of-Affairs (Progressive aspect, for instance, indicates that the situation is ongoing at the temporal reference point), whereas participant-internal modality (e.g. Intention or Ability) concerns “the relation between a participant in a State-of-Affairs and the potential realization of that State-of-Affairs” (Hengeveld & Mackenzie 2008: 212). Thus, lexical modifiers specifying duration (e.g. for a long time) represent the lexical counterpart of Progressive aspect operators, and lexical modifiers such as intentionally, involuntarily, reluctantly or easily, with some effort, with great difficulty correspond to the participant-oriented modality operators Intention and Ability, respectively. As is easily seen, modifiers typically express much more specific qualifications of the unit in their scope than are provided by grammatical operators.

It should be stressed that the FDG approach to modification is radically different from that entertained by strictly formal models such as Generative Grammar, where modification is accounted for, alternatively, in terms of adjunction or specification, and often with reference to the concept of syntactic movement (see Alexiadou 2013 for an overview). Note that, as far as the terminology is concerned, a conception of modification as a syntactic operation would also appear to surface with a certain frequency in various functionally-oriented frameworks such as Construction Grammar (e.g. Norde et al. 2014) and Davidse & Breban’s (2019) “cognitive-functional approach to the order of adjectives in the English noun phrase”, and even in work rooted in the F(D)G tradition: in all these frameworks, reference is often made to, e.g., the modification of a noun, verb or adjective (phrase).⁵ Similar notions are also invoked in authoritative works in formal semantics. For instance, McNally (2016: 442) begins her chapter on modification by defining the notion of “modifier” in semantic terms, as something that “adds additional, non-essential descriptive content to that contributed by the expression that it combines with”; however, in the conclusions of the chapter she states that “[t]he definition of a modifier as a word or phrase that combines with an expression to yield another of the same semantic type is perhaps the best definition currently available” (2016: 463; emphasis mine).

All such terms and definitions, which more or less explicitly allude to the syntactic nature of modification, are strictly speaking inaccurate from an FDG perspective. On the latter approach, in fact, modification is regarded as a strictly functional notion and as such is only relevant to the Levels of Formulation: at these levels, modification represents a communicative strategy to supply lexical information about a given pragmatic or semantic unit that is not already specified within the head of that unit. This means that the morphosyntactic constituents expressing the lexical information in question should in no case be thought of as modifiers but such constituents (whatever the morphosyntactic category and layer to which they belong) are regarded as merely formal devices for the encoding of interpersonal or representational modifiers. Once this all-important theoretical matter is adequately clarified, one may of course choose to retain such terms as, e.g., “noun (phrase) modification” as a useful shorthand. It is essential to bear in mind, however, that it is never a noun or noun phrase that is modified by an adjective or adjectival phrase: rather, a nominally-headed variable of the Interpersonal or the Representational Level may be modified by lexical means at the relevant level of functional analysis, in which case the modifier in question will surface at the Morphosyntactic Level as a morpheme, word, phrase or clause of the appropriate type, depending on the underlying unit of Formulation and on typological features of the language.

3.3 The relation between heads and modifiers

As we saw in Section 2.2, in FDG both heads and modifiers are characterized, in quite general terms, as restrictors. On a narrow interpretation, this means that both heads and modifiers restrict the denotation of the variable to which they apply, i.e. each restrictor “narrows down the set of potential referents” of the variable (Dik 1997: 133). This is in fact the conception of heads and modifiers enunciated by Dik’s Functional Grammar (FG), where heads and modifiers were referred to as first and second restrictors (or primary and secondary restrictors), respectively. In this way, the difference between heads and modifiers is not characterized as an ontological one (like that between modifiers and operators) but as a difference in the order in which each restrictor is applied to the variable in the dynamic construction of meaning. That is, in an expression like old elephant, “restrictors are successively “stacked” onto each other through the relation “:”, rather than being conjoined with each other” (Dik 1997: 133). It is precisely in the recognition of this dynamic mechanism that the F(D)G approach to the semantics of referential expressions differs from the one usually invoked in logic and truth-conditional semantics, “where expressions of this type are analysed in terms of conjunctions of predicates”, so that the referent of the overall expression is understood as an intersection of the sets denoted by the two attributes elephant and old.⁶ These two approaches can be represented as in (16a) and (16b), respectively (adapted from Dik 1997: 132–133):

Drawing on the arguments presented in Dahl (1971), Dik explains the difference between these two types of analysis by observing that “[i]f an expression such as pregnant women were to be analysed in terms of the intersection of two sets, the following paraphrases should represent its meaning equally well”:

