1 Introduction
Erlewine & Gould (2014; 2016) claim to identify a previously unrecognized interpretation for Japanese internallyheaded relative clauses (IHRCs): the salient set reading. Erlewine & Gould (2016) (henceforth E&G) develop an account of this interpretation within a unified theory of internally, externally, and socalled doubly headed relatives. To separate the salient set interpretation of IHRCs from their standard readings, E&G invoke the maximal informativeness semantics for definites of von Fintel et al. (2014). In this response, we argue that the mechanism developed to produce the salient set reading does not extend to deliver the standard readings for IHRCs with quantified internal heads even with the help of maximal informativeness: it either derives the wrong reading (for certain downward entailing quantifiers) or none at all (for nonmonotonic quantifiers and certain upward entailing quantifiers).^{1}
A secondary goal of this response is to suggest that the salient set reading does not arise from an IHRC but rather a headless relative with a gap, a conclusion independently supported by Grosu & Hoshi (2018). The confusion arises, we claim, because the postposition or case marker that introduces the gap site does not surface (Kameshima 1989; 1990; Matsumoto 1997), giving the illusion of an IHRC.
2 The socalled salient set reading and Inverse Trace Conversion
E&G focus on IHRCs of the form in (1). These have an external quantifier zenbu ‘all’ and an internal head (IH) modified by a numeral ringoo mittsu ‘three apples’.
 (1)
 Erlewine & Gould (2016: 7)
 Junyawa
 JunyaTOP
 [_{IHRC}
 [Ayakaga
 AyakaNOM
 [_{IH}
 ringoo
 appleACC
 mittsu]
 threeCL
 muita]no]o
 peelPASTNOACC
 zenbu
 all
 tabeta.
 eatPAST
 ‘(lit.) Junya ate all of [that Ayaka peeled three apples].’
 ‘Ayaka peeled three apples and Junya ate all of them.’
The expected interpretation of such an IHRC with an internal head is one in which the numeral takes internal scope, rather than taking the whole RC as its restrictor, as it would in an externally headed relative (Hoshi 1995; Shimoyama 1999; 2001). On the regular reading Ayaka peels three apples and Junya eats those three apples. The salient set reading, on the other hand, arises when the IHRC refers not to the three peeled apples but to a set of apples of which the peeled apples are three. The pictures in (2) and (3) from E&G demonstrate two scenarios where this larger set is made salient. The salient set reading is true in context (2) if Junya ate the entire group of apples that Ayaka peeled three of. That larger group numbers 12. Context can play a large role in making one particular superset more salient. For instance, in context (3), two groups or baskets of apples are presented, one of which Ayaka peeled three apples from. The salient set reading here is one in which the IHRC refers to this basket of six apples, hence the interpretation that Junya ate all of those six apples.
(2)  
Interpretation of (1) in Context (2):  
(i) Junya ate the three peeled apples.  Regular Reading  
(ii) Junya ate the twelve apples.  Salient Set Reading 
(3)  
Interpretation of (1) in Context (3):  
(i)  Junya ate the three peeled apples.  Regular Reading  
(ii)  Junya ate the six apples in the first group.  Salient Set Reading^{2} 
One popular analysis of Japanese (and Korean) IHRCs is an Etype anaphora analysis. Essentially, the IHRC is semantically interpreted as a clause coordinated with the matrix clause, resumed by crosssentential anaphora (Hoshi 1995; Shimoyama 1999; 2001; Kim 2007 but see Grosu 2012; Grosu & Hoshi 2016; Landman 2016 for different approaches). The paraphrase in (4) illustrates this analysis:
(4)  [Ayaka peeled three apples] and [Junya ate all of them]. 
It is claimed in E&G that the Japanese translation of (4) using crosssentential anaphora does not have the salient set reading.^{3}^{4} If the salient set reading is indeed a reading of IHRCs, this observation could possibly be added to the list of other differences between crosssentential anaphora and IHRCs, as discussed in, for example, Hoshi (1995); Shimoyama (1999; 2001); Kim (2007), and Grosu & Landman (2012). We will show in Section 5.3, however, that there is an alternative syntactic parse, a headless relative parse, that would derive the salient set reading without any special mechanisms. This syntactic parse is independently available and we show that it can account for cases that the proposal in E&G cannot.
As noted, E&G attempt to derive the regular and the salient set readings from a single structure for IHRCs. They propose that all relative clauses derive from a protorelative structure in which the head and its determiner are clauseinternal (regardless of whether they are externally, internally or doublyheaded, or in English or Japanese). In (5), the internal head is copied and merged external to the RC. (E&G assume a process of Late Merge that gives a resulting structure in which the head’s determiner is repositioned so the higher head NP and the CP predicate form a constituent to the exclusion of the higher copy of the determiner.)
(5)  Japanese protorelative structure after Copy and Late Merge (E&G: 18) 
In order to interpret such structures, with multiple copies of the determiner, the general strategy since Fox (2002) is to employ a rule of Trace Conversion. In standard Trace Conversion, a lower copy’s quantificational determiner is replaced by a definite determiner, which introduces a variable that is bound by the higher quantifier. Trace Conversion is demonstrated below with Quantifier Raising (QR). In (6), the generalized quantifier every apple QRs. As a movement process, QR leaves a copy in its merge position, giving rise to the narrow syntactic representation in (6b). As it stands, (6b) cannot be semantically interpreted. Fox (2002) (see also Sauerland 1998) proposes a rule that effectively converts the lower copy into a covarying definite description (6c) bound by the higher copy.
(6)  Interpreting movement (QR) with copies:  
a.  Ayaka peeled every apple.  
b.  Narrow syntax  
[every apple]_{1} Ayaka peeled [every apple]_{1} (QR with copy)  
c.  LF after Trace Conversion  
[every apple] λx. Ayaka peeled [the apple x] 
Trace Conversion has two components: variable insertion and determiner replacement. Variable insertion (7a) adds an (identity) predicate that introduces a variable, and this predicate intersects with the NP. This essentially equates the lower copy with the variable indexed by movement. The second component is determiner replacement: the quantifier in the lower copy is replaced by the definite determiner THE.
(7)  a.  Variable insertion 
every λy.apple(y) → every [ λy.apple(y) & y = x ]  
b.  Determiner replacement  
every [ λy.apple(y) & y = x ] → THE [ λy.apple(y) & y = x ] 
Erlewine (2014) proposes an innovation he calls Inverse Trace Conversion. In Inverse Trace Conversion (ITC), it is the higher copy that is converted into a definite description. Inverse Trace Conversion captures various properties of scope reconstruction. An example is shown below with the narrow scope reading of the universally quantified subject every apple with respect to negation (E&G 2016: 13):
(8)  Every apple isn’t rotten.  
a.  Narrow syntax  
[every apple]_{1} NEG [_{vP} [every apple]_{1} is rotten]  
b.  LF after Inverse Trace Conversion  
[THEλy.apples(y) ] λX. NEG [_{vP} [every [λy.apple(y) & y ⊑ X ]] is rotten] 
In this case, variable insertion introduces a predicate that mereologically relates the lower copy to the higher copy: the individuals in the restrictor of the quantified lower copy constitute an atomic individual part (⊑) of the plurality of apples that the higher copy refers to.
E&G use ITC to compose IHRCs, which as noted, involve movement on their account just as externally headed relatives do. The difference is in which copy is pronounced. In IHRCs, the lower copy is pronounced and the higher copy undergoes ITC. Applying the process to the syntactic representation in (5), this results in the higher copy as a definite description, while allowing the quantifier to be interpreted in the lower copy (Hoshi 1995; Shimoyama 1999; 2001).
