2.1 Syntactic accounts

Goodall (Reference Goodall1987), Moltmann (Reference Moltmann1992) and others propose that ‘respectively’ triggers a syntactic operation in the presence of coordination that transforms a sentence with phrasal conjunctions into clausal conjunctions. However, McCawley (Reference McCawley, Bach and Harms1968: 297), Chomsky (Reference Chomsky and Peters1972: 123, note 26), Stockwell, Schachter & Partee (Reference Stockwell, Schachter and Partee1973), Pullum & Gazdar (Reference Pullum and Gazdar1982), Kay (Reference Kay, Hall, Meacham and Shapiro1989), Dalrymple & Kehler (Reference Dalrymple and Kehler1995), and McCawley (Reference McCawley1998) noted data like (6), which refute this assumption.

1. (6)

   1. (a) Those five men are Polish, Irish, Armenian, Italian, and Chinese (respectively).

   2. (b) They live in Chicago and New York (respectively).

   3. (c) The successive descendants of my fruit fly will be heavier, respectively, than the successive descendants of yours.

Kay (Reference Kay, Hall, Meacham and Shapiro1989: 182) shows that ‘respectively’ can only establish a one-to-one mapping between denotata if there is an independent ranking. Thus, the sentence in (7a) is acceptable but the one in (7b) is not.Footnote ⁴

1. (7)

   1. (a) The three best students received the three best scores respectively.

   2. (b) #The students were pleased by their scores, respectively.

In (8) I provide more examples lacking conjunction altogether. As Kay (Reference Kay, Hall, Meacham and Shapiro1989, Reference Kay, Horn and Ward2004) observes, the correct generalization is that a one-to-one mapping between pluralities is established via some pragmatic ranking due to context, surface order, or world knowledge.

1. (8)

   1. (a) The following two sections will deal with these two issues respectively.

   2. (b) Two different formulas were tested, respectively, by two groups of patients with various periodontal disorders.

   3. (c) This controversy revolved around the place where a new meeting house should be set, and as the parties could not agree, they built two in places they respectively chose.

In sum, the data indicate that the so-called ‘respectively construction’ cannot be seen as a syntactic operation on coordinate structures. In these data, a bijection is established between plural-denoting nominal phrases, which strongly suggests that the correct account is semantic in nature.

2.2 Semantic accounts

Link (Reference Link1998) proposes a special conjunction meaning that forms ordered tuples using higher-dimensional λ-abstractions like λxy and allows predicates to distribute over them. For example, the conjunction George and Nick yields a tuple (g, n) which is then used to produce a propositional conjunction as shown in (9). Such an account is not monotonic because the i-sums corresponding to the NPs George and Nick and Martha and Honey are destroyed when combined with the verbal semantics.Footnote ⁵

1. (9)

   1. (a) George and Nick hate Martha and Honey, respectively.

      λx ₁x ₂.[λy ₁y ₂[hate(x ₁, y ₁)∧hate(x ₂, y ₂)](m, h)](g, n) hate(g, m)∧hate(n, h)

   2. (b) George and Martha are drinking and dancing, respectively.

      λxy[drink(x)∧dance(y)](g, m) drink(g)∧dance(m)

Gawron & Kehler (Reference Gawron and Kehler2004) propose a more comprehensive account, which scales up to non-coordinate cases. Conjunction is systematically viewed as sum formation: nominal conjunction yields individual sums while verbal conjunction yields property sums. A pragmatically established sequencing function f can access the surface of conjuncts and a Resp _f operator produces the respectively readings. The Resp _f operator is similar to a distribution operator Distr in the sense that it attaches to a pluralic entity and operates over its members. As (10) illustrates, Resp _f takes two arguments: a verbal representation and a nominal representation.

1. (10) Sue and Karen jog and drive.

   Resp _f(jog⊔drive)(s∨k)=jog(s)⊔drive(k)

The analysis of Sue and Karen love Bob and Tim proceeds as follows. The subject and object NPs correspond to the joins s∨k and b∨k. The distribution Distr copies the verb meaning over each conjunct Distr(love(b⊔t))=love(b)⊔love(t), and then the operator Resp _f combines this VP with the subject: Resp _f(loves(b)⊔loves(t))(s∨k)=loves(b)(s)⊔loves(t)(k). Basically the same applies to respectively readings without NP conjunction.

Like Link's, this account is non-monotonic. At some point in the process of semantic composition we have an i-sum for the representation of a conjoined NP, but later that representation disappears. There is no representation for the NP plurality in the final semantics of the sentence. This point is important because it reverberates in the syntax–semantics interface. If Gawron & Kehler (Reference Gawron and Kehler2004: 193) are right, no theory of grammar may assume that verbs directly predicate the NP valents that they subcategorize. There is aways the potential for a respectively reading, in which case Resp _f intervenes and causes a verb to predicate only a strict subset of its argument's denotation. My main goal is to show that there is an alternative analysis of respectively readings which does not create such a divide between syntax and semantics. Semantic composition can remain simple and monotonic, and subcategorization can operate in tandem with predication, as assumed by contemporary syntactic theories.

There seem to be three empirical problems with the account in Gawron & Kehler (Reference Gawron and Kehler2004) . First, although every occurrence of the adverb respectively is assumed to trigger the insertion of a Resp _f operator, it is possible for a Resp _f operator to be inserted without the overt presence of the adverb. The problem is that nothing but a stipulation captures the fact that the adverb can be optional in respectively interpretations. Ideally, the optionality of the adverb should follow from independently motivated factors.

A second problem is that the f mechanism that captures the contextually-determined orderings is claimed to only apply to non-atomic referents. According to Gawron & Kehler (Reference Gawron and Kehler2004: 174), this correctly captures the oddness of cases like #A/The/Every man smiled respectively, where at least one of the NPs is singular and therefore has no proper subparts in which to map anything. However, this stance may be excessive. The native speakers of English that I have consulted accept the example in (11) below, found in the Web 1T 5-gram Version 1 of Google's N-Gram Corpus (which is also the source of the other attested examples included in this paper, referenced by their URLs). In this data point the presence of ‘respectively’ is allowed because a ranking of parties and their costs is easy to conceptualize, even though each party is a singular NP. The adverb respectively is felicitous because the universally quantified subject can outscope the verb and obtain an interpretation where there is a different ‘bearing’ event for each party/cost.

1. (11) While the relationship of the parties remains amicable, it is essential that an agreement be reached as to the costs [that] each party will respectively bear.

   (www.innovasafe.com/doc/blatt.doc)

The third problem is that Gawron & Kehler (Reference Gawron and Kehler2004) predict that there should be no difference between respectively readings with or without the adverb, because in both cases the relevant interpretations are produced by Resp _f. Below I provide empirical evidence against this prediction. Consider the sentence in (12). There is a pragmatically odd first interpretation where the same twenty executives were promoted and demoted on the same day. This is the ‘strict identity’ reading, where each conjunct predicates exactly the same dependent. However, there is also a ‘weak identity’ reading, where out of twenty executives some were promoted and others were demoted. The latter seems to be what is usually called a respectively reading, but in fact this is not true. The example in (12) actually worsens with the overt presence of the adverb respectively.