However, Dik observes, “[17b] is redundant in a way in which [17a], and the expression pregnant women, are not”. This indicates that the way in which the meaning of referential expressions consisting of a head and a modifier is construed in natural languages is in fact quite different from a logic of conjoined predicates. Whereas the latter consists in identifying two sets and then picking up one or more referents that belong to both sets, “the normal way of defining a set in natural language is to choose a universe and a defining property which singles out a set within that universe” (Dahl 1971: 2, cited in Dik 1997: 134). This difference can also be represented in form of diagrams: while in a logical analysis the expressions Buddhist Japanese and Japanese Buddhist simply have the same designation (Figure 2), an F(D)G-style analysis in terms of stacked restrictors will represent the way in which the designation of the two expressions is dynamically construed, as shown in Figure 3 (where the larger sets represent the denotation of the head and the smaller subsets are identified by adding a modifier).

Figure 2

Logical analysis: designation of both Buddhist Japanese and Japanese Buddhist (Dik 1997: 135).

Figure 3

Functional (Discourse) Grammar analysis: dynamic construal of designation for Buddhist Japanese and Japanese Buddhist (Dik 1997: 135).

This example shows that, depending on the way in which lexeme classes are organized in each individual language, the same lexical expressions may in principle function either as heads or as modifiers: the difference between Buddhist Japanese and Japanese Buddhist, thus, does not consist in the extension set of each expression (which is ultimately the same, as shown in Figure 2) but only in the order in which each restrictor is applied to the variable in the dynamic construal of meaning.

The F(D)G-style representation of heads and modifiers as consecutively stacked restrictors has been criticized by González Escribano (2008: 131–133), mainly on the argument that once a variable has been restricted by a first (lexical) specification, its denotation will no longer be exactly the same as when the variable was first introduced in formal representation – that is, before the application of the first restrictor. As a consequence, he argues, “each successive occurrence of the “x” variable stands for a different (gradually decreasing) set of entities, i.e., it is a different variable” (2008: 131); in turn, this implies that each further restrictor will never apply to exactly the same variable as that to which the preceding restrictors apply. In my view, this criticism is ultimately motivated by González Escribano’s retention of a logical-semantic (rather than linguistic-semantic) approach to designation. This becomes particularly evident where he states that Dik’s analysis of the contrast Japanese Buddhist vs. Buddhist Japanese is based “on the extension of the respective sets involved” (2008: 167): as explained above, however, what Dik is concerned with is not the actual extension of linguistic expressions as such but the linguistic process of “choos[ing] a universe and a defining property which singles out a set within that universe”. Also recall that, as stressed in Note 1, the reason why FDG makes use of closing variables is that the variable itself can be understood as an argument of each head or modifier that may be specified for it: one corollary of this conception of the relation between a variable and its head and modifier(s) is that, although each restrictor represented to the right of a colon takes scope on all that occurs to its left (i.e. the variable as a whole), heads and modifiers are actually seen as applying to the relevant variable independently, so that in the end there is no direct relation between the head of a variable and any modifier that may be attached to it.⁷ On this approach, there is no reason why each occurrence of a given variable in the FDG metalanguage should be distinguished from the preceding occurrence(s) of the same variable (e.g. by providing each occurrence of the variable in question with a different index), as would follow from González Escribano’s argument. As we will see in the next section, this does not mean that the current FDG approach to headedness and modification is immune to any sort of theoretical or analytical problems: just, these problems are not of the type of those pointed out by González Escribano.

4.1 The problem

The view of the head/modifier distinction summarized above was originally developed in FG to account for the denotation of head+modifier expressions, and is essentially maintained as such in current FDG. Denotation being a strictly semantic notion, as far as FDG is concerned the conception of heads and modifiers as first and second restrictors most straightforwardly applies to the Representational Level: as we will see shortly, however, even at this level such a view of the head/modifier distinction is not entirely satisfactory. But before turning to the shortcomings of this view of heads and modifiers, it must be stressed that the same type of formal representation illustrated above is also used in FDG for the Interpersonal Level, which is designed to capture the interactional aspects of linguistic communication and as such does not deal with purely semantic matters such as denotation. Why, then, should a formalism which explicitly deals with denotation restriction be applied to the Interpersonal Level? The key to understanding this is provided by a somewhat broader definition of variable structure than has been described above. This is in fact suggested by Dik himself where he points out that, in a notational format such as (3), repeated here as (18), “[t]he colon “:” indicates that the information to the right gives a specification of, a restriction on, the possible values of [the variable], as it has been specified to the left” (1997: 132).