(9)  LF after Inverse Trace Conversion (E&G 2016: 26)^{5}  
a.  [_{DP} THE [_{NP} apple [_{CP} λX [Ayaka peeled [_{DP} three [_{NP} [_{NP} apple(s)] [λy. y ⊑ X]]]]]]]  
b.  [DP] = ⟦THE⟧ (λX. X apple(s) and Ayaka peeled three [atomic apple parts of X]) 
The IHRC as a whole, then, denotes the plurality of apples such that Ayaka peeled three apples in that plurality. This of course is the salient set reading: the IHRC denotes the larger set from which the lower quantifier takes its domain. The different contexts in (2)/(3) make different domain sets salient, hence the referent of the whole IHRC differs. The referent of the IHRC DP in (9) is the apple sum 1+2+3+4+5+6+7+8+9+10+11+12 in context (2), and 1+2+3+4+5+6 in context (3).
3 Derivation of the regular reading
To generate the regular reading—where the IHRC denotes just the three peeled apples in the examples above—from the LF after ITC, E&G invoke a recent analysis of definite descriptions found in von Fintel et al. (2014). To appreciate this innovation, it is helpful to review the earlier, standard analysis of definite descriptions (Sharvy 1980; Link 1983), which took them to denote the unique maximal object (or degree). For instance, the definite description in (10) denotes the maximal object that satisfies the restriction, i.e. the maximal number that is the number of children that John has. A formal definition is given in (11) following Sharvy (1980) and Link (1983).
(10)  the number of children that John has 
(11)  ⟦THEφ⟧ (where φ is of type ⟨α, ⟩) is defined only if there is a unique maximal object x such that φ(x) is true (based on the ordering on elements of type α). The reference of the φ (when defined) is this unique maximal element. 
The restrictor of THE is the set of degrees in (12a). Given the context in (12b), the unique maximal object in that set is 4 (12c, d): 5 and greater are not in the restrictor set, and 3 and below are not maximal compared to 4.
(12)  a.  λd. John has dmany children 
b.  Context: John has exactly four children.  
c.  1 < 2 < 3 < 4 < 

d.  THE returns the unique maximal object satisfying the restriction 
von Fintel et al. (2014) address a case where the maximal object/degree is not the denotation of the definite. Take (13) from von Fintel et al. (2014: 168).
(13)  the amount of walnuts sufficient to make a pan of baklava 
If we used the maximalitybased semantics of THE in (11), this definite would be undefined. For instance, if 150 g of walnuts is enough then so, technically, is 200 g, 700 g, and so on. But this means there’s no maximal object/degree that satisfies the restrictor, hence the definite is undefined. von Fintel et al.’s (2014) solution is to redefine maximality in terms of informativeness, not degrees or objects. This Maximal Informativeness Semantics (MIS) for THE is given below.
(14)  Maximal Informativeness Semantics for THE (von Fintel et al. 2014: 166)  
a.  ⟦THE⟧ (φ) is defined in w only if there is a uniquely maximal object x, based on the ordering ≥_{φ}, such that φ(w)(x) is true. The reference of “the φ” (when defined) is this maximal element.  
b.  For all x, y of type α and property φ of type ⟨s, ⟨α, t⟩⟩, x ≥_{φ} y iff λw.φ(w)(x) entails λw.φ(w)(y). 
The recipe here is, as before, to look at the individuals that satisfy the restrictor, but unlike the standard account, we collect the propositions each of these individuals/degrees form with the restrictor. The next step is then to identify the proposition that entails all the other propositions.
(15)  ⋮  
a.  “that 170 g of walnuts is sufficient to make a pan of baklava”  
b.  “that 160 g of walnuts is sufficient to make a pan of baklava”  
c.  “that 150 g of walnuts is sufficient to make a pan of baklava”  
d.  
e.  
⋮ 
Which propositions should we consider? Well, if 150 g is sufficient, then amounts higher than 150 are too simply because of what sufficient means.^{6} So all such numbers satisfy the property and generate propositions that are under consideration. But propositions that involve amounts lower than 150 g are not considered because they are false in this context—that’s why (15d, e) are crossed out.^{7} Now, of the remaining candidate propositions, we ask which one entails all others: well if 150 g is sufficient—again, just because of what sufficient means—this entails that any number greater than 150 is sufficient too. But this means the most informative proposition is the one that involves the lowest sufficient amount. What makes all the difference here compared to the original Sharvy/Link’s cases is that the restrictor property involves the predicate sufficient which reverses the entailment relations:
(16)  a.  Property: λd.dmuch walnuts is sufficient to make a pan of baklava 
b.  Context: 150 g of walnuts is sufficient to make a pan of baklava 
(17)  Whenever “dmuch walnuts is sufficient to make a pan of baklava” is true, it is necessarily true that “d’much walnuts is sufficient to make a pan of baklava” where d’ ≥ d. 
On the MIS account, the correct prediction is made that the referent of the definite description in (13) is the smallest amount of walnuts that would yield a true proposition, namely 150 g.
Returning to IHRCs, E&G claim that the MIS account of definites derives the regular reading from the LF after ITC. Consider the IHRC in (1), repeated below, on its regular reading where it denotes the three apples that are peeled by Ayaka. On the analysis of IHRCs as derived from the protoJapanese relative in (5), along with ITC, the IHRC is a definite description of the form in (19a), where the property φ is defined in (19b).
 (18)
 Junyawa
 JunyaTOP
 [_{IHRC}
 [Ayakaga
 AyakaNOM
 [_{IH}
 ringoo
 appleACC
 mittsu]
 threeCL
 muita]no]o
 peelPASTNOACC
 zenbu
 all
 tabeta.
 eatPAST
 ‘(lit.) Junya ate all of [that Ayaka peeled three apples].’
 ‘Ayaka peeled three apples and Junya ate all those three.’ (regular reading)
(19)  a.  THE φ 
b.  φ = “λX.λw. X apple(s) and Ayaka peeled three [atomic apple parts of X] in w” 
In the regular reading of example (1)/(18), the referent of the IHRC DP is 1+2+3 in either context (2) or context (3). The LF is repeated from (9) above.
(20)  LF after Inverse Trace Conversion (E&G 2016: 26)  
a.  [_{DP} THE [_{NP} apple [_{CP} λX [Ayaka peeled [_{DP} three [_{NP} [_{NP} apple(s)] [λy. y ⊑ X]]]]]]]  
b.  ⟦DP⟧ = ⟦THE⟧ (λX.λw. X apple(s) and Ayaka peeled three [atomic apple parts of X] in w) 
To determine the referent of (20b) on the MIS account of definite descriptions, we consider the propositions that result from saturating the property complement of THE with various pluralities—what we will call the whole or Xslot produced by ITC. Further, we consider only those propositions that are true in the context in which we evaluate the definite description. In context (2) (it works the same way with context (3)), the list of propositions in (21) are the true propositions generated by saturating the property with plural individuals. There are many such plural individuals, since not only are groups like 1+2+3+4 possible, but also 1+2+3+7 or 1+2+3+9+12. Vertical ellipses indicate a nonexhaustive list. We are also using an at least semantics for the numeral (deriving the exactly interpretation by implicature), because an exactly semantics, being nonmonotonic, would not generate any entailments. We take this up more fully in Section 4.1. The propositions listed below are a short hand: where we say A. peeled at least 3 apples of 1+2+3, we technically mean “1+2+3 are apples and Ayaka peeled at least 3 atomic apple parts of 1+2+3.”^{8}
(21)  a.  λw. A. peeled at least 3 apples of 1+2+3 in w 
b.  λw. A. peeled at least 3 apples of 1+2+3+4 in w  
c.  λw. A. peeled at least 3 apples of 1+2+3+5 in w  
d.  λw. A. peeled at least 3 apples of 1+2+3+5+6 in w  
⋮  
e.  λw. A. peeled at least 3 apples of 1+2+3+4+5 in w  
f.  λw. A. peeled at least 3 apples of 1+2+3+4+6 in w  
⋮  
g.  λw. A. peeled at least 3 apples of 1+2+3+4+5+6 in w  
h.  λw. A. peeled at least 3 apples of 1+2+3+10+11+12 in w  
⋮  
i.  λw. A. peeled at least 3 apples of 1+2+3+4+5+6+7 in w  
⋮  
j.  λw. A. peeled at least 3 apples of 1+2+3+4+5+6+7+8+…+12 in w 
We’ve collected all the true propositions in the relevant context. At this point the recipe says to step back from the particular context and identify the entailment relations among these propositions. As E&G show, there is an entailment relation among the propositions because whenever you peel at least three atomic apples in X, it is necessarily true that you peeled at least three apples in Y, when X is a part of Y:
(22)  “that Ayaka peeled at least three atomic apple parts of apples X” entails “that Ayaka peeled at least three atomic apple parts of apples Y” where X ⊑ Y 
So this means that the smaller the value of X is, the more informative the proposition ‘that Ayaka peeled at least three atomic apple parts of apples X’ is. The proposition in (21a) entails all other true propositions in (21).