1. (12) Today, twenty executives were promoted and demoted (#respectively).

This suggests that the reading in (12) is related to – but not the same as – a respectively reading. In this case the adverb respectively is not felicitous because there is no independent ranking of executives. Such data are unexpected in Gawron & Kehler (Reference Gawron and Kehler2004) . In (13) I provide more examples of sentences with interpretations that appear to be respectively readings but which cannot co–occur felicitously with the adverb.

1. (13)

   1. (a) The front and the back of the ship are called the bow and the stern, but which is which?

   2. (b) We know houses four and five are the Swede and the German, but which is which?

      (www.everything2.com/index.pl?node_id=984348)

   3. (c) Goodman and I are like Drysdale and Koufax. Yeah, but which one is which?

      (sfgate.com/cgi-bin/blogs/mlasalle/detail?blogid=38&entry_id=10946)

   4. (d) Caesar and I are as fire and water, but as to which is which you must judge for yourself.

      (Talbot Mundy, in Tros of Samothrace, Appleton-Century, 1934, Chapter XXXI, p. 168)

The data in (13) also indicate that the surface pairing between nominal conjuncts is not a grammatical constraint, since it can be canceled by certain continuations. In other words, respectively readings without an overt adverb are biased towards presentation order, whereas respectively readings with an overt adverb are restricted to it.Footnote ⁶

A third empirical argument against the notion that respectively readings with or without adverbs arise from the same Resp _f operator is based on the contrast in (14). Since both sentences are only felicitous if interpreted with a respectively reading, both should require the presence of a covert operator Resp _f. But whereas example (14a) tolerates the overt realization of the adverb, (14b) becomes less acceptable with the overt realization of the adverb. The function of the adverb respectively is to single out a certain kind of interpretation in a larger set of possible interpretations. In cases like (14b) – where no ambiguity is usually perceived because husband and wife is such a strong binomial expression – the presence of the adverb is unwarranted. For Gawron & Kehler (Reference Gawron and Kehler2004) it should not matter if Resp _f is overt or covert, and yet the data indicate that it does.

1. (14)

   1. (a) Tom and Mary are tall and short.

      (Tom and Mary are tall and short respectively.)

   2. (b) Tom and Mary are husband and wife.

      (?#Tom and Mary are husband and wife respectively.)

The fact that the adverb respectively can access surface order is not remarkable. Many other expressions like the former, the latter, in that order, conversely, contrariwise, and vice versa, seem to access surface order metalinguistically, even in the absence of conjunction, as the data in (15) illustrate. I conclude that such expressions are contextual in the sense of Kay (Reference Kay, Hall, Meacham and Shapiro1989), and have the means to access surface order. I do not discuss how such access works as this is an independent question beyond ‘respectively’.

1. (15)

   1. (a) [Craig Jones]_i is a different kind of film-maker from [Stuart Lee]_j.

      While [the former]_i is more comfortable with abstract concepts, [the latter]_j likes to convey his thoughts as directly as possible.

   2. (b) About 20% of [men]'s underwear is bought by [women], and vice versa.

      (=‘About 20% of women's underwear is bought by men.’)

   3. (c) [Worms] may have evolved from [arthropods] rather than the other way around.

   4. (d) [The Hobbit] was written before [The Lord of the Rings], but I didn't read them in that order.

I have tried to argue that so-called respectively readings are nothing but a subset of a more general kind of reading. The latter are often not compatible with the presence of the adverb, as shown in (12) and (13). This means that Resp _f is missing important generalizations: on the one hand, it is too restrictive to model the ‘respectively readings’ that arise without ‘respectively’ like in (12), while on the other, it does not correspond to the actual semantic contribution made by the adverb. I thus reject the existence of an optionally covert operator like Resp _f, and seek an alternative analysis.

3.1 Cumulative quantification readings

The sentences in (16) have an interpretation known as cumulative quantification, in the sense of Scha (Reference Scha, Groenendijk, Janssen and Stokhof1981: 497).Footnote ⁷ For instance, in a cumulative reading of (16a) there are many possible mappings between the soldiers and the targets they hit. As Beck & Sauerland (Reference Beck and Sauerland2000) and others note, data like (16) indicate that such readings also occur with conjunction.

1. (16)

   1. (a) Two soldiers hit two targets.

   2. (b) Several critics praised some movies.

   3. (c) 700 Dutch companies used 10 000 American computers.

   4. (d) These dots correspond to those cities.

   5. (e) Jim and Frank want to marry two dentists.

Various accounts have been proposed for such readings.Footnote ⁸ For example, Kroch (Reference Kroch1974: 205) and Scha (Reference Scha, Groenendijk, Janssen and Stokhof1981: 497) use meaning postulates like (17).

1. (17) P(α, β)=(∀α′∊α→∃β′∊β∧P(α′, β′))

   ∧

   (∀β′∊β→∃α′∊α∧P(α′, β′))

Krifka (1989), Sternefeld (Reference Sternefeld1998) and many others make use of a **-operator to a similar effect. For my purposes, it is not important which analysis is adopted, and I remain agnostic with respect to this choice.

Figure 1 shows the readings that (17) allows for a cumulative interpretation of (16a). Crucially, two of these are bijections. In fact, when discussing a sentence like (16a), Link (Reference Link1998: 56) writes that ‘the situation described (…) is a special cumulative reading where, in addition, the relation is a bijective function’.

I therefore propose that some respectively readings are simply a bijective cumulative interpretation. The functions of the adverb respectively is to single out one of the bijective relations in the set of possible independently obtained cumulative interpretations. This explains why the presence of the adverb is optional, and why respectively readings do not arise when the NPs are disjoined rather than conjoined.Footnote ⁹ To be clear, in this view respectively interpretations are a consequence of cumulative readings, and the function of the adverb is to pick a cumulative reading that is bijective and satisfies a pragmatic- or linearization-based ranking.

Figure 1

Respectively readings as an instance onf cumulative readings.

This approach makes several predictions. Cumulative quantification can apply to predicates with more than two arguments, and when it does, it does not force all arguments to be cumulatively interpreted. For example, any two NPs in (18) can be interpreted cumulatively (or all three).

1. (18) Two students sent four letters to three congressmen.

One student may have mailed two letters to the three congressmen, while the other student mailed the two other letters; or the two students collectively sent one letter to one congressman and another letter to the remaining congressmen. The total of combinations is 24 (=2×4×3), minus other readings due to scope ambiguity (e.g. Link Reference Link1998: 54).

This has consequences for the current account of respectively readings. It entails that sentences with three plural NPs are ambiguous in essentially the same way. This prediction is borne out in data like (19), from McCawley (Reference McCawley1998) and Gawron & Kehler (Reference Gawron and Kehler2004), where ‘respectively’ allows all or any two NPs to be mapped bijectively. This follows without stipulation if such readings are just special cases of cumulative readings, as proposed above.

1. (19) George and Martha sent a bomb and a nasty letter to the president and the governor respectively.

Cumulativity can also target NPs that are not co-arguments of the same verb, as shown by examples involving adjuncts like (20). In these cases the cumulation process seems to target the predicates in and causing, which can be argued to take as arguments an eventuality and a nominal. It is therefore tempting to assume that events are structured mereologically as in Bach (1986), and that in(e,x) and cause(e,y) can be interpreted cumulatively. For more discussion see for example Schein (Reference Schein1993: Chapter 9).

1. (20) Five alarms malfunctioned in two locations, causing four accidents.

Whatever the proper account of cumulative readings is, my claim is that it will predict the bijective mappings that ‘respectively’ requires, among many other readings. I use the admittedly oversimplified account in (17) for exposition purposes only, but more sophisticated approaches can be adopted, like the generalized PPart operator of Schwarzschild (Reference Schwarzschild, Stokhof and Torenvliet1990, Reference Schwarzschild1996).

Crucially, however, none of the above accounts can deal with respectively readings that involve predicate conjunction, like that in (21a).

1. (21)

   1. (a) For their talent exposition, the top three pageant winners sang, breathed fire, and played the kazoo (respectively).

   2. (b) For their talent exposition, [the top three pageant winners]_i sang, they_i breathed fire, and they_i played the kazoo (#respectively).

As Gawron & Kehler (Reference Gawron and Kehler2004: 200) show, even the PPart operator account in Schwarzschild (Reference Schwarzschild1996: 88) cannot explain why (21a) has a respectively reading but (21b) does not. Gawron & Kehler (Reference Gawron and Kehler2004: 199) also observe that the PPart operator is limited to relations among individual-level groups, and does not extend to cases that involve relations that include property-level groups. Gawron & Kehler take such data to show that predicates cannot directly predicate their dependents, but I argue otherwise below.

3.2 Dependent sharing in conjunction

As already discussed in Section 2.2, sentences like (12) above, repeated here in (22), can be said to have two kinds of readings.

1. (22) Today, twenty executives were promoted and demoted.

In the ‘strict identity’ reading, twenty executives were both promoted and demoted (perhaps because of a computer glitch). This reading is somewhat implausible because of world knowledge, but would otherwise be the preferred interpretation (compare with Twenty executives were interviewed and surveyed). The reading is one where the executives that were promoted are different from the executives who got demoted (a ‘weak identity’ reading). The latter interpretation is similar – but not identical to – a respectively reading: the presence of the adverb is not allowed because the denotata of the subject lack an independent ranking.

Arguably, the sentence in (23) also has strict/weak identity readings. The weak reading is what is usually called a respectively reading, since it can be forced by adding the adverb respectively. In the absence of the adverb, the strict reading is the preferred one.

1. (23) Tom and Fred sang and danced.

Technical differences aside, most accounts of VP coordination assume that conjuncts must necessarily share the subject (i.e. must have a ‘strict identity’ reading), as shown in Figure 2. This explains the observed subject–verb agreement and semantic interpretation.

Figure 2

Shared dependent identity.

However, in light of ambiguous examples like (22) above, I will revise the analysis of conjunction so that any dependents that are shared by conjuncts are either equated (‘strict identity’) or cumulated (‘weak identity’). The former yields the same result as in Figure 2 and the latter yields a kind of interpretation of which respectively readings are a subset. Fortunately, there is a simple operation that can succinctly capture all the relevant patterns: Link's ‘⊕’ sum operator. In order to model non-Boolean nominal conjunction, Link (Reference Link, Bauerle, Schwarze and Stechow1983) resorts to a sum relation ‘⊕’ interpreted as a join operator in a mereological domain: x⊕y =x ⊔y . This definition does not rule out the possibility that x=y, in which case the sum does not yield a plurality. This follows from the fact that the join ‘⊔’ is idempotent (i.e. ∀x[x⊔x=x]).Footnote ¹⁰ Formally, Link's sum encapsulates exactly what is necessary to deal with the two readings of (22a), and consequently, the respectively readings in (22b). The proposed generalization is stated in (24) as a condition on the syntax–semantics interface of conjunction. In Section 3.5 I argue that (24) should be generalized to other kinds of dependencies.

1. (24) Shared Dependent Condition for Conjunction (informal)

   Predication dependencies shared by conjuncts are combined via ‘⊕’.

I will formalize (24) by augmenting the semantic content of the conjunction and. For simplification, I assume that the semantics of conjunction only takes two conjuncts Q and P.Footnote ¹¹ I thus propose the lexical entry in (25). Here, P[x ₀, …, x _n] and Q[y ₀, …, y _n] are two conjuncts with exactly n (n⩾0) free variables, each corresponding to a shared dependent. The variables corresponding to the shared dependents are combined via ‘⊕’.

1. (25) Conjunction (preliminary)

   and : λP.λQ.λz ₀…λz _n. Q[x ₀…x _n]∧P[y ₀…y _n] ∧

   z ₀=(x ₀⊕y ₀) ∧…∧ z _n=(x _n⊕y _n)

In VP coordination there is exactly one shared dependent: the subject. This means that the variables of the subjects selected by each of the VP conjuncts are combined via ‘⊕’. As a consequence, these variables are either equated or cumulated. If equated, both VPs predicate the same entity (‘strict identity’). If cumulated, each VP predicates a different part of the plurality (‘weak identity’). This is shown in Figure 3.

Figure 3

Non-deterministic dependent sharing.

If the obtained coordinate VP combines with a subject like Tom and Sue, we obtain the representation in (26). The arrow ‘↦_β’ indicates beta-reduction.

1. (26) Tom and Sue sing and dance.

   (λz.sing(x ₁)∧dance(x ₂)∧z=(x ₁⊕x ₂)) (t⊕s) ↦_β

   sing(x ₁)∧dance(x ₂)∧(t⊕s)=(x ₁⊕x ₂)

if x ₁=x ₂ then ‘⊕’ does not form a plurality and we obtain a reading where Tom and Sue are both agents of singing and dancing (i.e. z=x ₁=x ₂). Conversely, if x ₁≠x ₂ then ‘⊕’ is interpreted as a plurality-forming cumulation and we obtain a reading where t⊔s=x ₁⊔x ₂. Because ‘⊕’ is a symmetric relation, in the latter case it is not known whether t=x ₁ and s=x ₂ or if t=x ₂ and s=x ₁. This indeterminacy can be resolved by adding the adverb respectively. That is the adverb's function.

The account scales to other kinds of plural NP subjects, like (27). As before, if ‘⊕’ is interpreted as equality then all students sang and danced, and if ‘⊕’ is interpreted as a plurality-forming cumulation then some students sang while other students danced.

1. (27) Several students sing and dance.

   (λP.∃x(student*(x)∧P(x)))

   (λz.dance(x ₁)∧sing(x ₂)∧z=(x ₁⊕x ₂))↦_β

   ∃x(student*(x)∧sing(x ₁)∧dance(x ₂)∧x=(x ₁⊕x ₂))

As discussed in Section 2.2, this falls short of being a true respectively reading because (27) does not allow the presence of the adverb respectively. There must be an independent ranking between denotata in order to obtain a true respectively reading. Out of the blue, (27) exhibits no such ranking, and therefore respectively is not allowed. Conversely, in cases like Robin and Kim are cousins. They live in Chicago and New York (respectively), an independent ranking is provided, and therefore the adverb can be used.

The bijection forced by the presence of the adverb respectively in sentences like Tom and Fred sang and danced holds between the atoms in the denotation of the plural NP and the eventualities denoted by the conjoined VP. I thus assume that verbs come with Neo-Davidsonian eventuality arguments, as argued by Kamp (Reference Kamp, Bäuerle, Egli and von Stechow1979), Partee (Reference Partee1984), Higginbotham (Reference Higginbotham1985), Bach (Reference Bach1986), Carlson (Reference Carlson1987), Krifka (1989b), Parsons (Reference Parsons1990), and Pustejovsky (Reference Pustejovsky1991), inter alia. Support for this comes from well-known evidence that events can be bound anaphorically, modified by a non-restrictive relative clause, distributed, compared, and intensified:

1. (28)

   1. (a) Fred stabbed Jones. It happened at midnight.

   2. (b) Kim evaded the police, which is not an easy thing to do.

   3. (c) John called every Monday.

      (=‘For all past Mondays John called.’)

   4. (d) I love you more than you love me.

   5. (e) Fred is so going to jail.

If such eventualities exist, then their conjunction should also form a plural eventuality. This is empirically supported by the adverbials in (29), which predicate the conjoined plural eventuality. I thus follow Carlson (Reference Carlson1987: 540–542), Lasersohn (Reference Lasersohn1995: Chapter 14), Gawron & Kehler (Reference Gawron and Kehler2004) and many others in assuming that plurality-forming conjunction can operate cross-categorially: nominal conjunction yields a nominal plurality and predicate conjunction yields a plural eventuality.

1. (29)

   1. (a)

   2. (b)

   3. (c)

   4. (d)

Accordingly, the conjunction in (25) above must be revised so that it forms a plurality with the denotata of the conjuncts. Drawing from Link (Reference Link, Bauerle, Schwarze and Stechow1983) and Krifka (Reference Krifka, Kálman and Pols1990), I adopt the conjunction in (30), where the α₁ and α₂ variables correspond to individuals or eventualities denoted by the conjuncts. If the conjuncts are verbal, so is the plurality, and if the conjuncts are nominal, so is the plurality. Nothing hinges on this particular formalization, however, as other approaches to conjunction like Lasersohn (Reference Lasersohn1995), Schwarzschild (Reference Schwarzschild1996), and Landman (Reference Landman2000) can be adopted instead.

1. (30) Conjunction (revised)

   and : λP.λQ.λz ₀ … λz _n.∃α(α=(α₁⊕α₂)∧

   Q(α₁)[x ₀ … x _n]∧P(α₂)[y ₀, … y _n]∧

   z ₀=(x ₀⊕y ₀)∧…∧z _n=(x _n⊕y _n))

The revised conjunction in (30) allows VP conjunction to be modeled as shown in (31). The α variables are renamed as e for perspicuity. In (31a) we combine and with danced. In (31b) we combine sang with and danced. Note that a sum is formed with the denotation of the conjuncts, and the shared subject dependent is combined via ‘⊕’. Finally, in (31c) we combine Tom and Fred with sang and danced.

1. (31) [[Tom and Fred]_NP [sang and danced]_VP]_S.

   1. (a) (λP.λQ.λz.∃e(e=(e ₁⊕e ₂)∧Q(x ₁)(e ₁)∧P(x ₂)(e ₂)∧z=(x ₁⊕x ₂)))

      λx.λe.dance(e, x))

      ↦_βλQ.λz.(e ₃=(e ₁⊕e ₂)∧Q(x ₁)(e ₁)∧dance(e ₂, x ₂)∧z=(x ₁⊕x ₂)

   2. (b) (λQ.λz.∃e(e=(e ₁⊕e ₂)∧Q(x ₁)(e ₁)∧dance(e ₂, x ₂)∧∃z(z=(x ₁⊕x ₂)))

      λx.λe.sing(e, x))

      ↦_βλz.(e=(e ₁⊕e ₂)∧sing(e ₁, x ₁)∧dance(e ₂, x ₂)∧z=(x ₁⊕x ₂))

   3. (c) (λz.∃e(e=(e ₁⊕e ₂)∧sing(e ₁, x ₁)∧dance(e ₂, x ₂)∧z=(x ₁⊕x ₂))) (t⊕f)

      ↦_β∃e(e=(e ₁⊕e ₂)∧sing(e ₁, x ₁)∧dance(e ₂, x ₂)∧(t⊕f)=(x ₁⊕x ₂))

Exactly as before, the semantic representation obtained in (31) allows two distinct interpretations. If ‘⊕’ is interpreted as equality then we have the reading where each conjunct predicates the same plurality (i.e. (t⊕s)=(x ₁=x ₂)), and if ‘⊕’ is interpreted as a plurality-forming cumulation then we have a reading where each verb predicates a different individual (i.e. (t=x ₁ & f=x ₂) or (t=x ₂ & f=x ₁)). The same account deals with sentences like (32). Although the preferential interpretation conveys that both men sing and dance, there is also an interpretation where different men do different things.Footnote ¹²

1. (32) Two men sing and dance.

   ∃z(men(z)∧|z|⩾2 ∧

   ∃e(e=(e ₁⊕e ₂)∧sing(e ₁, x ₁)∧dance(e ₂, x ₂)∧z=(x ₁⊕x ₂)))

3.3 On the role of the adverb respectively

In this account, respectively is similar to other adverbs in that it simply adds restrictions to the semantics of the verbal structure that it adjoins to. It does not alter it in any way. Let us tentatively assume that this adverb is represented as in (33). The sentence in (34) can thus obtain the intended representation.

1. (33) respectively: λP.λx.λe.(P(x)(e)∧respectively(e)) (preliminary)

1. (34) Tom and Fred sang and danced respectively.

   ∃e(e=e ₁⊕e ₂∧sing(e ₁, x ₁)∧dance(e ₂, x ₂) ∧

   (t⊕f)=(x ₁⊕x ₂))∧respectively(e))

The respectively reading only arises if ‘x ₁⊕x ₂’ is interpreted as a plurality-forming cumulation. In that case, each verb predicates a different referent. As before, this is the weak identity reading of the shared subjects. Let us assume that this sentence is true because e =e′₁⊔e′₂, where 〈e′₁, t〉∊I(sing) and 〈e′₂, f〉∊I(dance). Since the adverb respectively can access surface order, it can readily identify the rankings t≺f and e′₁≺e′₂. I use the relation ‘≺’ to mean ‘outranks’, in Kay's sense. The adverb can then detect the bijection {(e′₁, t), (e′₂, f)} by inspecting the tuples above (i.e. denoted by the verbs described by the plural eventuality e).

The adverb operates in basically the same way in sentences like (35). As discussed in Section 3.1, an independently motivated cumulative interpretation allows a reading where Tom loves Sue and Fred loves Mia.

1. (35) Tom and Fred love Sue and Mia respectively.

   ∃e (sing(e, t⊕f, s⊕m)∧respectively(e))

For example, assuming that e =e′₁⊔e′₂, where 〈e′₁, t, s〉∊I(love) and 〈e′₂, f, m〉∊I(love), then the rankings t≺f and s≺m allow the bijection {(t, s),(f, m)} to be evoked. The ‘love’ events denoted by e lack an independently established ranking, and therefore cannot enter any bijection. One can assume that existential closure applies to e in (35).

Now let us consider some negative examples. The sentence #A man smiled respectively is not felicitous because there is only one event and one individual participant. This follows from the fact that ‘≺’ is a relation (i.e. requires at least two entities). In (36), however, the oddness is caused by the absence of an independent ranking for the entities denoted by the subject and the predicate.

1. (36)

   1. (a) #The women smiled respectively.

   2. (b) #Every man smiled respectively.

   3. (c) ?#Each woman read a book respectively.

As discussed in Section 2.2, the same is true for #The students smiled respectively and #The students were pleased by their scores respectively. As Kay (Reference Kay, Hall, Meacham and Shapiro1989) correctly notes, The three best students received the three best scores respectively more readily elicits a contextually determined ranking and therefore can be felicitous.

I now turn to the adverb. This word combines with verbal expressions and predicates their eventuality variable, just like many other adverbials. The difference lies in the satisfaction conditions introduced by ‘respectively’. Put informally, (37a) states that ‘respectively’ construes two sets, X and Y. Set X contains all tuples describing an event e′, where e′⩽e. Here, e is the eventuality that the adverb predicates and ‘⩽’ is the event-mereological ‘part-of’ relation. Set Y is composed of all the pairs of entities occupying two fixed positions n and m in the tuples of X. Finally, Y must be a bijection according to a contextual ranking r, as defined in (37b). A ranking r is simply a set of contextually determined relations like ‘a ₁≺a ₂’. Although I omit rankings from (37b), I assume that they are established along the lines of Kay (Reference Kay, Hall, Meacham and Shapiro1989, Reference Kay, Horn and Ward2004) and Section 2.2.

1. (37)

   1. (a) respectively(e)=1 iff

      X={τ: ∃P∃e′(τ∊I(P) & τ=〈e′ ,… 〉 & e′⩽e )} &

      ∃n∃m(Y={(p _n, q _m): =〈…, p _n, …, q _m, …〉∊X} & Bij _r(Y))

   2. (b) Bij _r (Y)=1 iff

      ¬∃a∃b∃c(a≠b≠c & ({(a, b), (a, c)}⊆Y ∀ {(a, b), (c, b)}⊆Y))

For illustration, let us suppose that the sentence in (38) is true because 〈e ₁, t〉∊I(sing) and 〈e ₂, f 〉∊I(dance), for e =e′₁⊔e′₂. From the truth conditions in (37) it follows that X={〈e ₁, t〉, 〈e ₂, f)}, and from the surface order we extract the ranking t≺f and e ₁≺e ₂. This yields the bijection Y={(e ₁, t),(e ₂, f)}. The same analysis applies to These two men sing and dance, with the difference that the ranking is not due to surface order.

1. (38) Tom and Fred sang and danced respectively.

   ∃e(e=e ₁⊕e ₂∧sing(e ₁, x ₁)∧dance(e ₂, x ₂) ∧

   (t⊕f)=(x ₁⊕x ₂)∧respectively(e))

Now let us consider cases without verbal conjunction. As described in Section 3.1, a cumulative quantification reading of These two boys love these two girls can be interpreted so that 〈e′₁, b ₁, g ₂〉∊I(love) and 〈e′₂, b ₂, g ₁〉∊I(love). By adding the respectively adverb we can felicitously obtain Y={(b ₁, g ₁), (b ₂, g ₂)} if there is a metalinguistically determined ranking b ₁≺b ₂ and g ₁≺g ₂. As discussed above, e′₁ and e′₂ are not sufficiently distinct and salient in order to establish an independent ranking. In multiple ‘respectively’ sentences like (39) (McCawley Reference McCawley1998: 294), there is a cumulative quantification reading of the verb which yields X={〈e ₁, t, b, p〉, (e ₂, m, l, g〉}. This allows one adverb to evoke Y={(t, b), (m, l)} and the other to evoke Y′={(b, p), (l, g)}.

1. (39)

   1. (a) Tom and Martha respectively sent a bomb and a nasty letter to the president and the governor respectively.

   2. (b) Tom and Martha sent a bomb and a nasty letter respectively to the president and the governor respectively.

This analysis is also compatible with (11), repeated here as (40).

1. (40) While the relationship of the parties remains amicable, it is essential that an agreement be reached as to the costs that each party will respectively bear.

First, note that my account of raising will ensure that the verb bear predicates the subject referent each party and the object referent the costs.Footnote ¹³ Second, in order for the respectively reading to be felicitous, there must be a different event and a different cost for each party, and comprehenders must be able to conceptualize the respective rankings. This is possible because the eventuality e has narrow scope under the each party NP. Thus, for each party there will be a different ‘bear’ situation with a certain cost. One way to model the respectively reading of (40) is to assume that the adverb respectively has access to all assignments made to the eventuality variable e that it predicates. In this approach, the set X will contain the tuples formed by the various values for the e variable and for x and y that are relevant for the interpretation of the verb bear in the sentence. Consequently, set Y will contain pairs of costs and parties, which must form a bijection for the sentence to be felicitous, as usual, according to a pragmatically determined ranking.

There may be an alternative analysis, however. Schein (Reference Schein1993) and Kratzer (Reference Kratzer2003) note that quantifiers like every allow cumulative readings (e.g. Three copy-editors caught every mistake in the manuscript can have an interpretation where different editors caught different mistakes). A respectively reading like (40) would become unremarkable if each can have this kind of cumulative reading as well.

3.4 Other kinds of dependent cumulation with respectively readings

4.1 Basics

Head-driven Phrase Structure Grammar is a theory of grammar where phonology, syntax, semantics, and pragmatics co-exist locally in the same kind of linguistic structure, called a sign. Central to this non-derivational theory is the notion of constraint satisfaction (Carpenter Reference Carpenter1992). The grammar rules (which I interchangeably refer to as constructions) are nothing more than declarative statements that impose constraints about linguistic structure, rather than operations that alter, compare, or displace previously assembled representations. The information conveyed by a sign is encoded as sets of features. The feature phon encodes phonology, syn encodes various kinds of syntactic information (including part-of-speech, agreement, case, and subcategorization constraints), and sem contains semantic representations. Signs can be of type word or phrase. The sign for the proper name Tim is given in (66).

1. (66)

   

Valency is recorded in a list-valued feature val, which in this case is empty (i.e. 〈 〉). The empty set value for slash means that there are no extracted dependents. Semantics is represented as a list of restrictions restr containing predications. For exposition purposes I adopt the ‘flat’ semantic formalism of Copestake et al. (Reference Copestake, Flickinger, Sag and Pollard2005) and ignore the treatment of quantifier scope, but nothing hinges on this.Footnote ¹⁷ From now on I use the symbol ‘NP’ as a shorthand for a feature structure description of a nominal sign with an empty valence list, as in (67). Hence, NP^x is nothing but a nominal sign that has empty valence and a referential index x.

1. (67)

   

Lexical entries and grammar rules are partial descriptions of linguistic structure. For example, the verbs in (68) require a valent with nominal part-of-speech, empty valence, and a referential index, but say nothing about the phonology or semantic representation of that valent.

1. (68)

   

Any sign that satisfies the stated constraints will do. Drawing some inspiration from Kathol (Reference Kathol, Levine and Green1999), I assume that verbal morpho-syntactic agreement information is appropriate for the type verb. Hence, the verb form in (68a) is underspecified and can combine with any kind of subject (e.g. I/You/He/They left), whereas (68b) can only combine with third person singular subjects. The two occurrences of x indicate that the two indices must be unified. In other words, the variable contributed by the subject NP is exactly the same as the variable predicated by the verb; the two are inextricably identified in the lexical entry in (68). In general, variable binding is ensured lexically in HPSG: predicates directly link their logical arguments to the variables of their valents. This is one of the aspects of unification-based grammar that Gawron & Kehler (Reference Gawron and Kehler2004) identify as problematic for respectively readings.

Everything in the grammar is uniformly represented with features, and syntactic trees are no exception. This is shown in Figure 4. The feature mtr encodes information about the mother node and the list-valued feature dtrs encodes information about the daughters of the construction. I use the tree notation when talking about structures licensed by the grammar, and reserve the feature matrix notation when talking about the grammar.

Figure 4

A feature structure representation of a tree.

As (69) shows, VP is shorthand for a verbal sign that seeks an NP, and an S is shorthand for a verbal sign with empty valence.

1. (69)

   

All grammar rules are of the form τ⇒φ, where τ is a type and φ is a set of features that any entity of the type τ must satisfy. These rules can capture varying degrees of generalization because they are hierarchically organized. Any piece of linguistic information of a given type τ must not only satisfy any rule of the form τ⇒φ, but also every rule τ′⇒φ′, where τ′ is a (direct or indirect) supertype of τ. For example, consider the type hierarchy of phrasal rules in Figure 5. The constructions (cxt) in this fragment are phrasal, and can be either headed or non-headed. I view coordinate structures (and perhaps certain comparatives) as a non-headed constructions, but nothing hinges on this as far as this work is concerned.

Figure 5

Top nodes of the hierarchy of phrasal constructions.

For now, let us focus on the rules governing headed constructions, i.e. the types subsumed by headed-cxt in Figure 5. The head-comps-cxt type in Figure 5 models phrases formed by a lexical head and its complements, and the specifier-head-cxt models structures formed by a specifier and a head. Other rules will model adjunction constructions, and so on. The feature hd-dtr, which is only appropriate for headed constructions, is used to identify which of the daughters is the head. Throughout, I use X, Y, Z as variables over feature structures and L as a variable over lists of feature structures. The specifier-head-cxt construction allows a head phrase to combine with a specifier, as seen in (70).

1. (70) Specifier–Head Construction

   

The notation X:φ means that the variable X describes a feature structure with at least as much information as the feature structure description φ. Note that (70) is silent about semantics, part-of-speech, or phonology. This rule simply imposes (partial) syntactic constraints on a local tree, by stating that a construction of the type specifier-head-cxt can be formed by a sign X and a head that selects X as a valent. The resulting mother node has empty valence (i.e. [val 〈〉]), like the one shown above in Figure 4.

Following Malouf (Reference Malouf2000) and Sag (Reference Sag2010a), I assume that the type specifier-head-cxt has various subtypes, each capturing specific grammatical patterns for different specifier–head constructions. For example, Malouf (Reference Malouf2000: 15) models different case assignment patterns for gerund clauses with three different subtypes of specifier-head-cxt: an accusative subject rule deals with clauses like (71), a possessive subject rule deals with (72), and a nominative subject rule deals with cases like (73).

1. (71)

   1. (a) Pat disapproved of [[me] [quietly leaving before anyone noticed]].

   2. (b) Everyone was impressed by [[Pat] [artfully folding the napkins]].

1. (72)

   1. (a) Pat disapproved of [[my] [quietly leaving before anyone noticed]].

   2. (b) Everyone was impressed by [[Pat's] [artfully folding the napkins]].

1. (73)

   1. (a) [[He] [wears a tuxedo]].

   2. (b) I insist that [[Sandy] [leave me alone]].

   3. (c) [[Pat] [artfully folded the napkins]].

There are at least four other kinds of specifier–head construction that have their peculiar syntactic, semantic, pragmatic, and prosodic conditions. First, we have ‘Mad magazine’ sentences (Akmajian 1984), such as (74), which have a number of idiosyncratic properties that can be naturally captured constructionally (e.g. they cannot be embedded, their subject must be accusative, and the they never include tense, modals, and sentential adverbials).

1. (74)

   1. (a) [[Him] [wear a tuxedo]?! (Sure.)

   2. (b) [[My boss] [give me a raise]]?! (Ha.)

   3. (c) What! [[Larry] [be a doctor]]?

Second, consider absolute constructions like (75), which also differ from (71) in several ways: the preposition with is required, does not have an NP distribution, adjoins to a clause, and has variable semantic connectivity with the matrix (Stump Reference Stump1985, Culicover & Jackendoff Reference Culicover and Jackendoff2005).

1. (75) With [[him] [organizing things]], we'll never get anything done.

Third, structures like (76) appear to be a kind of specifier–head fragment construction with an accusative subject.

1. (76) Tom: I'd like to see that movie.

   Kim: [[Me] [too]].

Finally, directive and imperative sentences like (77) allow optional non-first person subjects (but are only compatible with second person reflexive objects), and come with special pragmatics and intonation.

1. (77)

   1. (a) [[You] [leave me alone]]!

   2. (b) [[Everyone] [seat yourselves/*themselves]]!

The above subtypes of specifier-head-cxt can be modeled by the hierarchy in Figure 6. Each type can introduce particular syntactic, semantic, pragmatic and prosodic constraints. For example, nom-subject-pred-cxt corresponds to the rule in (78) below, which licenses structures formed with a nominative subject phrase and an agreeing finite verbal head phrase. Since nom-subject-pred-cx is a subtype of specifier-head-cxt, it inherits all its constraints: the specifier sign is required to be the only member of the head's val, and the mother's val list is empty.

Figure 6

Hierarchy of specifier–head constructions (extending hierarchy in Figure 5).

Departing from Pollard & Sag (Reference Pollard and Sag1994) I propose that subject–verb agreement is computed at the phrasal level rather than lexically, as shown in (78), similarly to Sag, Wasow & Bender (Reference Sag, Wasow and Bender2003: 102).

1. (78) Nominative Subject–Predicate Construction

   

Basically, this rule ensures that whatever is the agreement information X in the VP daughter, it must unify with the information in the nominative subject daughter. The Headed Construction discussed below will ensure that the agreement specifications the verb's cat are percolated to its phrasal projections. Finally, the Coordination Construction in Section 4.2 will require that the conjuncts and the mother have the same cat value, and therefore, verbal conjuncts shall impose the same agreement specifications on their shared dependents.

Although the headed constructions in Figure 5 are different, they also have similarities which can be captured by the head-cxt supertype. For example, the Headed Construction in (79) ensures that for any headed construction the syn information of the head daughter and of the mother are the same, by default (see Ginzburg & Sag Reference Ginzburg and Sag2000: 33). The default constraint ‘/’ ensures that the features X of the head daughter and of its mother are identical, except if explicitly contradicted by the application of other rules. Thus, (70) does not allow the head daughter and the mother to have the same val information in specifier–head constructions, but does allow them to have the same part-of-speech. Thus, a VP projects an S, an N′ projects an NP, and so on.Footnote ¹⁸

1. (79) Headed Construction

   

Some rules are even more general and apply to all phrasal types in Figure 5, not just to headed constructions. One example is the semantic composition rule in (80), which states that the semantic content of the mother of any phrasal structure is the concatenation of the semantic contents of the n daughters. The ‘o’ symbol denotes list concatenation.Footnote ¹⁹

1. (80) Principle of Compositionality

   

The cumulative effect of all of these constraints is shown in Figure 7. As mentioned above, whenever the grammar requires two features to have the same value, the result is unification (or more accurately, structure-sharing, since in model-theoretical grammar there are no procedural operations). In the linguistic structures licensed by the grammar constraints, unification is represented via multiple occurrences of a boxed tag, such as the 1 in Figure 7. This states that the specifier-head-cxt has identified the valent subcategorized by the head with the non-head daughter.

Figure 7

Representation of the sentence [[Tim]_NP [left ]_VP]S.

Consider now how complementation constructions are modeled. A head–complement rule allows any lexical head to combine with its lexically subcategorized complements (i.e. the more oblique valents X ₂, …, X _n) as determined by (81).

1. (81) Predicational Head–Complement Construction

   

Note that nothing is said about the semantics or part-of-speech of the daughters or of the mother node. Any word with a non-empty valence can instantiate the head Y and impose the relevant lexically specified constraints on the X valents. The Headed Construction in (79) interacts with this rule and makes sure that the mother and the head daughter Y have the same cat information.

Following Pollard & Sag (Reference Pollard and Sag1994), a subject raising verb like continue lexically selects an infinitival VP complement and shares with its subject valent X. The infinitival auxiliary verb to is also a raising verb and therefore also shares the subject valent with its (base form) VP complement. The analysis of subject control verbs like want is similar in that the verb lexically selects a base form VP and shares with it the same subject index. The lexical entires for these verbs are shown in (82).

1. (82)

   

Semantically, want embeds the meaning of the complement VP phrase, labeled by l ₁. The two verbs differ in that continue does not assign any semantic role to its subject. Thus, the subject variable of help in (83) is bound to the subject variable of every dominating verb up to the matrix verb continue, and cumulative quantification can then apply to help(e, x, y) as discussed in Section 3.1 above, and yield the intended respectively interpretation.

1. (83) Fred and Sue continue to want to help Kim and Mia (respectively).

Let us now turn to how unbounded filler–gap dependencies are modeled in HPSG.Footnote ²⁰ Basically, certain lexical constraints allow valents to be located in slash rather than in val. The values of slash are percolated to the mother node by the Headed Construction in (79), since slash is a syn feature. This is why the dependency is unbounded, and can be iterated indefinitely. The percolation of gaps can be interrupted only by certain constructions, namely those that impose constraints on the value of slash. One such construction is the topicalization construction, shown in (84), in which some sign X is realized as a sister of the head clause that contains the corresponding gap.

1. (84) Filler–Head Construction

   

Other gaps (if any) are percolated to the mother, via the remainder set Y. The effect of this headed construction is illustrated in the topicalization sentence in Figure 8. For exposition purposes I abbreviate cat information.

Figure 8

A topicalization construction.

4.2 Conjunction and respectively

For the analysis of verbal coordination I assume a familiar binary branching rule like X→XX, which I formalize in HPSG terms as (85), drawing from Beavers & Sag (Reference Beavers, Sag and Müller2004). I leave open the possibility of revising this rule as proposed in Sag (Reference Sag, Kim and Wechsler2002) in order to generalize to other categories, and to deal with coordination of ‘unlike categories’ phenomena.Footnote ²¹

1. (85) Verbal Coordination Construction

   

This construction allows two verbal constituents to be coordinated as long as they have compatible cat values and first daughter is not marked with a coordinator (i.e. bears the specification [crd–]), and the second daughter is marked with a coordinator (i.e. bears [crdcrd-marked]). I assume that the crd-marked type has several subtypes, one for each kind of coordination marker. Thus, the lexical entry for and is lexically specified as [crdconj], the entry for disjunction or is lexically specified as [crddisj], and so on. Since the mother node in (85) is underspecified for the value of crd, this rule can apply recursively as intended.

There are various types of coordination, often with special syntactic, prosodic, semantic and pragmatic characteristics. To give some well-known examples, consider packaging conjunction (e.g. Eggs, cheese and bacon was all I needed), numeral conjunction (e.g. I counted five hundred and twenty-two cats), arithmetical conjunction (e.g. Two and two is four), conditional conjunction (e.g. Take one more step and I'll shoot you where you stand), and intensification conjunction (e.g. The sound became louder and louder), among others like Boolean conjunction, and plurality-forming non-Boolean conjunction (Tom and Mary agreed, You can't simultaneously drink and drive, Kim rarely sings and dances, or Robin read the newspaper and dried her hair in exactly twenty seconds). These and other coordination patterns can be modeled with a type hierarchy of constructions in which each type introduces the specific constraints associated with that kind of coordination.

In this work I focus on non-Boolean conjunction, and on the formalization of the Shared Dependent Condition for Conjunction. In order to describe sum indices in HPSG, I view ‘⊕’ as the function defined in (86).

1. (86) ⊕(x, y)=z iff z =(x ⊔y )

Plurality-forming conjunction must create a plurality from the denotation of the conjuncts. This is done by taking the indices α₁ and α₂ of the conjuncts and combining them via ‘⊕’. Similarly, the referents of every sign selected by the conjuncts must be combined via ‘⊕’. This is formalized in (87), which generalizes to any number m of shared dependents.Footnote ²²

1. (87) Non-Boolean Conjunction (m⩾0)

   

In this work I focus on the instances of v-coord-cxt which are also subtypes of nbool-cnj-cxt. Obviously, other kinds of verbal coordination do not form a plurality (e.g. Boolean coordinations like It is not raining and it is not snowing or disjunctions like I can sing or dance).

The current grammar thus licenses structures like the one in Figure 9, with the semantics ∃z(men(z)∧z=(x ₁⊕x ₂)∧sing(e ₁, x ₁)∧dance(e ₂, x ₂)). If x ₁⊕x ₂ is interpreted as equality then x ₁=x ₂ and therefore the men are both singing and dancing. However, if x ₁⊕x ₂ is interpreted as a sum then x ₁≠x ₂ and we obtain a reading where the men singing and dancing are different. This interpretation is consistent with the adverb respectively. Whereas Gawron & Kehler's (Reference Gawron and Kehler2004) account obtains these readings from different syntactic trees and different semantic representations, the present account does not. In fact, both strict and weak identity readings are the result of a single tree structure and a single semantic representation.

Figure 9

Valent sharing in conjunction: The men sang and danced.

A sentence like (88) is modeled in the same way, the only difference being that the subject is an i-sum rather than a plural.

1. (88) Sue and Karen sing and dance.

   z=(y ₁⊕y ₂)∧sue=y ₁∧karen=y ₂∧z=(x ₁⊕x ₂)∧ sing(e ₁, x ₁)∧dance(e ₂, x ₂)

This can be interpreted with or without a respectively reading because of ‘⊕’. Moreover, if z is a plurality then by the symmetry of ‘⊕’ it follows that either y ₁=x ₁ and y ₂=x ₂, or that y ₁=x ₂ and y ₂=x ₁.

Instances of respectively readings involving filler–gap dependencies like Which book and what magazine did you say that John bought and Bill read? and others discussed in Section 3.4.2 are captured in a very similar way, as shown in Figure 10. The only difference is that the dependents that are shared and summed are extracted signs rather than signs in situ. For simplicity, I omit the semantic representation but it should be clear that the verbs bought and read predicate x ₁ and x ₂ respectively, and that the two variables are combined via ‘⊕’ at the conjunction node. The signs in slash percolate in the tree as a consequence of the Headed Construction. Exactly as before, if ‘⊕’ is interpreted as equality then z=(x ₁=x ₂), but if it is interpreted as plurality-forming cumulation then z is a plurality formed with x ₁ and x ₂. For an account of interrogative and subject–auxiliary inversion constructions see Ginzburg & Sag (Reference Ginzburg and Sag2000) and Sag (Reference Sag, Boas and Sag2010b).

Figure 10

Gap sharing in an across-the-board extraction construction.

The adverb respectively operates essentially like any other verbal modifier, syntactically and semantically. The feature sel(ect) allows modifiers, determiners and markers to lexically impose constraints on the heads that they combine with. Such structures are modeled via the head-functor-cxt provided in (89).Footnote ²³ Since this rule does not impose any constraints on the mother node, the Headed Construction will force all the syntactic features of the head and the mother to be identical.

1. (89) Head–Functor Construction

   

The lexical entry for respectively is provided in (90), and consists of an adverb that selects a verbal head via sel and predicates the verb event.

1. (90)

   

Note that since no constraints are imposed on val, the verbal head can be a verb phrase or a sentence, coordinate or not. Figure 11 illustrates how (90) and (89) interact. I omit some features for exposition purposes. The adverb is interpreted as in Section 3.3.

Figure 11

Generic example of adverbial adjunction.

If we revise the conjunction rule in (4.2) so that the sign in sel that conjuncts share is combined via ‘⊕’ then we obtain an N′ like the one in (91) and the respective semantic representation. The analysis is shown in Figure 12, and is parallel to what happens to shared dependents recorded in val and slash as discussed above.

1. (91) The event will take place between [days six and seven]_N′

   days(z)∧∃s ₁order(s ₁, x ₁)=6∧∃s ₂order(s ₂, x ₂)=7∧

   s=s ₁⊕s ₂∧z=x ₁⊕x ₂

Figure 12

Conjunction of adnominal expressions.

Basically, when adjuncts are conjoined any sign selected via sel is combined via the ‘⊕’ relation. It is trivial to revise the conjunction rule in (87) to achieve this effect on sel values.

A similar approach can be adopted for shared dependent cumulation in RNR and in split–antecedent relative clause extraposition, as discussed in Section 3.4.2. There are several alternative analyses of RNR and Extraposition, which I cannot discuss because of space limitations (but see Chaves Reference Chaves2012a). My goal here is to suggest that Right-Node-Raised shared dependents are also subject to the shared dependent cumulation phenomenon: any signs selected as dependents by each of the conjuncts are combined via ‘⊕’, as shown, for the example in (55a) above, in Figure 13.

If ‘⊕’ is interpreted as equality then we obtain standard non-additive RNR (the strict identity reading, where the total of money under discussion is $20,000). If ‘⊕’ is interpreted as plurality-forming cumulation then we obtain the additive RNR cases (the weak identity reading, where the total of money is $10,000). I suspect that there are various kinds of RNR construction, and that the kind illustrated below is essentially across-the-board rightward extraction, along the lines of Gazdar (Reference Gazdar1981); see also Chaves (Reference Chaves2011). The adverb respectively can adjoin to the coordinate S^e and evoke a bijection as usual, provided that there is a ranking established by surface order. In the case of Figure 13 no such ranking is possible and therefore the presence of ‘respectively’ is not felicitous. If a ranking can be established, however, we obtain cases like (58) above.

Figure 13

Right-Node Raising.