Note that in this definition there is no explicit reference to denotation as such but only to a specification (or restriction) of the possible values of a variable. This view of the kind of relation indicated by the colon is neutral between denotation (i.e. semantics) and the interactional function of linguistic expressions (i.e. pragmatics, in the sense relevant to FDG’s Interpersonal Level). In this way, the colon simply indicates that the element to its right provides further information about the variable with respect to the element(s) to its left: that is, the head expands on the information indicated by the variable symbol (which already constitutes an initial categorization of the variable itself), and a modifier expands on the information indicated by the variable symbol, plus the head, plus any other, narrower-scope modifier that the expression may contain.

Now, the real problem with a definition of the head/modifier opposition as first vs. second restrictors is that it entails that it is impossible to modify a variable which does not have a head. This is because, in the absence of a head, a modifier would become the first restrictor, hence the head itself (Evelien Keizer, p.c.): accordingly, Keizer (2012: 403) claims that “it is not possible to modify Individual-designating units with an absent head”. This is also the position held by Hengeveld & Mackenzie (2008: 237–238), according to whom “[w]here the head is empty, it follows that no modification is possible”. The authors justify this assumption by observing that any qualification of an entity referred to by means of a proper name “can only be of the interpersonal type”, as for instance the adjective poor in (19); otherwise, the qualification can only be non-restrictive, as with the apposition in (20a) and the relative clause in (20b). This is because, as mentioned in Section 3.1, proper names are inserted at the Interpersonal Level, and therefore a NP headed by a proper name corresponds to a headless Individual at the Representational Level.

In actual practice, however, it is also possible to modify a proper name by means of an expression which is to all intents and purposes restrictive. Compare (21)–(22), where restrictive modification applies to a NP denoting a person and a place, respectively, both of which are syntactically headed by a proper name and thus correspond to headless variables at the Representational Level:

Other modifiers that may be attached to a NP headed by a proper name, such as wailing in (21), are not strictly speaking restrictive; their meaning, however, is still clearly representational and not interpersonal. Further examples of non-restrictive modification at the Representational Level are the NPs in boldface in (23)–(24):

The very same problem emerges with proforms, which, just like NPs headed by a personal name, are analyzed as absent-headed variables at the Representational Level (see Section 3.1). Nevertheless, many languages allow (certain types of) proforms to be modified by a restrictive relative clause. This is shown in (25a–d) for English third person pronouns, including the use of distal demonstrative pronouns exemplified by (25d), which is neither deictic nor anaphoric:⁹

4.2 Headedness and modification at the Representational Level

4.3 A revised approach to the modification of headless variables

The representations in (35)–(40) correctly capture the fact that the lexical, phrasal or clausal modifiers in the corresponding constructions apply to a variable whose head is left unspecified. All these representations, however, run into the notational problem that the number of opening and closing brackets for the overall variable does not add up (there are too many closing brackets).¹⁷ In theory, a possible solution could be to assume that the representation of absent-headed variables as (v₁) should actually be understood as a simplified form of the representation in (41):

where the head of the variable is in fact not absent, but is restricted by a null element. This would make it possible to represent the modification of such variables as shown in (42):

This, however, would be an ad hoc solution explicitly designed to solve a notational problem, which does not actually add any descriptive accuracy to FDG’s account of the relation between headedness and modification. It is clear, in fact, that for a variable to be restricted by a null head or not to be restricted by any head at all is exactly the same thing from a notional point of view.

Once the assumption that only headed variables can be modified is rejected, however, the problem does indeed become a merely notational one, and as such can be solved by introducing a minimal adjustment to the formalism: namely, whenever a representational or interpersonal variable takes a modifier but there is no grammatical evidence that a head of any type should be present in underlying representation, a second opening bracket will be added before the first occurrence of the modified variable, so as to show that this variable and its modifier belong together at the relevant level of Formulation. As a result, any modification of a headless variable will be accounted for as an instantiation of the general representational frame in (43):

Thus, the English constructions with a proper name and a representational adjective in (21)–(24) will all be derived from the frame in (44a), and the English and Yucatec Maya constructions with a pronoun and a relative clauses (i.e. (25a–d) and (31)–(33)) will be derived from the frame in (44b).