(23)  “that Ayaka peeled at least three atomic apple parts of apples 1+2+3” entails “that Ayaka peeled at least three atomic apple parts of apples Y” for all Y such that 1+2+3 ⊑ Y 
On the other hand, there is no entailment when going from a larger whole plurality to a smaller one. In general, if someone peels at least three atomic apple parts of X, it does not follow that she peeled at least three atomic apple parts of W, where W is a part of X. The proposition in (21j) for instance, can be true in many different situations other than the situation where apples 1, 2, 3 are peeled. For example, (21j) is true in a situation where apples 10, 11, 12 are peeled. In such a situation, however, not all of the other propositions in (21) are true. For instance, (21a)–(21g) are not true. (Other propositions might be: like (21h).)
(24)  “that Ayaka peeled at least three atomic apple parts of X” does not entail “that Ayaka peeled at least three atomic apple parts of W” where W ⊑ X 
What all this means is that putting (22) and (24) together, the strongest proposition is (21a), where the number of peeled apples is the same as the plurality from which they are taken, i.e. three. More technically, based on informativenessbased ordering (14b), then, the ordering in (25) follows.
(25)  E&G (2016: 27)  
a.  X ≥_{φ} Y if and only if “that Ayaka peeled at least three atomic apple parts of X” entails “that Ayaka peeled at least three atomic apple parts of Y”  
b.  X ≥_{φ} Y if and only if X ⊑ Y. 
The ITCMIS analysis of the regular reading thus correctly predicts that the referent of the IHRC DP in (9) in context (2) or (3) is the apple sum 1+2+3. Given (14a) and (25), the apple sum 1+2+3 is the uniquely maximal (or most informative) object in the sense that it creates the most informative or strongest true proposition, “that Ayaka peeled three apple parts of 1+2+3”. So despite the variable insertion that would initially only give us a referent plurality that properly includes the individuals denoted by the internal head, E&G’s proposal lets the entire IHRC denote just those individuals picked out by the internal head with MIS.
When the internal head is universally quantified, the ITCMIS account derives the observed reading. Assume a scenario in which Ayaka peels all 12 of 12 apples.
 (26)
 Junyawa
 JunyaTOP
 [[
 Ayakaga
 AyakaNOM
 dono
 every
 ringomo
 apple
 muita]no]o
 peelPASTNOACC
 zenbu
 all
 tabeta.
 eatPAST
 ‘(lit.) Junya ate all of [that Ayaka peeled every apple].’
 ‘Ayaka peeled every apple and Junya ate all of them.’
This is true if Ayaka peeled all 12 apples and Junya ate those apples. This is a regular reading, and the ITCMIS derives it. The true propositions in this case are:
(27)  a.  λw. A. peeled every apple of 1 in w 
b.  λw. A. peeled every apple of 1+2 in w  
⋮  
c.  λw. A. peeled every apple of 1+2+3+4+5+6+7+8+9+10+11+12 in w 
The entailment relations are such that since every is downward entailing, only the proposition with the largest value of X entails all other propositions here and is not entailed by any other propositions. So if one peels all of a set X, one does not necessarily peel all of a superset of X. But if one peels all of a set X, one necessarily peels all of a subset of X. So the referent of the IHRC DP will be the plural individual that constitutes all 12 apples, as desired.
4 Undergeneration
While the ITCMIS analysis correctly derives the regular reading in the two cases discussed above, there are other instances in which it does not. We divide these into three categories. The first involves internal heads with nonmonotonic quantifiers (exactly n, at least/exactly half). With these, no unique, maximally informative proposition can be identified, hence the IHRC DP is undefined on a regular reading. The second involves a closer look at upward entailing quantifiers, like more than n and at least n. We will show that by overtly signalling more than n or at least n on the internal head, this has undesired consequences in certain scenarios.
The third category of problems involves (nonuniversal) downward entailing quantifiers (fewer than n, at most n), where the ITCMIS delivers the wrong plural individual as the meaning for the IHRC DP. We will walk through each of these cases as explicitly as possible, illustrating the issues with informal prose.
4.1 Nonmonotonic quantifiers
4.1.1 Exactly n
In the last section we noted that it is crucial to interpret numerals with an at least semantics. With an exactly semantics, there is no unique strongest proposition. Consider (18) again with an exactly semantics for the numeral:
(28)  a.  λw. A. peeled exactly 3 apples of 1+2+3 in w 
b.  λw. A. peeled exactly 3 apples of 1+2+3+4 in w  
c.  λw. A. peeled exactly 3 apples of 1+2+3+5 in w  
d.  λw. A. peeled exactly 3 apples of 1+2+3+5+6 in w  
⋮  
e.  λw. A. peeled exactly 3 apples of 1+2+3+4+5 in w  
f.  λw. A. peeled exactly 3 apples of 1+2+3+4+6 in w  
⋮  
g.  λw. A. peeled exactly 3 apples of 1+2+3+4+5+6 in w  
h.  λw. A. peeled exactly 3 apples of 1+2+3+10+11+12 in w  
⋮  
i.  λw. A. peeled exactly 3 apples of 1+2+3+4+5+6+7 in w  
⋮  
j.  λw. A. peeled exactly 3 apples of 1+2+3+4+5+6+7+…+11+12 in w 
As the reader can verify, no proposition here will entail any other regardless of whether the whole is larger or smaller:
(29)  a.  “that Ayaka peeled exactly 3 apples of W” does not entail “that Ayaka peeled exactly 3 apples of X”, where W ⊑ X 
b.  “that Ayaka peeled exactly 3 apples of Y” does not entail “that Ayaka peeled exactly 3 apples of X”, where X ⊑ Y 
The proposition that one peels exactly n of some set does not entail that one peels exactly n of some super set. So not only does the semantics fail to pick out the plural individual that is required on the regular reading (18) but the IHRC DP will be undefined since there is no unique strongest proposition. As von Fintel et al. (2014: 169–170) explicitly state, this would lead to a presupposition failure if there is more than one object that fulfills the predicate description, which is the case here. Crucially, though, a Japanese IHRC with an internal head overtly quantified by choodo ‘exactly’ is possible and felicitously has a regular reading, so that the IHRC refers to the plurality of three apples peeled.^{9}
 (30)
 Junyawa
 JunyaTOP
 [ [Ayakaga
 AyakaNOM
 [_{IH}
 ringoo
 appleACC
 choodo
 exactly
 mittsu]
 threeCL
 muita]no]o
 peelPASTNOACC
 zenbu
 all
 tabeta.
 eatPAST
 ‘(lit.) Junya ate all of [that Ayaka peeled exactly three apples].’
 ‘Ayaka peeled exactly three apples and Junya ate all those three.’
We do not see, then, how a regular reading can arise with predicates that contain nonmonotonic quantifiers.
4.1.2 At least half/exactly half
The nonmonotonic quantifier half causes similar problems. Consider (31) with hanbun ‘half’ in the context in (32), namely 1+2+3+4+5+6 out of the twelve apples are peeled by Ayaka. Example (31) has the regular reading, namely the reading that is true when Junya ate the six peeled apples, 1+2+3+4+5+6. Under this reading, the denotation of the IHRC DP in (33) should be 1+2+3+4+5+6.
 (31)
 E&G (2016: 5)
 Junyawa
 JunyaTOP
 [[Ayakaga
 AyakaNOM
 ringoo
 appleACC
 hanbun
 half
 muita]no]o
 peelPASTNOACC
 zenbu
 all
 tabeta
 eatPAST
 ‘(lit.) Junya ate all of [that Ayaka peeled half of the apples].’
 ‘Ayaka peeled half of the apples and Junya ate all of them.’
 (32)
This is the representation that the syntax generates for the IHRC DP:
(33)  ⟦DP⟧ = ⟦THE⟧ (λX.λw. X apples and Ayaka peeled half of the [atomic apple parts of X] in w) 
The first step of the MIS recipe is to assemble the true propositions, given a context, in which the property φ—the argument of THE in (33)—is saturated by alternative individuals. In the case of ITCgenerated sentences, the alternative propositions to consider are ones where the value of the whole (X) are different. In the context in (32), where apples 1 through 6 out of twelve apples are peeled, plural individuals (filling the Xslot of (33), the whole) that yield true propositions under an at least semantics for hanbun ‘half’ are given in (34).
(34)  a.  λw. A. peeled at least 1/2 of 1 in w 
b.  λw. A. peeled at least 1/2 of 1+2 in w  
c.  λw. A. peeled at least 1/2 of 1+2+3 in w  
d.  λw. A. peeled at least 1/2 of 1+2+3+4 in w  
e.  λw. A. peeled at least 1/2 of 1+2+3+4+5 in w  
f.  λw. A. peeled at least 1/2 of 1+2+3+4+5+6 in w  
g.  λw. A. peeled at least 1/2 of 1+2+3+4+5+6+7 in w  
h.  λw. A. peeled at least 1/2 of 1+2+3+4+5+6+7+8 in w  
i.  λw. A. peeled at least 1/2 of 1+2+3+4+5+6+7+8+9+10 in w  
j.  λw. A. peeled at least 1/2 of 1+2+3+4+5+6+7+8+9+10+11 in w  
k.  λw. A. peeled at least 1/2 of 1+2+3+4+5+6+7+8+9+10+11+12 in w 
Note first off, that an at least semantics for hanbun ‘half’ is required, since on an exactly semantics, the only proposition in (34) that would be true would be (34k). This would make apples 1–12 the referent of the IHRC DP. We still do not generate the regular reading. So as with the case of overt exactly n we just saw above in (30), we can similarly overtly add exactly to half in (31). The referent of the IHRC DP remains the same (the six apples) but the ITCMIS derives twelve apples as its denotation.
Let’s get back to an at least semantics for half. There is a simplification in (34): there are, in fact, many other true propositions we could list as well. For instance the proposition that Ayaka peeled at least half of the atomic apple parts of 4+5+6+7+8+9. Or that Ayaka peeled at least half of 1+2+3+10+11+12, among many others. We do not list these here, because we can show that even without those additional propositions, the target proposition—the one that would generate the regular reading under ITCMIS—is not the uniquely maximally informative proposition. In order to derive the regular reading on the MIS, it would have to be the case the proposition (32f) be the most informative one—where the DP refers to the 6 peeled apples. (Intuitively, this is where the whole is equal to the part.) We ask what kind of entailment relations exist among these propositions, first going from a smaller to larger value of the whole (i.e. the X slot). In general, if someone peels at least half of the atomic apple parts of apples X, it does not follow that she peels at least half of the atomic apple parts of apples Y, where X is a part of Y. The proposition in (34f) can be true in a situation, for example, where apples 1, 2, 3 are peeled. None of the propositions with larger plural individuals in (34g)–(34k) is true in such situation. Thus,
(35)  “that Ayaka peeled at least half of the atomic apple parts of X” does not entail “that Ayaka peeled at least half of the atomic apple parts of Y” where X ⊑ Y 
So, (34f) does not entail the propositions (34g)–(34k).
Now let’s look at the entailment relations between propositions going from a larger value of the whole X to smaller. In general, if Ayaka peeled at least half of the atomic apple parts of apples X, it does NOT follow that Ayaka peeled at least half of the atomic apple parts of apples W, where W is a part of X. The proposition in (34f) can be true in a situation, for example, where apples 4, 5, 6 are peeled. None of the propositions with smaller plural individuals in (34a)–(34e) is true in the situation. Thus,
(36)  “that Ayaka peeled at least half of the atomic apple parts of X” does not entail 
“that Ayaka peeled at least half of the atomic apple parts of W” where W ⊑ X 
There is no entailment relation among these propositions and therefore there is no (unique) strongest, most informative proposition, and so no way for the definite to refer (without resorting to a salient set reading). So the ITCMIS analysis predicts that there is no regular reading available in sentence (31), contrary to fact. More specifically, the ITCMIS analysis fails to derive the fact that the six peeled apples 1+2+3+4+5+6 are the referent of the IHRC DP, because the proposition where the whole is 1+2+3+4+5+6 is not stronger than any other (true) proposition.
E&G (2016: 28–29) do discuss the half example in (31), but only in connection to the fact that it is false in context (2), where only apples 1, 2, 3 are peeled out of the twelve apples.^{10} Let’s assume that Junya ate these three peeled apples. The sentence in (31) is judged to be false or infelicitous, intuitively because it is not the case that Ayaka peeled half of the apples. But for E&G, the infelicity has a different source. E&G note that “there is no productive entailment pattern between ϕ_{half}propositions corresponding to different apple sums (p. 28)”. They note further that the uniqueness presupposition of THE is not met, as 1+2+3+4+5+6, as well as 1+2+3+7+8+9, for example, make the relevant proposition true. There is thus no uniquely maximal object as required by MIS in (14), and the denotation of the IHRC DP is undefined. In any event, while E&G’s account may correctly rule out (31) in context (2), so will the existing Etype account (Hoshi 1995; Shimoyama 1999; 2001) or the alternative approaches of Grosu & Landman (2012), Grosu & Hoshi (2016) and Landman (2016). And to repeat the salient result of this section, we have shown that the ITCMIS approach cannot deliver the regular reading in context (32) as shown above.
4.2 An alternative possibility
A reviewer has recommended that alternative propositions be considered when evaluating informativeness, in which case E&G’s ITCMIS analysis delivers the regular reading. The reviewer claims that (modified) numeral expressions such as at least/more than/exactly/fewer than/at most n are extensional predicates. Further the reviewer assumes that such predicates can be factored out of the evaluation of MIS and do not influence informativity (qua monotonicity) (see the cases referred to in footnote 8 and discussion in von Fintel et al. 2014: 172–173). One way to play this out in the present case of IHRCs is to decompose the numeral predicates into an existential quantifier portion and a conjunct that expresses the numeral properties.^{11}
(37)  λX.λw [X are apples in w_{0} & [∃Y. Y are at least 6/exactly 6/at least half in w_{0} & Y are apples in w & Y ⊑ X & Ayaka peeled Y in w ]]^{12} 
As a conjunct that is, in the reviewer’s terminology, extensional, the numeral expression simply need not be part of the propositions that are evaluated for informativity and instead we have those in (38), which are weaker existential claims. So, as before, we replace the whole X with pluralities which are compatible with Y being in the extension of, for example, exactly 6, and render the proposition true (assume the context in (32)), generating the propositions below:
(38)  a.  λw.∃Y [ A. peeled Y apples of 1+2+3+4+5+6 in w] 
b.  λw.∃Y [ A. peeled Y apples of 1+2+3+4+5+6+7 in w]  
⋮  
c.  λw.∃Y [ A. peeled Y apples of 1+2+3+4+5+6+7+8+9+10+11+12 in w] 
In this case the entailment relations are indeed such that the proposition in (38a) entails all others, making it the strongest proposition, which in turn means that the plurality 1+2+3+4+5+6 is the referent for the DP. This is the regular reading. Essentially, this strategy removes entirely the force of the numeral expression from the proposition and the result is simply very weak existential claims. MIS will then always favour the smallest true plurality.
At present, we think this alternative interpretation of E&G faces a number of challenges. First, we note that factoring numeral predicates out of the evaluation of informativity (if doing so is legitimate at all in (37)) does not solve the undergeneration problems we will discuss in sections 4.3 and 4.4 below, contrary to the reviewer’s claim. The regular reading still would not be derived from E&G’s uniform LF for both the salient set and regular readings (i.e. with a partof relation).^{13} It should be emphasized that we are not against deriving the regular reading from a different LF using MIS with the modification suggested by the reviewer. Our point is that E&G’s LF with ITC (crucially the variable insertion with a partof relation) is not suitable for deriving appropriate interpretations.
Another way to demonstrate the challenges that the alternative interpretation of E&G faces involves English DPs that have the same LF that E&G propose for Japanese IHRCs, as in (39):^{14}
(39)  the apples that Ayaka peeled at least 6/half of 
(λX.Ayaka peeled at least 6/half of X) 
While rather formal, the English DP in (39) is grammatical. And it would have to be given the same basic LF as E&G propose for Japanese LFs (except not derived via Inverse Trace Conversion). However, (39) cannot refer to the 6 apples in the context in (32), unlike an IHRC on the regular reading; (39) can only refer to the larger set of apples, six/half of which Ayaka peeled. So if we allow MIS to give the Japanese IHRC this meaning (from the LF E&G propose) we also make a false prediction about the meaning of (39).
4.3 Upward monotonic quantifiers: More than/at least n
At the beginning of Section 3 we showed how the ITCMIS analysis derived the regular reading for internal heads quantified by bare numerals (crucially under an at least semantics). However, here we show that the ITCMIS analysis does not deliver the correct regular reading with a slightly modified sentence and context—indeed, it turns out to deliver no regular reading at all due to presupposition failure.
Consider the sentence in (40), where the internal head is quantified by a modified numeral nyori takusan ‘more than n’.
 (40)
 Junyawa
 JunyaTOP
 [
 [Ayakaga
 AyakaNOM
 [_{IH}
 ringoo
 appleACC
 hutatsu
 twoCL
 yori
 than
 takusan]
 many
 muita]no]o
 peelPASTNOACC
 zenbu
 all
 tabeta.
 eatPAST
 ‘(lit.) Junya ate all of [that Ayaka peeled more than two apples].’
 ‘Ayaka peeled more than 2 apples and Junya ate all of them.’
Let’s consider the scenario in (41), where Ayaka peeled four apples. For sentence (40) to be true in this scenario, Junya would have to eat all of the peeled apples. It will be shown below that the ITCMIS analysis fails to produce the correct truth conditions for (40).
 (41)
Given the meaning of the quantifier, it turns out that in this context there are several equally informative true propositions. This renders the IHRC DP undefined, as it fails to satisfy the uniqueness presupposition of THE in (14). In the context in (41), true propositions of the appropriate form derived by ITC are listed below, with replacement of the X slot by various pluralities constituting three through twelve apples. Note that we include various pluralities for which it is true that Ayaka peeled more than two (atomic) parts of that plurality: this means there are four propositions where the plurality itself is constituted by just three apples: apples 1+2+3 (42a), apples 1+3+4 (42b), apples 2+3+4 (42c), and apples 1+2+4 (42d). There is of course the target proposition in (42e), where the plurality filling the whole slot is four, and this should be the proposition that MIS would need to pick out so that these four apples are the denotation of the IHRC DP in (43). In addition, we need to include larger pluralities that are extensions for each of (42a–d) as well as (42e), which are given—with many elided—in (42f) onward.
(42)  a.  λw. A. peeled more than 2 apples of 1+2+3 in w 
b.  λw. A. peeled more than 2 apples of 1+3+4 in w  
c.  λw. A. peeled more than 2 apples of 2+3+4 in w  
d.  λw. A. peeled more than 2 apples of 1+2+4 in w  
e.  λw. A. peeled more than 2 apples of 1+2+3+4 in w  
f.  λw. A. peeled more than 2 apples of 1+2+3+5 in w  
g.  λw. A. peeled more than 2 apples of 1+2+3+6 in w  
⋮  
h.  λw. A. peeled more than 2 apples of 1+2+3+8+10 in w  
⋮  
i.  λw. A. peeled more than 2 apples of 1+2+3+4+5 in w  
j.  λw. A. peeled more than 2 apples of 1+2+3+4+5+6 in w  
k.  λw. A. peeled more than 2 apples of 1+2+3+4+5+6+7 in w  
⋮  
l.  λw. A. peeled more than 2 apples of 1+2+3+4+5+6+7+8+9+…+12 in w 
The first problem is that the target proposition (42e) is not the strongest, since (42e) is entailed by all of (42a)–(42d) and (42e) does not itself entail any of (42a)–(42d).
Moreover, there is no unique strongest proposition since no proposition in (42a)–(42d) entails any of the other in that group. So on the MIS, a definite would be undefined (and even if it were, it would deliver a plurality of three as the denotation, which is not the meaning of the IHRC DP in (40) in the context in (41)). And it turns out that the propositions in (42a–d) are not necessarily stronger than all of the other propositions: for instance, (42b) does not entail (42f). (And we would find similar instances for (42a, c, d).) So not only are (42a–d) not stronger than each other—nor the correct proposition to generate the denotation of the IHRC DP in (40)—none are necessarily stronger than all of the other propositions. (They may entail some—(42a) entails (42f), for instance—but not all.) The most salient problem, to repeat, is that (42e) (which would allow the IHRC DP to denote the four apples) is not the strongest proposition and therefore we cannot account for the meaning of (40).
Similar considerations will arise with sukunakutomo n ‘at least n’, as expected. We provide a brief sketch using sentence (43), which is just like (18) except that sukunakutomo ‘at least’ is overtly included.
 (43)
 Junyawa
 JunyaTOP
 [
 [Ayakaga
 AyakaNOM
 [ringoo
 appleACC
 sukunakutomo
 at.least
 mittsu]
 threeCL
 muita]no]o
 peelPASTNOACC
 zenbu
 all
 tabeta.
 eatPAST
 ‘(lit.) Junya ate all of [that Ayaka peeled at least three apples].’
 ‘Ayaka peeled at least three apples and Junya ate all of them.’
The sentence in (43) is true in the context above in (41), where Ayaka peels four apples of a set of twelve, if Junya ate those four apples (i.e. the maximal apples Akaya peeled) but not if Junya ate just three.^{15} As in the case with hutatsuyori takusan ‘more than two’ above, we fail to generate the apple sum 1+2+3+4 as the denotation of the IHRC DP. We can see this by replacing ‘more than two’ with ‘at least three’ in the propositions in (42): “that Ayaka peeled at least three atomic apple parts of 1+2+3+4” is not the strongest proposition among the true propositions.
In the next section we will turn to left downward monotonic quantifiers. Aside from the left downward monotonic quantifier donoNmo ‘every’, the ITCMIS procedure fails to derive the regular reading in each of these cases.
4.4 Downward monotonic quantifiers: Fewer than/at most n
When the internal head bears the quantificational expression ringoo muttsu miman/muttsuyori sukunai (kazuno) ringo ‘fewer than six apples’ as shown in (44), the ITCMIS account cannot derive the regular reading and in fact produces a largerset reading due to the entailment patterns of fewer than n.
 (44)
 Junyawa
 Junyawa
 [Ayakaga
 AyakaNOM
 {ringoo
 appleACC
 muttsu
 sixCL
 miman
 fewer.than
 /muttsuyori
 /sixCLthan
 sukunai
 few
 (kazuno)
 numberGEN
 ringoo}
 appleACC
 muitano]o
 peelednoACC
 zenbu
 all
 tabeta.
 ate
 ‘(lit.) Junya ate all of [that Ayaka peeled fewer than six apples]’
 ‘Ayaka peeled fewer than six apples and Junya ate all of them.’
 (45)
This sentence, on the regular reading, is true in context (45), where Ayaka peels four apples of a larger set of twelve, and where Junya eats those four apples. Following the ITCMIS procedure, we collect the true propositions that vary over the whole slot as provided by ITC. In this context, examples of true propositions are in (46):
(46)  a.  λw. A. peeled fewer than 6 apples of 1 in w 
b.  λw. A. peeled fewer than 6 apples of 2 in w  
⋮  
c.  λw. A. peeled fewer than 6 apples of 1+2 in w  
d.  λw. A. peeled fewer than 6 apples of 1+3 in w  
⋮  
e.  λw. A. peeled fewer than 6 apples of 1+2+3 in w  
f.  λw. A. peeled fewer than 6 apples of 1+2+4 in w  
⋮  
g.  λw. A. peeled fewer than 6 apples of 1+2+3+4 in w  
⋮  
h.  λw. A. peeled fewer than 6 apples of 1+2+3+4+5 in w  
⋮  
i.  λw. A. peeled fewer than 6 apples of 1+2+3+4+5+6+7+8+9+10+11+12 in w 
To obtain the regular reading with MIS, we would want the proposition in (46g) to be the maximally informative one; that would set the denotation of the IHRC DP to be the X, where X = 1+2+3+4. There is a unique strongest proposition in (46) but it is not (46g). If Ayaka peeled fewer than six apples of apples 1+2+3+4, it does not follow that she peeled fewer than six apples when we add two more apples—maybe she peeled those two apples. Now, if Ayaka peeled fewer than six apples of apples 1–12, it does follow that she peeled fewer than six apples in a subset of those twelve. That makes (46i) the most informative proposition, and so under MIS the referent of the IHRC DP is the twelve apples that Ayaka peeled fewer than six of. That is what E&G call the salient set reading, which is possible. But the ITCMIS account does not derive the correct regular reading.
As expected, similar considerations apply when the internal head is quantified by ookutomo itsutsu ‘at most five’. The sentence in (47) can be true in the context in (45) with a regular reading (that Junya ate all the apples that Ayaka peeled—in this case 1+2+3+4):
 (47)
 Junyawa
 Junyawa
 [Ayakaga
 AyakaNOM
 ringoo
 appleACC
 ookutomo
 at.most
 itsutsu
 fiveCL
 muitano]o
 peelednoACC
 zenbu
 all
 tabeta.
 ate
 ‘(lit.) Junya ate all of [that Ayaka peeled at most five apples]’
 ‘Ayaka peeled at most five apples and Junya ate all of them.’
We can use the list in (46) above, with ‘fewer than 6’ replaced by ‘at most 5’. To derive the regular reading, we would want ‘that Ayaka peeled at most 5 atomic apple parts of 1+2+3+4’, our target proposition, to be the most informative, so that 1+2+3+4 can be the denotation of IHRC DP (viz., ‘the four apples that Ayaka peeled at most 5 of.’). But as with fewer than n above, the entailment relations are such that the proposition with the largest value for the Xslot entails all others. So if one peels at most five apples in a set X, it follows that one peels at most five apples in a subset of X. On the other hand, if one peels at most five apples in a set Y, it does not follow that one peels at most five apples in a superset of Y—maybe that superset includes a peeled apple. So once again, ITCMIS derives a referent for the IHRC DP that is far larger than the regular reading actually is.
In sum, for the downward entailing quantifiers that we have looked at, we derive the regular reading only in the case of the universal. That is because the strongest proposition involves the largest plural individual that meets the description “which Ayaka peeled.” This is equivalent to all the apples. For the other downward entailing quantifiers, fewer than n, at most n, the strongest proposition—the largest plurality for the whole—does not correspond to the maximal individual that fulfills the description “that which Ayaka peeled.” The Etype strategy (Hoshi 1995; Shimoyama 1999) and the nonEtype strategy (Grosu & Landman 2012; Grosu & Hoshi 2016; Landman 2016), on the other hand, deliver just this: the referent of the IHRC (or its associated Etype pronoun in the matrix clause) is the maximal individual that is apples that Ayaka peeled—not the maximal individual that Ayaka peeled a subset of apples from.
4.5 Conclusion
Using various types of quantifiers on internal heads, we have argued that the ITCMIS analysis is not successful in generating the regular reading. In the next section, we address the salient set interpretation. We first point out problems in deriving this reading from an LF that is shared with the regular reading. We then argue that the sentences that have the salient set reading are not IHRCs at all but surface identical headless relatives.
5 The salient set reading and an alternative
5.1 Salient Sets Restriction
We will now turn to the salient set reading. Notice that with the introduction of MIS, the salient set reading is no longer generated. In order to derive the salient set reading of (1)/(18), the pragmatic condition in (48) is added in E&G. After this condition applies, the referent of the IHRC DP in (9) is the apple sum 1+2+3+4+5+6+7+8+9+10+11+12 in context (2), and 1+2+3+4+5+6 in context (3).
(48)  Salient Sets Restriction (E&G 2016: 29) 
The existence of salient sets in the context allows for limiting the set of possible outputs of ⟦THE⟧ to those salient sets (represented as plural sums). 
The Salient Sets Restriction (SSR) is intended to “formalize the effect of context in definite description evaluation” (E&G 2016: 3), applying generally beyond cases with Inverse Trace Conversion. According to E&G (2016: 29), the sentence in (49) is unambiguously interpreted in context (50) to mean Junya will eat apples 1 and 2 because of the Salient Sets Restriction: it allows us to consider only 1+2 versus 3 as possible referents.^{16}
(49)  Junya will eat the two apples. 
 (50)
5.2 A problem in deriving the salient set reading
In the ITCMIS analysis, the regular reading and the salient set reading are assumed to be associated with a single syntactic structure for IHRC. A maximal informativeness semantics for definites is supposed to derive the regular reading, while the salient set reading is derived only if there are salient sets in the context, which triggers the pragmatic constraint, the Salient Sets Restriction, to kick in. We will now show (i) that the ITC analysis of the salient set reading makes wrong predictions with certain examples; and in the next subsection (ii) that there is a syntactic parse already available in the narrow syntax from which the salient set reading can be derived without any additional mechanisms, and which does not run into the problem in (i).
The salient set reading of (1), repeated below, is compatible with a scenario where there are some peeled or unpeeled pears in the pile, in addition to peeled and unpeeled apples. The sentence is true in context (52).
 (51)
 Junyawa
 JunyaTOP
 [_{IHRC}
 [Ayakaga
 AyakaNOM
 [_{IH}
 ringoo
 appleACC
 mittsu]
 threeCL
 muita]no]o
 peelPASTNOACC
 zenbu
 all
 tabeta.
 eatPAST
 ‘(lit.) Junya ate all of [that Ayaka peeled three apples].’
 ‘Junya ate all of that which Ayaka peeled three apples of.’
(52)  There are two mixed baskets, each containing six apples and six pears. Ayaka peeled three out of six apples in basket 1, while Tsubasa peeled six out of six apples in basket 2. Junya ate all the twelve apples and pears in basket 1. 
The ITC analysis does not predict this reading because in the LF syntax (see (9)) there is a copy of the internal head in the external position, limiting the referent to apple objects. A related problem arises with the example in (53).
 (53)
 Junyawa
 JunyaTOP
 [_{IHRC}
 [Ayakaga
 AyakaNOM
 [_{IH}
 ringoo
 appleACC
 mittsu]
 threeCL
 muita]no]kara
 peelPASTNOfrom
 nashio
 pearACC
 hutatsu
 twoCL
 totte
 take
 tabeta.
 eatPAST
 ‘(lit.) Junya took and ate two pears from [that Ayaka peeled three apples].’
 ‘Junya took and ate two pears from what Ayaka peeled three apples of/in.’
 #‘Junya took and ate two pears from the apples that Ayaka peeled three of.’
The relative here, on the ITC account, has a silent external head ringo ‘apple’. But then that would mean that this sentence says that Junya ate two pears from a salient set of apples, which is nonsensical.^{17} Instead, (53) expresses that Junya took two pears from a group of apples and pears. In sum, E&G’s analysis where the constituents labeled as “IHRC” in (51) and (53) are indeed considered to be IHRCs does not predict the observed readings. In the next subsection we suggest a different syntactic source for what is described as a salient set reading in E&G.
5.3 Alternative analysis: Headless relative parse
We suggest that the interpretation of sentences like (1), repeated below in (54), which was taken to represent a new species of internallyheaded relative with the parse in (54a) in E&G, are really independently attested headless relatives with the parse in (54b). (See also Grosu & Hoshi 2018).^{18}
 (54)
 Junyawa
 JunyaTOP
 [_{IHRC}
 [Ayakaga
 AyakaNOM
 [_{IH}
 ringoo
 appleACC
 mittsu]
 threeCL
 muita]no]o
 peelPASTNOACC
 zenbu
 all
 tabeta.
 eatPAST
 a.
 ITC parse
 [_{IHRC} THE [_{NP} apple(s) [_{CP} λX. Ayaka peeled [_{IH} three apple(s) ⊑ X]]]
 b.
 Headless relative parse
 [THE (one) [_{CP} λx. Ayaka peeled [ three apple(s) of x]]]
 ≈‘that which Ayaka peeled three apples of’
The headless relative parse in (54b) does not involve ITC, but rather a standard gap that is the object of a postposition or case marker. It is easy to mistake (54) as something more exotic because postpositions and case markers are deleted as a response to the fact that stranding of postpositions and case markers by relativization is not possible in Japanese. This is shown by the standard externallyheaded relative clause in (56) (built from the independent sentence in (55)). As there is no overt relative pronoun, piedpiping is not an option either, as shown in (57). The result is that the postposition or case marker does not surface, as in (58).^{19}
 (55)
 Yokoga
 YokoNOM
 Sotakara
 Sotafrom
 hono
 bookACC
 moratta.
 receivePAST
 ‘Yoko received a book from Sota.’
 (56)
 *[Yokoga
 YokoNOM
 <gap>
 kara
 from
 hono
 bookACC
 moratta]
 receivePAST
 hito
 person
 ‘the person who Yoko received a book from’
 (57)
 *[∅
 kara
 from
 [Yokoga
 YokoNOM
 <gap>
 hono
 bookACC
 moratta]]
 receivePAST
 hito
 person
 ‘the person from whom Yoko received a book’
 (58)
 [Yokoga
 YokoNOM
 <gap>
 hono
 bookACC
 moratta]
 receivePAST
 hito
 person
 ‘the person who Yoko received a book from’
So a gap is sometimes hidden, and this leads to potential ambiguity as in the following example. For interpreting such sentences, context and pragmatic knowledge and expectations play a role.
 (59)
 Matsumoto (1997: 40)
 [hono
 bookACC
 katta]
 bought
 gakusei
 student
 a.
 ‘the student who bought a book’
 b.
 ‘the student from whom (I) bought a book’
 c.
 ‘the student for/to whom (I) bought a book’
Looking at an example closer to (54), note that in the externallyheaded relativization (61) of sentence (60), the genitive case marker no that is translated as ‘in’ or ‘of’ in the sentence does not surface.
 (60)
 Ayakaga
 AyakaNOM
 sono
 that
 yamano
 mountainNO
 ringoo
 appleACC
 mittsu
 threeCL
 muita.
 peelPAST
 ‘Ayaka peeled three apples in/of the pile.’
 (61)
 [Ayakaga
 AyakaNOM
 <gap>
 ringoo
 appleACC
 mittsu
 threeCL
 muita]
 peelPAST
 yama
 mountain
 ‘the/a pile that Ayaka peeled three apples in/of’
The headless relative counterpart of (61) is (62), and its LF in (63).^{20}
 (62)
 Headless relative clause
 [Ayakaga
 AyakaNOM
 <gap>
 ringoo
 appleACC
 mittsu
 threeCL
 muita]no
 peelPASTNO
 ‘{that which/the one that/what} Ayaka peeled three apples in/of’
(63)  [_{DP} THE [_{NP} (one) [_{CP} λx [Ayaka peeled three apples in/of x]]]] 
Notice that the headless relative in (62) is stringidentical to the IHRC DP in (54). What is claimed to be a previously unidentified reading of IHRC in E&G is in fact an expected interpretation of independently available headless relatives such as (62). (Grosu & Hoshi 2018 come to a similar conclusion, based on these and other novel data and arguments they introduce; we refer the reader to their work for further arguments against a uniform LF).^{21} The sentence is thus syntactically ambiguous between the IHRC parse and the headless relative parse. Naturally, context plays a role in interpreting headless relatives like (62). There is only one group of apples in context (2), where Ayaka peeled three apples. The entire group of apples is thus the referent of the DP in (62). In context (3), on the other hand, there are two groups: one where Ayaka peeled three apples, and the other one where no one peeled any. The referent of the DP in (62) in this context is the first group of apples.
We note here that even though we used yama ‘pile’ (lit. ‘mountain’) as the external head in (61) to illustrate one possible externallyheaded counterpart of the headless relative in (62), there are of course other possibilities as shown in (64)–(66).^{22} We are not suggesting, however, that these externallyheaded relative clauses underlie the headless relatives. Depending on the context, the referent of the headless relative in (62) could be the apples themselves (as in E&G’s example), or their container (a cardboard box, for example, as in (67)).
 (64)
 [Ayakaga
 AyakaNOM
 <gap>
 ringoo
 appleACC
 mittsu
 threeCL
 muita]
 peelPAST
 hako/kago
 box/basket
 ‘the/a box/basket that Ayaka peeled three apples in’
 (65)
 [Ayakaga
 AyakaNOM
 <gap>
 ringoo
 appleACC
 mittsu
 threeCL
 muita]
 peelPAST
 {ringo/kudamono}no
 apple/fruitNO
 yama
 mountain
 ‘the/a pile of apples/fruits that Ayaka peeled three apples in’
 (66)
 [Ayakaga
 AyakaNOM
 <gap>
 ringoo
 appleACC
 mittsu
 threeCL
 muita]
 peelPAST
 sorerano
 those
 ringo/kudamono
 apples/fruits
 ‘those apples/fruits that Ayaka peeled three apples of’
 (67)
 [Ayakaga
 AyakaNOM
 <gap>
 ringoo
 appleACC
 mittsu
 threeCL
 muita]noni
 peelPASTNODAT
 atenao
 addresseeACC
 kaita.
 writePAST
 ‘(I) wrote the addressee’s name and address on the one that Ayaka peeled three apples in.’
Examples such as (51) and (53) from Section 5.2—cases of mixed baskets of pears and apples—whose interpretations are not generated in the ITC analysis, are no longer a problem in the headless relative parse. This is because the interpretation of the head is determined by the context, in more or less the same manner in which English headless/free relatives are interpreted (that which/what Ayaka peeled three apples of). The gapped headless relative parse is always available independently, and this structure gives us the salient set reading without recourse to any additional mechanisms.
A partwhole relation plays a big role in the ITC analysis of IHRC. Relativization that involves a partwhole relation has been discussed in the literature.^{23} In each of (68) and (69), a phrase in the relative clause and the referent of the entire DP containing the relative clause form a partwhole relation (e.g., ‘the tail’ and ‘the cat’). Thus, the headless relative (i.e. no) versions of these examples may at first sight appear to be amenable to the ITC analysis of IHRCs. However, the property we could derive within the ITC analysis in (70) for (68) is not the right one.
 (68)
 [Yuyaga
 YuyaNOM
 shippoo
 tailACC
 hippatta]
 pullPAST
 neko/noga
 cat/NONOM
 okotta.
 get.angryPAST
 ‘The cat/one that Yuya pulled the tail of got angry.’
 (69)
 Yokoga
 YokoNOM
 [Yuyaga
 YuyaNOM
 ashio
 legACC
 nihon
 twoCL
 kowashita]
 breakPAST
 isu/noo
 chair/NOACC
 naoshita.
 fixPAST
 ‘Yoko fixed the chair/one that Yuya broke two legs of.’
(70)  λX. X is a tail and Yuya pulled the tail of X 
For example, the whole IHRC of (68) would denote the tail, such that Yuya pulled the (or a) tail part of it. And the sentence would assert that the larger tail whole got angry. This is not the right interpretation of (68). Clearly, then, we do not want to apply the ITC analysis to such examples. So even within the ITC analysis, it needs to be admitted that such examples should receive a gapped headless relative parse and not an IHRC parse. In sum, even if the ITC analyses exists for some IHRCs, the headless relative parse would have to be assumed in order to generate sentences such as (51), (53), (67), (68) and (69) with appropriate interpretations, as these cannot be handled by the ITC analysis. What remains to be seen is independent evidence for additionally generating IHRCs with ITC, which derives the salient set reading for only a subset of the cases we have discussed.
Finally, recall that on the ITCMIS analysis, the salient set reading is derived when the Salient Sets Restriction (SSR) is invoked by the presence of one or more salient sets in the context. E&G report that three survey participants consistently only had access to the regular reading. Their explanation is that these speakers did not use the SSR, and they speculate that these speakers “did not perceive the schematic grouping as salient enough in the written survey, but would use the SSR for definite description reference in a real world context” (E&G: 31). Related to this, it is further noted that “[t]he directional light noun hoo is often used when salient sets are referred to or contrasted. Therefore another possibility for the speakers described in this section is that the use of the SSR is lexically tied more closely to the use of hoo for these speakers than for others” (E&G: 31).
In the headless relative analysis, the socalled salient set reading is an expected reading when the whole position of a partwhole relation is relativized. As we briefly touched upon in relation to sentence (59), contextual information and pragmatic knowledge is crucial as there is no overt indicator of a gap. In the headless relative analysis, we can view the use of hoo, a contrastive focus marker (Schwarzschild 2011; Matsui & Kubota 2012; Bamba 2016), as facilitating the interpretation of such relatives. The use of hoo in (71) clearly highlights the socalled salient set reading, by making the property ‘λx.Ayaka peeled three apples in/of x’ contrastive with some other property, for example, with ‘λx.no one peeled any apples in/of x’, in the case of context (3) with two groups of apples. Other types of properties can be contrasted, depending on which smaller focused constituents hoo associates with, as shown in (72).
 (71)
 [Ayakaga
 AyakaNOM
 <gap>
 ringoo
 appleACC
 mittsu
 threeCL
 muita]hoono
 peelPASTHOONO
 ‘the one that/that which/what Ayaka peeled three apples in/of’
(72)  a.  that which no one peeled any apples in/of  
b.  that which [Yoko]_{F} peeled three apples in/of  (if [Ayaka]_{F} in (71))  
c.  that which Ayaka peeled three [pears]_{F} in/of  (if [apples]_{F} in (71))  
d.  that which Ayaka peeld [five]_{F} apples in/of  (if [three]_{F} in (71)) 
The presence of the marker hoo in these examples actually provides a further argument against treating the salient set reading as derived from an IHRC. In unambiguous IHRCs as in (73), hoo does not quite sit well in the IHRCfinal position.
 (73)
 [[Shiroi
 White
 inuga
 dogNOM
 hoeteita](??hoo)no]ga
 was.barkingHOONONOM
 nekoo
 catACC
 oikaketa.
 ran.after
 ‘A white dog that was barking ran after a cat.’
This tells us that the structure that the salient set reading is derived from is not an IHRC.
To summarize, we have shown in this section that there are problems in deriving the salient set reading from an IHRC parse using ITC, and that these problems do not arise with an independently available headless relative parse. The ITC analysis would then need to assume a headless relative parse for those cases, and the question boils down to whether we have independent motivation for generating an IHRC parse with ITC.
We have arrived at a picture where the sources of the regular and salient set readings are dissociated. The regular reading is left to be handled by the existing analyses of IHRCs (see Grosu 2012; Grosu & Landman 2012; Grosu & Hoshi 2016; Landman 2016, and references therein), as the ITCMIS analysis largely undergenerates the regular reading as we saw in Section 4. The salient set reading should have a different syntactic source, a headless relative, because an IHRC analysis fails to generate correct interpretations as we saw in Section 5.2. Other than these empirical reasons, let us note here an additional conceptual reason to dissociate the syntactic sources of the two readings. The LF structure within the ITCMIS analysis always includes a partof relation, regardless of whether the salient set reading is present or not. This is counterintuitive in ways we briefly sketch below.
In example (74), the whole part in the partof relation, ‘in that pile of apples’, is syntactically overt. In the context in (2) with one group of apples, readings (75a) and (75b) are available, while reading (76a)—which invokes an even larger plurality of apples of which the pile is part—is not. In the ITC analysis, the reading in (76a) is not generated as the SSR would not apply without a larger salient set in the context. And yet, since variable insertion is an obligatory process in the derivation of IHRC from a protorelative, the property in (76b) is always generated. Tying the regular reading to the LF with a partof relation in this way seems counterintuitive. It is also not immediately obvious how the ITC analysis generates the reading in (75b).
 (74)
 Junyaga
 JunyaNOM
 [Ayakaga
 AyakaNOM
 sono
 that
 ringono
 appleNO
 yamano
 mountainNO
 ringoo
 appleACC
 mittsu
 threeCL
 muita]noo
 peelPASTNOACC
 zenbu
 all
 tabeta.
 eatPAST
 ‘(lit.) Junya ate all of [that Ayaka peeled three apples in that pile of apples].’
(75)  a.  Ayaka peeled three apples of that pile of apples and Junya ate all three of them. 
b.  ?Ayaka peeled three apples of that pile of apples and Junya ate all the apples in the pile. 
(76)  a.  Ayaka peeled three apples of that pile of apples and Junya ate all the apples in some salient set the the pile of apples Ayaka peeled three apples from are part of. 
b.  ⟦DP⟧ = ⟦THE⟧(λX. X apples in that pile of apples and Ayaka peeled [three atomic parts of apples of that pile of apples of X]) 
Another example comes from Erlewine & Gould (2014), and it generally has to do with IHRCs with creation verbs. It is reported there (p. 169) that sentence (77), credited to Chris Davis (p.c.), has a regular reading where the teacher reads what I wrote, half a paper. The expected salient set reading, where the entire paper is read is not possible, as it has not been written. And yet, the LF structure for (77) would still contain a partof relation (‘λX. I wrote half paper of paper X’).
 (77)
 [[Tetsuyashite
 all.nighterdid
 ronbuno
 paperACC
 hanbun
 half
 kaita]no]o
 wroteNOACC
 senseiga
 teacherNOM
 yondekureta.
 readBEN.PAST
 (lit.) ‘The teacher read (for me) [(I) pulled an allnighter and wrote half paper].’
 ‘(I) pulled an allnighter and wrote half a paper and the teacher read it for me.’
Again, the salient set reading may be blocked here as there is no salient set (the whole paper) in the context and the SSR does not kick in. However, such LF with a partof relation is counterintuitive for creation verbs in general (e.g., ‘λX. I built three houses of houses X’).
6 Conclusion
In this response, we have demonstrated that the ITCMIS analysis of IHRCs does not derive the observed interpretations for relatives with a number of quantified internal heads. Furthermore, a unified account of IHRCs with the regular reading and those relatives with the socalled salient set reading cannot be maintained. Among other problems, the unification cannot capture readings where the individuals denoted by the internal head are not subparts of denotation of the IHRC as a whole—e.g. the mixed basket scenario of apples and pears. We have suggested that the socalled salient set reading does not arise from a IHRC but rather is just the expected interpretation of a headless relative. Because postposition/case markers stranded by movement are deleted, these headless relatives are surface identical to some IHRCs. When the constructions are disambiguated, it becomes clear that the salient set interpretation needn’t arise from an IHRC.