A Choice Function Analysis of Either in the Either/or Construction

Abstract

In this paper, I propose an analysis that covers both the wide scope or reading of the either/or construction and the availability of Alternative Question and Yes/No Question readings, namely a hybrid of an ellipsis analysis and a choice function analysis of either. After presenting two sets of data, I introduce two hybrid analyses that combine an ellipsis analysis and a choice function analysis. The two differ from each other in terms of the item that introduces the choice function variable: in the first analysis, the disjunction particle or introduces the choice function variable while in the second analysis, either has that semantic role. It is demonstrated that the two analyses both account for the either/or construction data, whereas only the second hybrid analysis, in which either introduces the choice function variable, explains the Alternative Question and Yes/No Question data. Finally, I review another account proposed in previous research, namely the focus alternative semantics analysis, and point out its problems.

I am indebted to Akira Watanabe, Noriko Imanishi, Christopher Tancredi, audience at LENLS 16, and two anonymous reviewers for helpful comments and suggestions. All remaining errors are my own. This study was supported by Grant-in-Aid for Young Scientists (#18K12414) from the Japan Society for the Promotion of Science, to which I am grateful.

1 The Data

As noted in [1] as a problematic case and discussed in [2] in more detail, when disjunction is combined with certain kinds of elements in a sentence the sentence is (at least) two-ways ambiguous:

• (1) The department is looking for a phonologist or a phonetician. (cf. [1])

  • a. $\left[\kern-0.15em\left[ {\,{\text{look}}\,{\text{for}}\,} \right]\kern-0.15em\right]\,\left( {\left[\kern-0.15em\left[ {\,{\text{a}}\,{\text{phonologist}}\,{\text{or}}\,{\text{a}}\,{\text{phonetician}}\,} \right]\kern-0.15em\right]} \right)\,\left( {\text{d}} \right)\quad \quad \quad \quad \left( {\text{narrow scope}} \right)$

  • b. $\begin{aligned} \left[\kern-0.15em\left[ {\,{\text{look}}\,{\text{for}}\,} \right]\kern-0.15em\right]\,\left( {\left[\kern-0.15em\left[ {\,{\text{a}}\,{\text{phonologist}}\,} \right]\kern-0.15em\right]} \right)\,\left( {\text{d}} \right)\, \vee \,\left[\kern-0.15em\left[ {\,{\text{look}}\,{\text{for}}\,} \right]\kern-0.15em\right]\, & \left( {\left[\kern-0.15em\left[ {\,{\text{a}}\,{\text{phonetician}}\,} \right]\kern-0.15em\right]} \right)\,\left( {\text{d}} \right)\, \\ & \quad \quad \quad \quad \quad \quad \left( {\text{wide scope}} \right) \\ \end{aligned}$

There are two de dicto readings of or in relation to the intentional predicate. The narrow scope or de dicto reading is in (1a), and under this reading the department would be satisfied by finding either a phonologist or a phonetician. The “problematic” de dicto reading, which I am interested in, is described in (1b). On this reading, the department does not yet necessarily have a specific candidate in mind. They do already have in mind which of the two types of specialist they are going to look for, but the speaker forgot which it was. This reading becomes clearer when continued with “… but I don’t know which.” Thus, the overall meaning is as if the disjunction is connecting two propositions, taking widest scope, even though the indefinite in each disjunct takes narrow scope. This is called the “wide scope or” reading in [2].

[3] observes that the possible readings of a sentence change when either comes into the structure. He states a generalization:

• (2) [3]’s generalization (from [4]):

  • a. In or coordinations without either, as well as in either…or… coordinations with either undisplaced, the scope of or is confined to those positions where either can potentially appear.

  • b. When either is displaced it specifies the scope of or to be at that displaced position.

(2a) is based on the assumption that the base position of either is next to the left edge of the Disjunction Phrase (DisjP). Thus, when either is adjacent to the DisjP all three readings are available (3), whereas when either floats to a higher position the narrow scope or de dicto reading disappears (4). Note that sentences with either floated higher than that in (4) behave in the same way as (4).

• (3) Mary is looking for (either) a maid or a cook.

  • a. $\left[\kern-0.15em\left[ {\,{\text{look}}\,{\text{for}}\,} \right]\kern-0.15em\right]\,\left( {\left[\kern-0.15em\left[ {\,{\text{a}}\,{\text{maid}}\,{\text{or}}\,{\text{a}}\,{\text{cook}}\,} \right]\kern-0.15em\right]} \right)\,\left( {\text{m}} \right)\quad \quad \quad \quad \quad \quad \quad \quad \quad \left( {\text{narrow scope}} \right)$

  • b. $\left[\kern-0.15em\left[ {\,{\text{look}}\,{\text{for}}\,} \right]\kern-0.15em\right]\,\left( {\left[\kern-0.15em\left[ {\,{\text{a}}\,{\text{maid}}\,} \right]\kern-0.15em\right]} \right)\,\left( {\text{m}} \right)\, \vee \,\left[\kern-0.15em\left[ {\,{\text{look}}\,{\text{for}}\,} \right]\kern-0.15em\right]\,\left( {\left[\kern-0.15em\left[ {\,{\text{a}}\,{\text{cook}}\,} \right]\kern-0.15em\right]} \right)\,\left( {\text{m}} \right)\quad \left( {\text{wide scope}} \right)$

• (4) Mary is either looking for a maid or a cook.

  • a. $* ?\left[\kern-0.15em\left[ {\,{\text{look}}\,{\text{for}}\,} \right]\kern-0.15em\right]\,\left( {\left[\kern-0.15em\left[ {\,{\text{a}}\,{\text{maid}}\,{\text{or}}\,{\text{a}}\,{\text{cook}}\,} \right]\kern-0.15em\right]} \right)\,\left( {\text{m}} \right)\quad \quad \quad \quad \quad \quad \quad \quad \left( {\text{narrow scope}} \right)$

  • b. $\left[\kern-0.15em\left[ {\,{\text{look}}\,{\text{for}}\,} \right]\kern-0.15em\right]\,\left( {\left[\kern-0.15em\left[ {\,{\text{a}}\,{\text{maid}}\,} \right]\kern-0.15em\right]} \right)\left( {\text{m}} \right)\, \vee \,\left[\kern-0.15em\left[ {\,{\text{look}}\,{\text{for}}\,} \right]\kern-0.15em\right]\,\left( {\left[\kern-0.15em\left[ {\,{\text{a}}\,{\text{cook}}\,} \right]\kern-0.15em\right]} \right)\,\left( {\text{m}} \right)\quad \left( {\text{wide scope}} \right)$

[4] and [5] report data which at first glance look like an exception to [3]’s generalization, where disjunction can take wide scope over an island as in (5) but either cannot appear out of the island as in (6). In (5), or can take either narrow or wide scope with respect to if. Note that either can appear inside the island and the disjunction can take narrow or wide scope with respect to if as in (7). ((5) and (6) are taken from [4]).

• (5) If Bill praises Mary or Sue then John will be happy.

  • a. If Bill praises Mary then John will be happy and if Bill praises Sue then John will be happy. (narrow scope)

  • b. If Bill praises Mary then John will be happy or if Bill praises Sue then John will be happy. (wide scope)

• (6) *Either if Bill praises Mary or Sue then John will be happy.

• (7) If Bill praises either Mary or Sue then John will be happy. (^OKNS/^OKWS)

The fact that sentences with either inside an island do have wide scope or readings as in (7) conforms to the generalization in (2a), since sentences with either in its base position can have the scope of or higher than the surface position of either. In contrast, it goes against the generalization in (2b), since floated either does not mark the exact scope of or but allows the scope of or to be in a higher position.

To sum up, [3]’s generalization in (2) states that (i) in sentences with no either or with either in its base position, or can take both narrow scope and wide scope, while (ii) in sentences with floated either, only the wide scope or reading is available. We have also reviewed additional data reported by [4] and [5], in which or can take scope over an island but either cannot overtly appear outside the island.

Lastly, consider (8). An interrogative sentence with a DisjP without either is ambiguous between an Alternative Question (AltQ) and a Yes/No Question (YNQ) as in (8a). Once either comes in, however, an AltQ reading is no longer available and the sentence is unambiguously a YNQ regardless of the position of either, as shown in (8b, c).

• (8) Availability of question readings and the position of either

  • a. Did John see a maid or a cook? (AltQ/YNQ)

  • b. Did John see either a maid or a cook? (*AltQ/YNQ)

  • c. Did John either see a maid or a cook? (*AltQ/YNQ)

In the rest of this paper, I propose an analysis that accounts for the wide scope or reading of the either/or construction and the availability of AltQ/YNQ readings introduced above. In Sect. 2, I first introduce two hybrid analyses that combine an ellipsis analysis and a choice function analysis. The two differ from each other in terms of the item that introduces the choice function variable: in the first analysis, the disjunction particle or introduces the choice function variable while in the second analysis, either has that semantic role. It is demonstrated that the two analyses both account for the either/or construction data, whereas only the second hybrid analysis, in which either introduces the choice function variable, explains the AltQ/YNQ data. I thus eventually propose a hybrid analysis of an ellipsis analysis and a choice function analysis of either. In Sect. 3, I review a previous study and point out its problems. Section 4 concludes.

2 Proposal

2.1 Two Hybrid Analyses

I first introduce two possible analyses combining an ellipsis analysis and a choice function analysis, and examine the wide scope or data. Both of the analyses combine an ellipsis analysis, and a choice function analysis in which an item introduces a choice function variable and the wide scope or reading is obtained through Existential Closure of the choice function variable.

The first hypothesis is that or introduces a choice function variable (cf. [5, 6]) and either only has a syntactic role of marking the left edge of the first disjunct (cf. [7]). The choice function variable that or introduces takes the set of disjuncts, the denotation of the DisjP, as its argument and the position of Existential Closure determines the scope position of or. With the work of either, it is guaranteed that the scope position of or is never lower than the position of either, since either determines the size of the DisjP.

Let us look at the examples. In sentences with no either or with either in its base position (9) (= (3)), where there is an ambiguity between narrow scope and wide scope or, no ellipsis is involved in the derivation of the examples. Thus there are multiple possible positions for Existential Closure which correspond to the multiple possible scope positions of or.

• (9) Ambiguous between NS and WS or

  • a. Mary is looking for a maid or a cook.

  • b. Mary is looking for either a maid or a cook.

    ○ [∃f] Mary is looking for [∃f] PRO to FIND f({a maid, a cook}) (cf. [8])

In sentences with floated either as in (10), where the wide scope or reading is forced, either marks the left edge of the first disjunct and ellipsis is involved in the derivation. Since the choice function variable is introduced with the disjunction, Existential Closure is restricted to a position above the DisjP. With this analysis, we can account for the fact that only the wide scope or reading is available in the sentences.

• (10) Unambiguous: only WS or

  • a. Mary is either looking for a maid or a cook (= (4)).

    ○ ∃f. Mary is f({looking for a maid, looking for a cook})

  • b. Mary either is looking for a maid or a cook.

    ○ ∃f. Mary f({is looking for a maid, is looking for a cook})

  • c. Either Mary is looking for a maid or a cook.

    ○ ∃f. f({Mary is looking for a maid, Mary is looking for a cook})

The second hypothesis is that either introduces a choice function variable and or forms a set of disjuncts that serves as its argument. The claim that a DisjP denotes the set of its disjuncts is not new (cf. [9] among others). I adopt a compositional semantics of DisjP with the denotation of or in (11). (For detailed discussion, see Sect. 3.2).

• (11) $\left[\kern-0.15em\left[ {\,{\text{or}}\,} \right]\kern-0.15em\right]^{\text{w,g}} \,{ = }\,\uplambda{\text{x}}_{{ < {\text{s,}}\upsigma > }} .\,\uplambda{\text{y}}_{{ < {\text{s,}}\upsigma > }} .\,\left\{ {\left[\kern-0.15em\left[ {\,{\text{x}}\,} \right]\kern-0.15em\right]^{\text{w}} ,\,\left[\kern-0.15em\left[ {\,{\text{y}}\,} \right]\kern-0.15em\right]^{\text{w}} } \right\}$

• (12) $\left[\kern-0.15em\left[ {\,{\text{a}}\,{\text{maid}}\,{\text{or}}\,{\text{a}}\,{\text{cook}}\,} \right]\kern-0.15em\right]^{\text{w,g}} \,{ = }\,\left\{ {\left[\kern-0.15em\left[ {\,{\text{a}}\,{\text{maid}}\,} \right]\kern-0.15em\right]^{\text{w}} ,\left[\kern-0.15em\left[ {\,{\text{a}}\,{\text{cook}}\,} \right]\kern-0.15em\right]^{\text{w}} } \right\}$

Or has a set-forming function as its denotation. It takes two arguments of the same type and forms a set of them. The result of combining or with the disjuncts is the set of the disjuncts, as in (12).

The second version of the hybrid analysis can equally capture the facts in (9) and (10) since, as we can observe from the data, the position where the choice function variable is placed coincides with the overt position of either. In this analysis, we assign either the semantic work of introducing the choice function variable, as in (13). To get this to work out formally, I analyze this as involving a covert operator coindexed with either, whose sole semantic work is to modify the assignment function g so that it assigns to its index a choice function variable f_i, as in (14).

• (13) $\left[\kern-0.15em\left[ {\,{\text{either}}_{\text{i}} \,} \right]\kern-0.15em\right]^{\text{w,g}} \,{ = }\,{\text{g}}\left( {\text{i}} \right)$

• (14) $\begin{aligned} & \left[\kern-0.15em\left[ {\,{\text{Op}}_{\text{i}} \,\left[ {{\text{either}}_{\text{i}} \,{\text{DisjP}}} \right]\,} \right]\kern-0.15em\right]^{\text{w,g}}\,{ = }\,\left[\kern-0.15em\left[ {\,{\text{either}}_{\text{i}} \,{\text{DisjP}}\,} \right]\kern-0.15em\right]^{{{\text{w,}}\,{\text{g[i}}\, \to \,{\text{f}}_{\text{i}} ]}} ,\,{\text{where}}\,{\text{f}}_{\text{i}} \, \in \,{\text{D}}_{\text{Chf}} \,{\text{is}}\,{\text{a}}\,{\text{choice}}\, \\ & {\text{function}}\,{\text{Chf}}\left( {{\text{f}}_{\text{i}} } \right)\,{\text{iff}}\,{\text{for}}\,{\text{all}}\,{\text{P}}\,{\text{in}}\,{\text{dom}}\left( {{\text{f}}_{\text{i}} } \right){:}\,{\text{f}}_{\text{i}} \left( {\text{P}} \right)\, \in \,{\text{P}} \\ \end{aligned}$

With the items, the NS reading of the sentence with either in its base position is derived as in (15) and the WS reading of the sentence with floated either is derived as in (16).

• (15) Mary is looking for [_TP PRO TO FIND [_XP Op_i either_i [_DisjP a maid or a cook]]].

  • a. $\begin{aligned} \left[\kern-0.15em\left[ {\,{\text{XP}}\,} \right]\kern-0.15em\right]^{\text{w,g }} \, & { = }\,\left[\kern-0.15em\left[ {\,{\text{either}}_{\text{i}} \,{\text{DisjP}}\,} \right]\kern-0.15em\right]^{{{\text{w,}}\,{\text{g[i}}\, \to {\text{f}}_{\text{i}} ]}} \, :\,{\text{f}}_{\text{i}} \, \in \,{\text{D}}_{\text{Chf}} \\ & \,{ = }\,\left[\kern-0.15em\left[ {\,{\text{either}}_{\text{i}} \,} \right]\kern-0.15em\right]^{{{\text{w,}}\,{\text{g[i}}\, \to \,{\text{f}}_{\text{i}} ]}} \, \, \left( {\left[\kern-0.15em\left[ {\,{\text{DisjP}}\,} \right]\kern-0.15em\right]^{{{\text{w,}}\,{\text{g[i}}\, \to \,{\text{f}}_{\text{i}} ]}} } \right)\, :\,{\text{f}}_{\text{i}} \, \in \,{\text{D}}_{\text{Chf}} \\ & \,{ = }\,{\text{f}}_{\text{i}} \,\left( {\{ {\text{a}}\,{\text{maid}}\,{\text{in}}\,{\text{w,}}\,{\text{a}}\,{\text{cook}}\,{\text{in}}\,{\text{w}}\} } \right)\, :\,{\text{f}}_{\text{i}} \, \in \,{\text{D}}_{\text{Chf}} \\ \end{aligned}$

  • b. $\left[\kern-0.15em\left[ {\,{\text{TP}}\,} \right]\kern-0.15em\right]^{\text{w,g}} \,{ = }\,\uplambda\,{\text{w}} .\,\exists {\text{f}}_{\text{i}} .\,{\text{Chf}}\,\left( {{\text{f}}_{\text{i}} } \right)\,\& \,{\text{Mary}}\,{\text{to}}\,{\text{find}}\,{\text{f}}_{\text{i}} \,\left( {\left\{ {{\text{a}}\,{\text{maid}}\,{\text{in}}\,{\text{w,}}\,{\text{a}}\,{\text{cook}}\,{\text{in}}\,{\text{w}}\,{\text{a}}\,{\text{maid}}\,{\text{in}}\,{\text{w,}}\,{\text{a}}\,{\text{cook}}\,{\text{in}}\,{\text{w}}} \right\}} \right)\,{\text{in}}\,{\text{w}}$

  • c. $\begin{aligned} & \left[\kern-0.15em\left[ {\,\left( { 1 5} \right)\,} \right]\kern-0.15em\right]^{\text{w,g}} { = }\uplambda{\text{w}}^{{\prime }} .\,{\text{Mary}}\,{\text{is}}\,{\text{looking}}\,{\text{for}}\,\left[ {\uplambda{\text{w}} .\,\exists {\text{f}}_{\text{i}} .\,{\text{Chf}}\left( {{\text{f}}_{\text{i}} } \right)\,\& \,{\text{Mary}}\,{\text{to}}\,{\text{find f}}_{\text{i}} \left( {\left\{ {\text{a}} \right.} \right.} \right. \\ & \left. {\left. {\left. {{\text{maid}}\,{\text{in}}\,{\text{w,}}\,{\text{a}}\,{\text{cook}}\,{\text{in}}\,{\text{w}}} \right\}} \right)\,{\text{in}}\,{\text{w}}} \right]{\text{in}}\,{\text{w}}^{{\prime }} \\ \end{aligned}$

• (16) Mary is [_XP Op_i either_i [_DisjP looking for PRO TO FIND a maid or looking for PRO TO FIND a cook]].

  • a. $\begin{aligned} \left[\kern-0.15em\left[ {\,{\text{XP}}\,} \right]\kern-0.15em\right]^{\text{w,g}} & \,{ = }\,\left[\kern-0.15em\left[ {\,{\text{either}}_{\text{i}} \,{\text{DisjP}}\,} \right]\kern-0.15em\right]^{{{\text{w,}}\,{\text{g[i}}\, \to \,{\text{f}}_{\text{i}} ]}} \, :\,{\text{f}}_{\text{i}} \, \in \,{\text{D}}_{\text{Chf}} \\ & = \,\left[\kern-0.15em\left[ {\,{\text{either}}_{\text{i}} \,} \right]\kern-0.15em\right]^{{{\text{w,}}\,{\text{g[i}}\, \to \,{\text{f}}_{\text{i}} ]}} (\left[\kern-0.15em\left[ {{\text{DisjP}}\,} \right]\kern-0.15em\right]^{{{\text{w,}}\,{\text{g[i}}\, \to \,{\text{f}}_{\text{i}} ]}} )\, :\,{\text{f}}_{\text{i}} \, \in \,{\text{D}}_{\text{Chf}} \\ & { = }\,{\text{f}}_{\text{i}} (\{\uplambda{\text{w}}^{{\prime }} .\,\uplambda{\text{x}} .\,{\text{x}}\,{\text{is}}\,{\text{looking}}\,{\text{for}}\,\left[ {\uplambda{\text{w}} .\,{\text{Mary}}\,{\text{to}}\,{\text{find}}\,{\text{a}}\,{\text{maid}}\,{\text{in}}\,{\text{w}}} \right]\,{\text{in}}\,{\text{w}}^{{\prime }} ,\\ & \;\,\uplambda{\text{w}}^{{\prime }} .\,\uplambda{\text{x}} .\,{\text{x}}\,{\text{is}}\,{\text{looking}}\,{\text{for}}\,\left[ {\uplambda{\text{w}} .\,{\text{Mary}}\,{\text{to}}\,{\text{find}}\,{\text{a}}\,{\text{cook}}\,{\text{in}}\,{\text{w}}} \right]\,{\text{in}}\,{\text{w}}^{{\prime }} \} ) {:} \\ & \;\,{\text{f}}_{\text{i}} \, \in \,{\text{D}}_{\text{Chf}} \\ \end{aligned}$

  • b. $\begin{aligned} \left[\kern-0.15em\left[ {\,\left( { 1 6} \right)\,} \right]\kern-0.15em\right]^{\text{w,g}} & \,{ = }\uplambda{\text{w}}^{{\prime \prime }} .\,\exists {\text{f}}_{\text{i}} .\,{\text{Chf}}\,\left( {{\text{f}}_{\text{i}} } \right)\,\& \,{\text{f}}_{\text{i}} \left( {\left\{ {\uplambda{\text{w}}^{{\prime }} .\,\uplambda{\text{x}} .\,{\text{x}}\,{\text{is}}\,{\text{looking}}\,{\text{for}}\,\left[ {\uplambda{\text{w}} .\,{\text{Mary}}\,{\text{to}}\,{\text{find}}\,} \right.} \right.} \right. \\ & \left. {{\text{a}}\,{\text{maid}}\,{\text{in}}\,{\text{w}}} \right]\,{\text{in}}\,{\text{w}}^{{\prime }} ,\,\uplambda{\text{w}}^{{\prime }} .\,\uplambda{\text{x}} .\,{\text{x}}\,{\text{is}}\,{\text{looking}}\,{\text{for}}\,\left[ {\uplambda{\text{w}} .\,{\text{Mary}}\,{\text{to}}\,{\text{find}}\,{\text{a}}\,} \right. \\ & \left. {{\text{cook}}\,{\text{in}}\,{\text{w}}} \right]\,\left. {\left. {{\text{in}}\,{\text{w}}^{{\prime }} } \right\}} \right) ( {\text{w}}^{{\prime \prime }} )\left( {\text{Mary}} \right) \\ \end{aligned}$

So far, the first and the second hypotheses both account for the set of data examined. In order to tease apart the two hypotheses, I consider AltQ and YNQ data in the next section.

2.2 AltQ/YNQ Data Distinguish Hybrid Analyses

In this section, I turn to AltQ and YNQ data. As for the semantics of AltQs and YNQs, it is assumed here that the Question (Q) operator existent in the CP level in interrogatives has a different denotation in the two constructions. For AltQs, I adopt [10]’s wh operator that moves to take CP scope and leaves its restrictor in situ.¹

[10]’s claim is that there is a wh operator (and/or the Q morpheme in C) that moves to the CP domain and takes CP scope while its trace is interpreted as a choice function variable. An AltQ (17a) has the LF representation in (17b). They propose that the Q operator and the wh operator in AltQs have the denotations in (18). The wh operator combined with the index does the work of rewriting the assignment function.

• (17) AltQ example and its LF representation

  • a. Did John drink coffee or tea?

  • b. [_CP wh i [_C′ Q [_IP John drank [t_i coffee or tea]]]] (cf. [10])

• (18) Denotations of items

  • a. $\left[\kern-0.15em\left[ {\,{\text{Q}}\,} \right]\kern-0.15em\right] =\uplambda{\text{q}}_{\text{st}} .\,\uplambda{\text{w}} .\,\uplambda{\text{p}}_{\text{st}} .\,{\text{p}}\, = \,{\text{q}}$

  • b. $\left[\kern-0.15em\left[ {\,{\text{wh}}\,} \right]\kern-0.15em\right]^{\text{w,g}} =\uplambda{\text{R}}_{{{<}{\text{Chf,}}\,{<}{\text{s,}}\,{<}{\text{st,}}\,{\text{t}}{>}{>}{>}}}\uplambda{\text{w}} .\,\uplambda{\text{p}} .\, [\exists {\text{f}} .\,{\text{Chf}}\left( {\text{f}} \right)\,\& \,{\text{R}}\left( {\text{f}} \right)\left( {\text{w}} \right)\left( {\text{p}} \right) ] ,\,{\text{where}}\,{\text{f}}\, \in \,{\text{D}}_{\text{Chf}}$

The derivation of (17b) proceeds as in (19) in the notation adopted here. The DisjP denotes a set of the disjuncts and the trace of the wh operator is taken to be a choice function variable that takes that set as its argument.

• (19) Derivation of (17b)

  • a. $\left[\kern-0.15em\left[ {\,{\text{t}}_{\text{i}} \,{\text{coffee}}\,{\text{or}}\,{\text{tea}}\,} \right]\kern-0.15em\right]^{{{\text{g[i}}\, \to \,{\text{f}}_{\text{i}} ]}} \,{ = }\,{\text{f}}_{\text{i}} \left( {\{ {\text{coffee,}}\,{\text{tea}}\} } \right)$

  • b. $\left[\kern-0.15em\left[ {\,{\text{IP}}\,} \right]\kern-0.15em\right]^{{{\text{w,}}\,{\text{g[i}}\, \to \,{\text{f}}_{\text{i}} ]}} \,{ = }\,\uplambda{\text{w}}^{{\prime }} .\,{\text{John}}\,{\text{drank}}\,{\text{f}}_{\text{i}} \left( {\{ {\text{coffee,}}\,{\text{tea}}\} } \right)\,{\text{in}}\,{\text{w}}^{{\prime }}$

  • c. $\left[\kern-0.15em\left[ {\,{\text{C}}^{{\prime }} \,} \right]\kern-0.15em\right]^{{{\text{w,}}\,{\text{g[i}}\, \to \,{\text{f}}_{\text{i}} ]}} \,{ = }\,\uplambda{\text{w}} .\,\uplambda{\text{p}} .\,{\text{p = }}\uplambda{\text{w}}^{{\prime }} .\,{\text{John}}\,{\text{drank}}\,{\text{f}}_{\text{i}} \,\left( {\{ {\text{coffee,}}\,{\text{tea}}\} } \right)\,{\text{in}}\,{\text{w}}^{{\prime }}$

  • d. $\left[\kern-0.15em\left[ {\,{\text{CP}}\,} \right]\kern-0.15em\right]^{\text{w,g}} \,{ = }\,\uplambda{\text{w}} .\,\uplambda{\text{p}} .\,\exists {\text{f}}_{\text{i}} .\,{\text{Chf}}\left( {{\text{f}}_{\text{i}} } \right)\,\& \,{\text{p}}\,{ = }\,\uplambda{\text{w}}^{{\prime }} .\,{\text{John}}\,{\text{drank}}\,{\text{f}}_{\text{i}} \left( {\{ {\text{coffee,}}\,{\text{tea}}\} } \right)\,{\text{in}}\,{\text{w}}^{{\prime }}$

For YNQs, I assume that there is a distinct Q operator that derives a YNQ reading when the denotation of the IP in a question is a single proposition. This Q operator has a special semantic denotation which takes a single proposition and gives back the set of it and its negation as the question interpretation as in (20).² With this operator in the CP level, the interpretation of a YNQ (21a) is as in (21b). The Q operator takes the proposition denoted by the IP, and the meaning of the whole sentence is the set of the proposition and its negation, successfully deriving the YNQ reading.

• (20) $\left[\kern-0.15em\left[ {\,{\text{Q}}_{\text{YNQ}} \,} \right]\kern-0.15em\right]^{\text{w,g}} \,{ = }\,\uplambda{\text{p}} .\,\left\{ {\uplambda{\text{w}} .\,{\text{p}}\left( {\text{w}} \right) ,\,\uplambda{\text{w}} .\,\neg {\text{p}}\left( {\text{w}} \right)} \right\}$

• (21) A YNQ and its denotation

  • a. Q_YNQ Did John come?

  • b. $\left[\kern-0.15em\left[ {\,\left( { 2 1 {\text{a}}} \right)\,} \right]\kern-0.15em\right]^{\text{w,g}} \,{ = }\,\left\{ {\uplambda{\text{w}} .\,{\text{John}}\,{\text{came}}\,{\text{in}}\,{\text{w,}}\,\uplambda{\text{w}} .\,\neg {\text{John}}\,{\text{came}}\,{\text{in}}\,{\text{w}}} \right\}$

Let us now proceed to the discussion of whether the two hybrid analyses can handle the AltQ/YNQ data (8), repeated in (22).

• (22) Availability of question readings and the position of either

  • a. Did John see a maid or a cook? (AltQ/YNQ)

  • b. Did John see either a maid or a cook? (*AltQ/YNQ)

  • c. Did John either see a maid or a cook? (*AltQ/YNQ)

According to the first version of the hybrid analysis, there is no difference between (22a–c) in that the IPs in all of the sentences denote a single proposition. This is because of the choice function variable introduced by or, which is present in all of the sentences. The choice function variable takes the set denoted by the DisjP and gives back a single member of the set, and thus the denotation of the IP ends up as a single proposition. We can derive the YNQ reading for these sentences with the Q operator for YNQs in (20). For (23), which is the LF representation of (22b) with an overt either in its base position, the Q operator takes the proposition that the IP denotes and gives back the set of it and its negation, as in (24). The same account applies to the availability of the YNQ reading in (22a) and (22c).

• (23) Q_YNQ John saw either a maid or a cook

• (24) $\begin{aligned} & \left[\kern-0.15em\left[ {\,\left( { 2 3} \right)\,} \right]\kern-0.15em\right]^{\text{w,g}} \,{ = }\,\{\uplambda{\text{w}} .\,\exists {\text{f}} .\,{\text{John}}\,{\text{saw}}\,{\text{f}}\left( {\{ {\text{a}}\,{\text{maid,}}\,{\text{a}}\,{\text{cook}}\} } \right)\,{\text{in}}\,{\text{w,}}\,\uplambda{\text{w}} .\,\neg \exists {\text{f}} .\,{\text{John}}\,{\text{saw}}\, \\ & {\text{f}}(\{ {\text{a}}\,{\text{maid,}}\,{\text{a}}\,{\text{cook}}\} )\,{\text{in}}\,{\text{w}}\} \\ \end{aligned}$

However, the first version of the hybrid analysis cannot derive the AltQ reading in (22a). Making use of the wh operator that moves to take CP scope and whose trace is interpreted as a choice function variable (18) will give rise to two choice function variables in the LF structure of (22a): one originating from the wh operator and another from or. Once one of the two variables takes the set of the disjuncts as its argument, the result is a single member that the other variable is unable to operate over.

For example, consider example (17a) repeated in (25), whose underlying structure is as in (17b) repeated in (25a). If we combine this structure with the first version of the hybrid analysis, the denotation of the DisjP is as in (25b), where or introduces a choice function variable that takes as its argument the set of the disjuncts.

• (25) Did John drink coffee or tea?

  • a. [_CP wh i [_C′ Q [_IP John drank [t_i coffee or tea]]]]

  • b. $\left[\kern-0.15em\left[ {\,{\text{coffee}}\,{\text{or}}\,{\text{tea}}\,} \right]\kern-0.15em\right]^{\text{w,g}} \,{ = }\,\uplambda{\text{w}} .\,{\text{f}}\,\left( {\{ {\text{coffee}}\,{\text{in}}\,{\text{w,}}\,{\text{tea}}\,{\text{in}}\,{\text{w}}\} } \right)$

It is clear that (25b) cannot be the argument of the choice function variable introduced in the position of the trace of the wh operator, since (25b) is a single semantic interpretation that is not a set. It is impossible to derive the AltQ interpretation of (25) with the first version of the hybrid analysis.

How about the second version of the hybrid analysis? I take there to be two possible structures for (22a): either being completely absent and either being covert. When there is a covert either adjacent to the DisjP in (22a), the structure is identical to (22b), analyzed in (24) above. The existence of either accounts for the availability of the YNQ reading in (22a–c) in a way similar to (24). The second version of the hybrid analysis also successfully obtains the AltQ reading of (22a) with the Q operator and the wh operator in (18). Since either is absent, the only choice function variable in the structure is the one originating from the wh operator. The computation thus proceeds in exactly the same manner as (19):

• (26) Did John see a maid or a cook? (= (22a))

  • a. [cp Wh i [c′ Q [ip John saw [t_i a maid or a cook]]]]

  • b. $\begin{aligned} \left[\kern-0.15em\left[ {\,{\text{CP}}\,} \right]\kern-0.15em\right]^{\text{w,g}} & \,{ = }\,\left[\kern-0.15em\left[ {\,{\text{Wh}}\,} \right]\kern-0.15em\right]^{\text{w,g}} \,\left( {\left[\kern-0.15em\left[ {\,{\text{i}}\,{\text{C}}^{{\prime }} \,} \right]\kern-0.15em\right]^{\text{w,g}} } \right) \\ & \,{ = }\,\left[\kern-0.15em\left[ {\,{\text{Wh}}\,} \right]\kern-0.15em\right]^{\text{w,g}} \,\left( {\uplambda{\text{f}}_{\text{i}} .\,\left[\kern-0.15em\left[ {\,{\text{C}}^{{\prime }} \,} \right]\kern-0.15em\right]^{{{\text{w,g[i}}\, \to \,{\text{f}}_{\text{i}} ]}} } \right) \\ & \,{ = }\,\left[\kern-0.15em\left[ {\,{\text{Wh}}\,} \right]\kern-0.15em\right]^{\text{w,g}} \,\left( {\uplambda{\text{f}}_{\text{i}} .\,\left[\kern-0.15em\left[ {\,{\text{Q}}\,} \right]\kern-0.15em\right]^{{{\text{w,g[i}}\, \to \,{\text{f}}_{\text{i}} ]}} } \right)\,\left( {\left[\kern-0.15em\left[ {\,{\text{IP}}\,} \right]\kern-0.15em\right]^{{{\text{w,g[i}}\, \to \,{\text{f}}_{\text{i}} ]}} } \right) \\ & \,{ = }\,\left[\kern-0.15em\left[ {\,{\text{Wh}}\,} \right]\kern-0.15em\right]^{\text{w,g}} \left( {\uplambda{\text{f}}_{\text{i}} .\,\left[\kern-0.15em\left[ {\,{\text{Q}}\,} \right]\kern-0.15em\right]^{{{\text{w,g[i}}\, \to \,{\text{f}}_{\text{i}} ]}} } \right.\, (\uplambda{\text{w}}^{{\prime }} .\,{\text{John}}\,{\text{saw}}\,{\text{f}}_{\text{i}} \left( {\{ {\text{a}}\,{\text{maid,}}\,{\text{a}}\,{\text{cook}}\} } \right)\,{\text{in}}\,{\text{w}}^{{\prime }} ) )\\ & \,{ = }\,\left[\kern-0.15em\left[ {\,{\text{Wh}}\,} \right]\kern-0.15em\right]^{\text{w,g}} (\uplambda{\text{f}}_{\text{i}} .\uplambda{\text{w}} .\,\uplambda{\text{p}} .\,{\text{p = }}\uplambda{\text{w}}^{{\prime }} .\,{\text{John}}\,{\text{saw}}\,{\text{f}}_{\text{i}} \left( {\{ {\text{a}}\,{\text{maid,}}\,{\text{a}}\,{\text{cook}}\} } \right)\,{\text{in}}\,{\text{w}}^{{\prime }} )\\ & \,{ = }\,\uplambda{\text{w}} .\,\uplambda{\text{p}} . { [}\exists {\text{f}}_{\text{i}} .\,{\text{Chf}}\left( {{\text{f}}_{\text{i}} } \right)\,\& \,{\text{p}}\,{ = }\,\uplambda{\text{w}}^{{\prime }} .\,{\text{John}}\,{\text{saw}}\,{\text{f}}_{\text{i}} \left( {\{ {\text{a}}\,{\text{maid,}}\,{\text{a}}\,{\text{cook}}\} } \right)\,{\text{in}} \\ & {\text{w}}^{{\prime }} ]\, :\,{\text{f}}_{\text{i}} \, \in \,{\text{D}}_{\text{Chf}} \\ \end{aligned}$

Now it has been shown that the second version of the hybrid analysis, namely the ellipsis analysis + the choice function analysis of either, accounts for both the wide scope or reading data and the AltQ/YNQ data.

3 Comparison with Other Analysis

3.1 The Focus Alternative Semantics Analysis

[11] investigate the semantics of AltQs based on the focus alternative semantics of [13] and comment on the role of either in the either/or construction.

The basic idea of focus alternative semantics is that focused items have two semantic values: an ordinary semantic value and a focus semantic value. For example, in sentence (27), the focused item John has its ordinary denotation as its ordinary semantic value (27a) and a set of alternatives (of the same semantic type) as its focus semantic value (27b). A sentence that has a focused item in it also has an ordinary semantic value (27c) and a focus semantic value (27d), which is a set of propositions in which the position of the focused item varies according to the focus semantic value of the focused item.

• (27) [John]_F left.

  • a. $\left[\kern-0.15em\left[ {\,{\text{John}}_{\text{F}} \,} \right]\kern-0.15em\right]^{\text{o}} \,{ = }\,{\text{John}}$

  • b. $\left[\kern-0.15em\left[ {\,{\text{John}}_{\text{F}} \,} \right]\kern-0.15em\right]^{\text{f}} \,{ = }\,\left\{ {{\text{John,}}\,{\text{Bill,}}\,{\text{Amelie,}}\, \ldots } \right\}$

  • c. $\left[\kern-0.15em\left[ {\,{\text{John}}_{\text{F}} \,{\text{left}}\,} \right]\kern-0.15em\right]^{\text{o}} \,{ = }\,\uplambda{\text{w}} .\,{\text{John}}\,{\text{left}}\,{\text{in}}\,{\text{w}}$

  • d. $\begin{aligned} \left[\kern-0.15em\left[ {\,{\text{John}}_{\text{F}} \,{\text{left}}\,} \right]\kern-0.15em\right]^{\text{f}} & \,{ = }\,\left\{ {{\text{p:}}\,{\text{p}}\,{ = }\,\uplambda{\text{w}} .\,{\text{x}}\,{\text{left}}\,{\text{in}}\,{\text{w}}\, |\,{\text{x}}\, \in \,{\text{D}}} \right\} \\ & \,{ = }\,\left\{ {\uplambda{\text{w}} .\,{\text{John}}\,{\text{left}}\,{\text{in}}\,{\text{w,}}\,\uplambda{\text{w}} .\,{\text{Bill}}\,{\text{left}}\,{\text{in}}\,{\text{w,}}\,\uplambda{\text{w}} .\,{\text{Amelie}}\,{\text{left}}\,{\text{in}}\,{\text{w,}}\, \ldots } \right\} \\ \end{aligned}$

Building on the idea that either is focus-sensitive [11, 14, 15] propose that either operates over the focus semantic value of its sister, just like focus-sensitive items like only do, as we see immediately below. Their denotation of either XP is in (28), where either is proposed to be a focus sensitive operator that takes its sister DisjP as its argument as in (29) and gives rise to “closure” as in (30) (note that this denotation is primarily aimed to capture the “epistemic” reading of or discussed in [16] among others).

• (28) $\begin{aligned} & \left[\kern-0.15em\left[ {\,{\text{either}}\,{\text{XP}}\,} \right]\kern-0.15em\right]^{\text{o}} \,{ = }\,{\text{for}}\,{\text{all}}\,{\text{q}}\,{\text{in}}\,\left[\kern-0.15em\left[ {\,{\text{XP}}\,} \right]\kern-0.15em\right]^{\text{f}} \, :\,{\text{may}}\,{\text{q}}\,\& \,\neg \exists {\text{p}}\,\left[ {{\text{for}}\,{\text{all}}\,{\text{q}}\,{\text{in}}\,\left[\kern-0.15em\left[ {\,{\text{XP}}\,} \right]\kern-0.15em\right]^{\text{f}} \, :\,{\text{p}}\, \cap \,{\text{q}}\,} \right. \\ & \left. {{ = }\,\left\{ {} \right\}\,\& \,{\text{may p}}} \right] \\ \end{aligned}$

• (29) $\begin{aligned} & \left[\kern-0.15em\left[ {\,{\text{either}}\,{\text{it}}\,{\text{is}}\,{\text{raining}}\,{\text{or}}\,{\text{it}}\,{\text{is}}\,{\text{snowing}}\,} \right]\kern-0.15em\right]^{\text{o}} \,{ = }\,{\text{may}}\,{\text{r}}\,\& \,{\text{may}}\,{\text{s}}\,\& \,\neg \exists {\text{p}}\,\left[ {{\text{p}}\, \cap \,{\text{r}}\,{ = }\,\left\{ {} \right\}\,\& \,{\text{p}}\,} \right. \\ & \left. { \cap \,{\text{s}}\,{ = }\,\left\{ {} \right\}\,\& \,{\text{may}}\,{\text{p}}} \right] \\ \end{aligned}$

• (30) Either it is raining or it is snowing. ≈ It is possible that it is raining and it is possible that it is snowing and there are no other relevant possibilities.

In order to support the claim that focus-sensitive items such as only access the focus alternatives of their sister, [11] present an analysis based on focus alternative semantics to account for intervention effects in AltQs. Only, a focus-sensitive item, has the semantics in (31).

• (31) $\left[\kern-0.15em\left[ {\,{\text{only}}\,\upphi\,} \right]\kern-0.15em\right]^{\text{o}} \,{ = }\,\left[ {\uplambda{\text{w}} .\,{\text{for}}\,{\text{all}}\,{\text{p}}\,{\text{such}}\,{\text{that}}\,{\text{p}}\left( {\text{w}} \right)\,{ = }\, 1\,\& \,{\text{p}}\, \in \,\left[\kern-0.15em\left[ {\,\upphi\,} \right]\kern-0.15em\right]^{\text{f}} } \right]\,\left( {{\text{p}}\,{ = }\,\left[\kern-0.15em\left[ {\,\upphi\,} \right]\kern-0.15em\right]^{\text{o}} } \right)$

(31) means that of all the alternative propositions introduced by the focus semantic value of the sister of only, the only true one is the ordinary semantic value of the sister. Thus in the sample sentence (32), the overall meaning is equivalent to (32b).

• (32) $\left[\kern-0.15em\left[ {\,{\text{Only}}\,{\text{John}}_{\text{F}} \,{\text{left}}\, ]} \right]\kern-0.15em\right]^{\text{o}}$

  • a. $\begin{aligned} & { = }\,\uplambda{\text{w}} .\,\left[ {{\text{for}}\,{\text{all}}\,{\text{p}}\,{\text{such}}\,{\text{that}}\,{\text{p}}\left( {\text{w}} \right)\,{ = }\, 1\,\& \,{\text{p}}\, \in \,} \right.\left. {\left\{ {\uplambda{\text{w}} .\,{\text{x}}\,{\text{left}}\,{\text{in}}\,{\text{w}}\, |\,{\text{x}}\, \in \,{\text{D}}} \right\}} \right]\,\left( {{\text{p}}\,{ = }\,\left\{ {\uplambda{\text{w}} .\,} \right.} \right. \\ & \left. {\left. {{\text{John}}\,{\text{left}}\,{\text{in}}\,{\text{w}}} \right\}} \right) \\ \end{aligned}$

  • b. ${ = }\,\uplambda{\text{w}} .\,\left[ {{\text{for}}\,{\text{all}}\,{\text{x}}\,{\text{such}}\,{\text{that}}\,{\text{x}}\,{\text{left}}\,{\text{in}}\,{\text{w}}} \right]\,\left( {{\text{x}}\,{ = }\,{\text{John}}} \right)$

[11] present data like (33a, b) to show that only gives rise to intervention effects in AltQs. When only structurally intervenes between the DisjP and the Q operator in the CP layer, an AltQ reading is unavailable.

• (33) Intervention effects of only

  • a. ? Did John or Susan introduce Sue to only Mary_F?

  • b. ?* Did only Mary_F introduce Sue to Bill or to Tom?

According to [11], the Q operator in wh-Questions takes the focus semantic value of its sister and makes it the ordinary semantic value of the whole sentence as in (34).³

• (34) Two semantic values of wh-Questions

  • a. $\left[\kern-0.15em\left[ {\,{\text{Q}}\,\upphi\,} \right]\kern-0.15em\right]^{\text{o}} \,{ = }\,\left[\kern-0.15em\left[ {\,\upphi\,} \right]\kern-0.15em\right]^{\text{f}}$

  • b. $\left[\kern-0.15em\left[ {\,{\text{Q}}\,\upphi\,} \right]\kern-0.15em\right]^{\text{f}} \,{ = }\,\left\{ {\,\left[\kern-0.15em\left[ {\,{\text{Q}}\,\upphi\,} \right]\kern-0.15em\right]^{\text{o}} \,} \right\}$

[11] argue that this Q operator derives the AltQ reading of (35) (although they argue against the idea that AltQs are a kind of wh-Question).

• (35) Did the program execute or the computer crash?

  = [_CP Q [_DisjP [the program executed] or [the computer crashed]]] (cf. [11])

[11] further claim that the ordinary semantic value of a DisjP is the union of the disjuncts while the focus semantic value is the set of the disjuncts. Based on [17]’s analysis, [11] argue for an ordinary semantic value in (36a) and a focus semantic value in (36b) for the DisjP in the AltQ in (36).⁴

• (36) Did [_DisjP the program execute or the computer crash]?

  • a. $\left[\kern-0.15em\left[ {\,{\text{DisjP}}\, } \right]\kern-0.15em\right]^{\text{o}} \,{ = }\,\uplambda{\text{w}} .\,{\text{the}}\,{\text{program}}\,{\text{executed}}\,{\text{in}}\,{\text{w}}\,{\text{or}}\,{\text{the}}\,{\text{computer}}\,{\text{crashed}}\,{\text{in}}\,{\text{w}}$

  • b. $\left[\kern-0.15em\left[ {\,{\text{DisjP}}\, } \right]\kern-0.15em\right]^{\text{f}} \,{ = }\,\left\{ {\uplambda{\text{w}} .\,{\text{the}}\,{\text{program}}\,{\text{executed}}\,{\text{in}}\,{\text{w,}}\uplambda{\text{w}} .\,{\text{the}}\,{\text{computer}}\,{\text{crashed}}\,{\text{in}}\,{\text{w}}} \right\}$

By combining the denotation of DisjPs with that of the Q operator in wh-Questions, the AltQ reading of (35) is obtained. From (34a), the ordinary semantic value of the interrogative (35) is the focus semantic value of the DisjP, i.e. (36b). Based on previous research that analyzes the semantic denotation of questions as the set of their possible answers (cf. [18]), this is equivalent to the AltQ interpretation.

Now let us proceed to the discussion of how only intervenes between the DisjP and the Q operator. For (33b), [11] assume a (simpified) underlying structure (37), in which the DisjP has two VPs as disjuncts. Then, the denotation of the DisjP is the set of the denotations of the disjuncts as in (38) and the DisjP combines with Mary, which is the associate of only and carries a Focus intonation. The result is as in (39). After that, only makes use of the two semantic values to derive the semantics in (40). With the semantic work of the Q operator in (34), the overall interpretation of the whole sentence results in (41). Since this is not an AltQ interpretation, [11] account for the intervention effects of only in AltQs.

• (37) Q [_XP only [_IP Mary_F [_DisjP [introduce Sue to Bill] or [ to Tom]]]?

• (38) Denotation of DisjP

  • a. $\begin{aligned} & \left[\kern-0.15em\left[ {\,{\text{DisjP}}\, ]} \right]\kern-0.15em\right]^{\text{o}} \,{ = }\,\uplambda{\text{x}} .\,\uplambda{\text{w}} .\,{\text{x}}\,{\text{introduces}}\,{\text{Sue}}\,{\text{to}}\,{\text{Bill}}\,{\text{in}}\,{\text{w}}\,{\text{or}}\,{\text{x}}\,{\text{introduces}}\,{\text{Sue}}\,{\text{to}}\,{\text{Tom}}\,{\text{in}}\, \\ & {\text{w}} \\ \end{aligned}$

  • b. $\begin{aligned} & \left[\kern-0.15em\left[ {\,{\text{DisjP}}\,} \right]\kern-0.15em\right]^{\text{f}} \,{ = }\,\left\{ {\uplambda{\text{x}} .\,\uplambda{\text{w}} .\,{\text{x}}\,{\text{introduces}}\,{\text{Sue}}\,{\text{to}}\,{\text{Bill}}\,{\text{in}}\,{\text{w,}}\,\uplambda{\text{x}} .\,\uplambda{\text{w}} .\,{\text{x}}\,{\text{introduces}}\,{\text{Sue}}\,{\text{to}}\,{\text{Tom}}\,{\text{in}}\,} \right. \\ & \left. {\text{w}} \right\} \\ \end{aligned}$

• (39) Denotation of IP

  • a. $\begin{aligned} & \left[\kern-0.15em\left[ {\,{\text{IP}}\,} \right]\kern-0.15em\right]^{\text{o}} \,{ = }\,\uplambda{\text{w}} .\,{\text{Mary}}\,{\text{introduces}}\,{\text{Sue}}\,{\text{to}}\,{\text{Bill}}\,{\text{in}}\,{\text{w}}\,{\text{or}}\,{\text{Mary}}\,{\text{introduces}}\,{\text{Sue}}\,{\text{to}}\,{\text{Tom}}\, \\ & {\text{in}}\,{\text{w}} \\ \end{aligned}$

  • b. $\begin{aligned} & \left[\kern-0.15em\left[ {\,{\text{IP}}\,} \right]\kern-0.15em\right]^{{\rm{f}}} \,{ = }\,\left\{ {\uplambda{\rm{w}} .\,{\rm{x}}\,{\text{introduces}}\,{\rm{Sue}}\,{\text{to}}\,{\rm{Bill}}\,{\text{in}}\,{\rm{w,}}\,\uplambda{\text{w}}{\rm{.}}\,{\text{x}}\,{\rm{introduces}}\,{\text{Sue}}\,{\rm{to}}\,{\text{Tom}}\,{\rm{in}}\,{\text{w}}\,{\rm{|}}\,} \right. \\ & \left. {{\text{x}}\, \in \,{\rm{D}}_{\text{e}} } \right\}\,{\rm{ = }}\,\left\{ {\uplambda{\text{w}}{\rm{.}}\,{\text{Mary}}\,{\rm{introduces}}\,{\text{Sue}}\,{\rm{to}}\,{\text{Bill}}\,{\rm{in}}\,{\text{w,}}\,\uplambda{\rm{w}} .\,{\rm{Jane}}\,{\text{introduces}}\,{\rm{Sue}}\,{\text{to}}\,} \right. \\ & {\text{Bill}}\,{\rm{in}}\,{\text{w}}\,{\rm{,}}\,\uplambda{\rm{w}} .\,{\rm{Mary}}\,{\text{introduces}}\,{\rm{Sue}}\,{\text{to}}\,{\rm{Tom}}\,{\text{in}}\,{\text{w,}}\,\uplambda{\rm{w}} .\,{\rm{Amy}}\,{\text{introduces}}\,{\rm{Sue}}\,{\text{to}}\, \\ & \left. {{\rm{Tom}}\,{\text{in}}\,{\rm{w, \ldots}}\,} \right\} \\ \end{aligned}$

• (40) Denotation of XP

  • a. $\begin{aligned} & \left[\kern-0.15em\left[ {\,{\text{XP}}\,} \right]\kern-0.15em\right]^{\text{o}} \,{\rm{ = }}\,\uplambda{\text{w}}{\rm{.}}\,\left[ {{\text{for}}\,{\rm{all}}\,{\text{p}}\,{\rm{such}}\,{\text{that}}\,{\rm{p}}\left( {\text{w}} \right)\,{\rm{ = }}\, 1\,\& \,{\rm{p}}\, \in \,} \right.\left\{ {\uplambda{\rm{w}} .\,{\rm{x}}\,{\text{introduces}}\,{\rm{Sue}}\,{\text{to}}\,{\rm{Bill}}\,} \right. \\ & \left. {\left. {{\text{in}}\,{\rm{w,}}\,\uplambda{\text{w}}{\rm{.}}\,{\text{x}}\,{\rm{introduces}}\,{\rm{Sue}}\,{\text{to}}\,{\rm{Tom}}\,{\rm{in}}\,{\text{w}}\,{\rm{|}}\,{\text{x}}\, \in \,{\rm{D}}_{\text{e}} } \right\}} \right]\,\left( {{\rm{p}}\,{ = }\,\uplambda{\rm{w}} .\,{\rm{Mary}}\,{\text{introduces}}\,} \right. \\ & \left. {{\text{Sue}}\,{\rm{to}}\,{\text{Bill}}\,{\rm{in}}\,{\text{w}}\,{\rm{or}}\,{\text{Mary}}\,{\rm{introduces}}\,{\text{Sue}}\,{\rm{to}}\,{\text{Tom}}\,{\rm{in}}\,{\text{w}}} \right) \\ \end{aligned}$

  • b. $\left[\kern-0.15em\left[ {\,{\text{XP}}\,} \right]\kern-0.15em\right]^{\text{f}} \,{ = }\,\left\{ {\,\left[\kern-0.15em\left[ {\,{\text{XP}}\,} \right]\kern-0.15em\right]^{\text{o}} \,} \right\}$

• (41) $\left[\kern-0.15em\left[ {\,\left( { 3 7} \right)\,} \right]\kern-0.15em\right]^{\text{o}} \,{ = }\,\left\{ {\,\left[\kern-0.15em\left[ {\,{\text{XP}}\,} \right]\kern-0.15em\right]^{\text{o}} \,} \right\}$

Thus, adopting focus alternative semantics enables us to account for the semantics of only and the intervention effect it induces in AltQs, and [11] suggest extending this analysis to either.

There are, however, problems in the analysis. I discuss them next.

3.2 Problems of the Focus Alternative Semantics Analysis

The focus alternative semantics analysis faces several difficulties when we try to explain the data introduced in the previous section. I first describe an empirical problem, and then move on to theoretical problems.

The first problem is that the analysis makes a wrong prediction for the scope of disjunction. Recall that, as repeated below, (42a) is ambiguous between wide scope and narrow scope or readings whereas (42b–d) only have a wide scope or reading.

• (42) Narrow and wide scope or

  • a. Mary is looking for either a maid or a cook.

  • b. Mary is either looking for a maid or looking for a cook.

  • c. Mary either is looking for a maid or is looking for a cook.

  • d. Either Mary is looking for a maid or Mary is looking for a cook.

According to the denotation in (28), either makes use of the focus semantic value of its sister and gives back an ordinary semantic value. We thus have no way to get the wide scope or reading of (42a). Claiming that either projects up the focus semantic value is not a possible move, taking into consideration AltQ/YNQ data:

• (43) Availability of question readings and the position of either

  • a. Did John see a maid or a cook? (AltQ/YNQ)

  • b. Did John see either a maid or a cook? (*AltQYNQ)

  • c. Did John either see a maid or a cook? (*AltQ/YNQ)

According to [11], the AltQ reading available for sentences like (43a) comes from the focus semantic value that projects up to the TP level and is lifted to the ordinary semantic value by the work of the covert Q operator in the C position. Given that the AltQ reading is unavailable when either comes in, it is clear that either does not pass up the focus semantic value of its sister node but closes the alternatives in the position it occupies. It thus seems difficult to explain the availability of the wide scope or reading available for sentences with either adjacent to the DisjP by giving either some semantic role related to focus.

Aside from the empirical problem, the focus alternative semantics analysis in the form introduced in the previous section has a theoretical problem in the semantics of the DisjP and only. Consider again the derivation of (37) in (38)–(41) above. Two problems exist in this derivation. First, the ordinary semantic value of the XP shown in (40) means that, of all p such that p is true and p is a member of the focus semantic value of the IP (a set of propositions of the form λw. x introduces Sue to Bill in w or of the form λw. x introduces Sue to Tom in w, where x is a focus alternative to Mary), the only true one is the proposition λw. Mary introduces Sue to Bill in w or Mary introduces Sue to Tom in w. However, note that in the focus semantic value of the IP, there are the propositions λw. Mary introduces Sue to Bill in w and λw. Mary introduces Sue to Tom in w. These propositions are presupposed to be false according to the semantics of only, since they are not equal to the proposition that is asserted to be true, λw. Mary introduces Sue to Bill in w or Mary introduces Sue to Tom in w. Thus, for the disjoined proposition to be true, at least one of its disjuncts has to be true, but the given semantics requires both disjuncts to be false. This renders (37) necessarily false. Although (37) is a degraded example, it is not intuitively necessarily false. It is easy to see that the problem lies at least in part in the semantics of only itself and the problem is carried over to acceptable sentences.

Following [19], we can avoid this problem by modifying the semantics of only to make use of entailment (cf. [20]):

• (44) $\left[\kern-0.15em\left[ {\,{\text{only}}\,\upphi\,} \right]\kern-0.15em\right]^{\text{o}} \,{ = }\,\uplambda{\text{w}} .\,\forall {\text{p}}\left[ {{\text{p}}\left( {\text{w}} \right)\,{ = }\, 1\,\& \,{\text{p}}\, \in \,\left[\kern-0.15em\left[ {\,\upphi\,} \right]\kern-0.15em\right]^{\text{f}} } \right]\,\left( {{\text{p}}\, \supseteq \,\left[\kern-0.15em\left[ {\,\upphi\,} \right]\kern-0.15em\right]^{\text{o}} } \right){:}\,\left[\kern-0.15em\left[ {\,\upphi\,} \right]\kern-0.15em\right]^{\text{o}} \,{ = }\, 1$

I next turn to the second problem of (38)–(41). Adopting the revised interpretation of only in (44), the ordinary semantic value of the XP shown in (40) means that, of all p such that p is true and p is a member of the focus semantic value of the IP (a set of propositions of the form λw. x introduces Sue to Bill in w or of the form λw. x introduces Sue to Tom in w, where x is a focus alternative to Mary), the only true one(s) is entailed by the proposition λw. Mary introduces Sue to Bill in w or Mary introduces Sue to Tom in w. However, neither the proposition λw. Mary introduces Sue to Bill in w or the proposition λw. Mary introduces Sue to Tom in w is entailed by the proposition λw. Mary introduces Sue to Bill in w or Mary introduces Sue to Tom in w (and in fact, there is no such proposition in the focus semantic value of IP that is entailed by λw. Mary introduces Sue to Bill in w or Mary introduces Sue to Tom in w). Thus there is no p which is a member of the focus semantic value of the IP and, at the same time, is entailed by the ordinary semantic value of IP, and the semantics in (40) then comes out true only if nobody introduced Sue to Bill and nobody introduced Sue to Tom. This difficulty, which still exists when the revised denotation of only is adopted, arises from the semantics of DisjPs, in which the ordinary semantic value is not a member of the focus semantic value.

Notice that this problem is avoided by adopting the compositional semantics of DisjPs proposed in this paper. I have proposed that or has a set-forming function. In its ordinary semantic value, or takes two arguments of the same type and forms a set of them as in (45a). Its focus semantic value is a singleton set of this function, as in (45b).

• (45) Compositional semantics of DisjPs

  • a. $\left[\kern-0.15em\left[ {\,{\text{or}}\,} \right]\kern-0.15em\right]^{{{\text{g,}}\,{\text{o}}}} \,{ = }\,\uplambda{\text{x}}_{\upsigma} .\,\uplambda{\text{y}}_{\upsigma} .\,\left\{ {{\text{x,}}\,{\text{y}}} \right\}$

  • b. $\left[\kern-0.15em\left[ {\,{\text{or}}\,} \right]\kern-0.15em\right]^{{{\text{g,}}\,{\text{f}}}} \,{ = }\,\left\{ {\uplambda{\text{x}}_{\upsigma} .\,\uplambda{\text{y}}_{\upsigma} .\,\left\{ {{\text{x,}}\,{\text{y}}} \right\}} \right\}$

With this semantics, the semantics of sentence (46) can be computed fully compositionally with the semantics of either in the present proposal. On the assumption that subjects reconstruct at LF in their base position, inside the VP, the LF representation has John below only and covert either in its base position.⁵

• (46) ${\text{John}}\,{\rm{only}}\,{\text{saw}}\,{\rm{Bill}}_{\text{F}} \,{\rm{or}}\,{\text{Sue}}_{{\rm{F}}} . \,{\rm{ = }}\,{\text{Only}}\,\left[ {_{{{\rm{IP}}}} \,{\text{John}}\,{\rm{saw}}\,\left[ {_{\text{XP}} {\rm{Op}}_{\text{i}} \,{\rm{either}}_{\text{i}} \,{\rm{Bill}}_{\text{F}} \,{\rm{or}}\,{\text{Sue}}_{{\rm{F}}} } \right]} \right] .$

The ordinary semantic value is the result of applying the choice function introduced by either to the set of disjuncts, as in (47a). The focus semantic value is the result of combining a singleton set of a choice function, which is the focus semantic value of either, and the set of the sets of the alternatives of the disjuncts, which is the focus semantic value of the DisjP, via pointwise function application. This is shown in (47b). The alternatives of the focus semantic value expand up to the IP level, resulting in (48b). Notice that either and Op are necessarily above the DisjP to resolve a type mismatch that would occur without them when saw combines with the DisjP.

• (47) Denotation of XP

  • a. $\left[\kern-0.15em\left[ {\,{\text{XP}}\,} \right]\kern-0.15em\right]^{{{\text{g[i}}\, \to \,{\text{f}}_{\text{i}} ] , {\text{o}}}} \,{ = }\,{\text{f}}_{\text{i}} \left( {\left\{ {{\text{Bill,}}\,{\text{Sue}}} \right\}} \right) : {\text{ f}}_{\text{i}} \, \in \,{\text{D}}_{\text{Chf}}$

  • b.

• (48) Denotation of IP

  • a. $\left[\kern-0.15em\left[ {\,{\text{IP}}\,} \right]\kern-0.15em\right]^{{{\text{g[i}}\, \to \,{\text{f}}_{\text{i}} ] , {\text{o}}}} \,{ = }\,{\text{John}}\,{\text{saw}}\,{\text{f}}_{\text{i}} \left( {\left\{ {{\text{Bill,}}\,{\text{Sue}}} \right\}} \right) :\,{\text{f}}_{\text{i}} \, \in \,{\text{D}}_{\text{Chf}}$

  • b.

Now the revised denotation of only with entailment in (44) comes into the structure. The interpretation of (46) is given in (49), assuming that Existential Closure of the choice function variable takes place above the whole proposition. This means that, for a particular way of picking out a value from a pair of disjuncts, every true sentence of the form John saw A or B using that way of picking values is entailed by the result of using that way of picking values in John saw Bill or Sue. This is the intended reading.

• (49) $\begin{aligned} & \left[\kern-0.15em\left[ {\,\left( { 4 6} \right)\,} \right]\kern-0.15em\right]^{{{\rm{g[i}}\, \to \,{\text{f}}_{{\rm{i}}} ] , {\text{o}}}} \,{\rm{ = }}\,\exists {\text{f}}_{{\rm{i}}} .\,{\rm{f}}_{\text{i}} \, \in \,{\rm{D}}_{\text{Chf}} \,{\rm{and}}\,{\text{for}}\,{\rm{all}}\,{\text{p}}\,{\rm{such}}\,{\text{that}}\,{\rm{p}}\left( {\text{w}} \right)\,{\rm{ = }}\, 1\,\& \,{\rm{p}}\, \in \,\left\{ {{\text{John}}\,{\rm{saw}}\,} \right. \\ & {\text{x: x}}\, \in \,\left. {\left\{ {{\rm{f}}\left( {\text{A}} \right){\rm{:}}\,{\text{f}}\, \in \,\left\{ {{\rm{f}}_{\text{i}} } \right\}\,{\rm{\& }}\,{\text{A}}\, \in \,} \right.\left. {\left\{ {\left\{ {{\rm{x,y}}} \right\}\, |\,{\rm{x}}\, \in \,\left[\kern-0.15em\left[ {{\text{Bill}}_{{\rm{F}}} } \right]\kern-0.15em\right]^{\text{g,f}} \,{\rm{and}}\,{\text{y}}\, \in \,\left[\kern-0.15em\left[ {{\rm{Sue}}_{\text{F}} } \right]\kern-0.15em\right]^{{{\rm{g,f}}}} \,} \right\}} \right\}} \right\} :\,{\rm{p}}\, \\ & \supseteq \,{\text{John}}\,{\rm{saw}}\,{\text{f}}_{{\rm{i}}} \left( {\left\{ {{\text{Bill,}}\,{\rm{Sue}}} \right\}} \right) \\ \end{aligned}$

From the discussion in this section, it is clear that the present claim not only covers most of the data but also has theoretical advantages over [11]’s claim reviewed in the previous section.⁶

4 Conclusion

In this paper, I have investigated in detail the semantics of the either/or construction and AltQs in English. It has been shown that the proposed analysis, namely a hybrid analysis of an ellipsis analysis and a choice function analysis of either, accounts for the availability of the wide scope or reading and the distribution of AltQ/YNQ readings in English.