Regardless of the restrictive or non-restrictive nature of the modifier, the nominal expressions in (21)–(25) and (31)–(33), repeated below, will all be derived from one of these two general frames, and can thus be analyzed at the Representational Level as shown in (45)–(52):

As regards the fact that proper names are not available for modification by a restrictive relative clause, this does not depend on the fact that the corresponding representational variable does not have a head but naturally follows from the discourse-pragmatic status of the referent. In fact, the raison d’être of restrictive relative clauses is that they are required in order to help the addressee identify a referent; entities referred to by means of proper names, however, are by definition uniquely identifiable: it is therefore entirely to be expected that proper names should be incompatible with restrictive relative clauses (unlike the personal and demonstrative pronouns in (49)–(52), whose referents are not uniquely identifiable from the start).²⁰

Finally, it must be remarked that the possibility of headless variable modification is not limited to (pro)nominal expressions, as might be inferred from the foregoing discussion. First, I pointed out above that pro-adverbs like here or there may also be modified at the Representational Level, as in the case of appositional or relative structures such as here in Portugal and here where I live. Applying the same formalism as I have used in (45)–(52) for the modification of absent-headed (pro)nominal expressions, such constructions may be represented as in (53a–b) (where (l_i) = here and (l_j) = Portugal):

Furthermore, verbal proforms like do, do so or do it are liable to modification by the same types of adverbs as apply to lexical predicates. Thus, if a verbal predicate is retrieved anaphorically by means of one of these proforms, nothing prevents the proform in question from being modified by a manner adverb that may as well apply to the retrieved predicate, as for instance in The young girl dances: she does it beautifully. The same analysis applies to (54), where the lexical predicate act is retrieved by the pro-predicate do so and this is modified by the manner adverb effectively:

Once again, the modification of verbal proforms is a case of headless-variable modification, since, like any other proform, pro-predicates have no lexical content whatsoever, nor may they be argued to be headed by any type of abstract feature. Accordingly, expressions like does it beautifully or do so effectively will be represented as in (55), where the absent-headed Lexical Property (f_i) corresponds to the proform and the lexically specified adjectival Property (f_j) is attached to the former in modifier function:²²

Last but not least, note that the case of verbal pro-predicates is not the only one in which an absent-headed Lexical Property may be modified. Among the adjectival lexemes that may be inserted at the Representational Level, there is a subset of adjectives which can be used to modify the nominal head of a higher-layer variable of whatever type: the specific function of such adjectives is that of restricting the extent or the domain in which the modified nominal Property applies to the variable it heads. Compare for instance the adjectives rich and former, as used in (56)–(57) (adapted from Hengeveld & Mackenzie 2008: 220). In (56), rich serves as a qualification of the Individual the neighbour, not of the Lexical Property neighbour (that is, one may be rich as a person, but not in one’s quality of being a neighbour); in (57), by contrast, former only modifies the Lexical Property neighbour, without affecting the Individual as such (that is, the person in question may no longer be a neighbour, but will in any case still be an individual!):

As noted by Hengeveld & Mackenzie (2008: 254), a distinctive feature of adjectives that modify nominal Properties is that they may not be used in predicative position (cf. *The neighbour is former), whereas this is perfectly possible with other types of adjectives (e.g. The neighbour is rich). Similarly, in the Location-designating expression the main road and the Proposition-designating expression linguistic theory, the adjectival Properties main and linguistic do not modify the Location or Propositional Content as a whole but only the nominal Properties in the head of those expressions (cf. *The road is main, *This theory is linguistic). Once it is established that the adjectives former, main and linguistic function as modifiers of the Lexical Property in the nominal expressions considered here, it may be asked whether such adjectives may also be combined with Lexical Properties whose head is left unspecified. The answer is clearly affirmative. Recall, in fact, that nominal Lexical Properties may be retrieved by such anaphoric proforms as English one (see (9)–(10) in Section 3.1): it is thus immediately evident that in a sentence like (58) the adjectival Property former only modifies the retrieved nominal Property neighbour:

As shown in this representation, such structures constitute yet another case of headless variable modification at the Representational Level.

4.4 Headedness and modification at the Interpersonal Level

At the Interpersonal Level, the modification of headless variables is maximally relevant to the layer of Subacts of Ascription. Like any other layer of the Interpersonal Level, Subacts of Ascription are defined in interactional terms, more specifically, as attempts made by the speaker to ascribe a given property – as opposed to Subacts of Reference, which, as explained in Section 3.1, are understood as attempts to evoke a referent. In the default case, a property will be ascribed to a referent (of whatever type):²³ for instance, a simple nominal expression like the horse will be analyzed at the Interpersonal Level as a Subact of Reference whose head is restricted by a Subact of Ascription (that is, a referent is evoked to which the property horse is ascribed). This analysis is formalized in (59a), where each Subact is explicitly connected to the corresponding representational unit (operators of both levels are omitted for ease of reading). Similarly, if more than one property is ascribed to the same referent, the head of the corresponding Subact of Reference will be displayed as a configuration of equipollent Subacts of Ascription, as shown in (59b) for the NP the white horse: this reflects the fact that the different statuses of the properties horse and white is not a matter for the Interpersonal Level but only emerges at the Representational Level, where the Lexical Property white is analyzed as a modifier of the Individual whose head is restricted by the Property horse:

(59)	a.	the horse
	b.	the white horse

As is evident in these representations, the head of a Subact of Ascription is by default absent: this is because the semantic category to which each Subact corresponds can only be specified at the Representational Level, since this aspect of meaning concerns the descriptive, not the interactional content of the expression. Yet, modification of a Subact of Ascription is a very common lexical operation. Typical modifiers of this layer are interpersonal adverbs or adjectives with emphatic, reportative or evaluative meaning such as those exemplified in (60a–d):

To obviate the problem posited by such expressions, Hengeveld & Mackenzie depart from FDG’s usual avoidance of null categories (see Section 1) and suggest that the modified Subact of Ascription be provided with a null head, represented graphically as “[ ]”:

It is clear, however, that this representation is merely a variant of the ø-strategy considered in (41)–(42): as argued above, such analyses may avoid the notational inconvenient stemming from the assumption that headless variables are not liable to any kind of modification, but, in actual fact, they do not help solve the theoretical problem inherent in that assumption. Now, as pointed out in Section 4.1, the assumption in question is a direct consequence of the fact that in the F(D)G tradition heads and modifiers are conceived of as first and second restrictors respectively, so that the absence of a first restrictor logically entails that no second restrictor may be present either. This approach to headedness and modification, however, is specifically designed to account for the mechanisms whereby the denotation of linguistic expressions may be restricted by lexical means: there is therefore no reason why the same approach should be extended to situations to which the very notion of denotation is irrelevant – that is, first and foremost, at the Interpersonal Level. It follows that rejecting the assumption that headless variables cannot be modified not only eliminates all the theoretical and notational problems connected to that notion but also ties in very well with the very nature of FDG’s Interpersonal Level, where linguistic expressions are analyzed in terms of their interactional properties and not with respect to their denotation. In this way, it becomes possible to formalize the modification of headless interpersonal variables in the very same way as was done for representational variables in the previous section: accordingly, the modified Subacts of Ascription in (60) can unproblematically be analyzed as shown in (62), in accordance with the notational convention introduced in Section 4.3:

A further case of headless variable modification at the Interpersonal Level concerns the layer of Subacts of Reference. As shown in (59), lexical expressions which are used as Subacts of Reference are analyzed at the Interpersonal Level as being headed by a (configuration of) Subact(s) of Ascription.²⁴ Referential proforms, on the other hand, do not express any lexical information but, in their deictic uses, are triggered by an appropriate configuration of abstract features [±S, ±A], as explained in Section 3.1 for personal pronouns. For anaphoric, cataphoric and logophoric proforms, however, there is no need to assume that the head of the Referential Subact is restricted by such abstract features: what triggers such proforms, in fact, is merely the presence of a semantic unit coreferential with that designated by the proform in the preceding or following discourse (coreferentiality being captured at the Representational Level by assigning the same index to the two variables, see Hengeveld & Mackenzie 2008: 119–121). It follows that any phoric use of a referential proform will be represented at the Interpersonal Level as an absent-headed Subact of Reference, that is as (R₁). Now, phorically used proforms are available for modification by interpersonal lexemes which provide additional – typically, evaluative – information on the Subact of Reference, such as the non-descriptive uses of adjectives like poor, lucky, dear, little or (good) old. In (63a–b), for instance, poor and old are clearly evaluative interpersonal modifiers expressing commiseration and endearment respectively, and do not refer to the financial situation or age of the referent (see Butler 2008: 227):

Once again, the only way in which such instances of interpersonal modification may be accounted for without invoking ad hoc theoretical constructs such as that of “null head” is by accepting the idea that modification does not necessarily presuppose headedness. Formally, the modification of absent-headed Referential Subacts will again be represented by inserting an additional opening bracket at the beginning of the interpersonal unit formed by the Subact and its modifier(s), as shown in (64a–b). Note that this solution can also account for cases such as (63b), where two interpersonal modifiers are attached to the same Subact: in such cases, the two modifiers will be distributed in such a way that the one with higher scope (here, lucky) takes the outermost position in formal representation: