JOURNAL OF SEMANTICS Volume 24 Number 3
CONTENTS SIGRID BECK AND ARNIM VON STECHOW Pluractional Adverbials
215
MARTINA FALLER The Ingredients of Reciprocity in Cuzco Quechua
255
ROBERT VAN ROOIJ Strengthening Conditional Presuppositions
289
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Journal of Semantics 24: 215–254 doi:10.1093/jos/ffm003 Advance Access publication May 17, 2007
Pluractional Adverbials SIGRID BECK Universita¨t Tu¨bingen ARNIM VON STECHOW Universita¨t Tu¨bingen
This paper investigates the semantics of adverbials like ‘page by page’ and ‘stone upon stone’. An analysis is developed in which sentences containing such adverbials have a pluractional semantics; that is, pluralization affects simultaneously the event- and the individual-argument slot of a predicate. Sternefeld’s (1998) system of plural operators is used and extended for this purpose. The adverbial constrains the relation that is pluralized and makes visible a higher plural operator. In the case of ‘page by page’-type adverbials, this is a fairly standard operator that leads to a simple divisional interpretation. In the case of ‘stone upon stone’-type adverbials, the operator has a stronger semantics that leads to a sequence interpretation. The generality of our theory permits straightforward extension to data like ‘she ran and ran’ and ‘she climbed higher and higher’, among others. Finally we propose that inclusive alternative ordering reciprocals (Dalrymple et al. 1998) have a pluractional sequence interpretation as well.
1 INTRODUCTION The topic of this paper is the semantic analysis of the sentences in (1). Example (1a,b) contain the adverbial modifiers ‘one after the other’ and ‘dog after dog’, respectively, which add to the simple (1#) information on how the overall event of the dogs entering the room is to be divided into subevents based on a division of the group of dogs into individual dogs. We call these adverbials pluractional adverbials, following, for example, Lasersohn’s (1995) use of the term pluractionality for the division of larger eventualities into subeventualities. (1) a. These three dogs entered the room one after the other. b. They entered the room dog after dog. (1#) These three dogs entered the room. The type of situation described by (1a) [and also by (1b) if the referent of ‘they’ is the same as the referent of ‘the three dogs’] is depicted informally in (2). We will aim to derive this fact by associating with (1a,b) (roughly) the truth conditions in (3); that is, we will propose a The Author 2007. Published by Oxford University Press. All rights reserved. For Permissions, please email:
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Abstract
216 Pluractional Adverbials compositional semantics for (1a,b) that derives approximately the truth conditions in (3), and (3) serves to capture our intuitions about the situations in which (1a,b) would be considered true. (2) a. These three dogs entered the room one after the other. b. D3 / D2 / D1 #x / y# ¼ x enters the room after y (3) These three dogs entered the room, and the entering can be divided into a sequence of subevents in each of which one of the dogs enters, and the dogs can be divided into a sequence of individual dogs each of which entered in one of the subevents.
(4) a. This mystery offers puzzle within puzzle. b. She laid book upon book and built a staircase long enough to climb up and look over the wall. c. The Wall of Tears is a very big wall that was built, stone over stone by the prisoners when Isabela was a penal colony back in 1946. (5) a. Because life’s interaction is like a series of boxes one within the other, ecological studies are organized in hierarchical levels. b. In storing textiles, rugs, or other large-sized weavings, these should never be folded and piled one upon the other. c. My grandmother had on not just one skirt, but four, one over the other. There is a semantically simpler type of pluractional adverbial exemplified by (6) below. The analysis of the ‘piece by piece’ type of adverbial will be our basis for formulating the semantics for pseudoreciprocal ‘dog after dog’-type adverbials. (6) a. Sally ate the cake piece by piece. (6#) a. Sally ate the cake. (7) a. Cigar wrapper is harvested leaf by leaf. b. The traditional visual apple-by-apple inspection is labour intensive and prone to human error.
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While we largely concentrate on the particular examples in (1), the phenomenon as such is of course more general. Other examples of reduplicative adverbials like ‘dog after dog’ are given in (4), and other examples of the ‘one . . . the other’ type are provided in (5). These data were collected informally from the Web. All of these adverbials, we claim, have a semantics analogous to (1), which we call pseudoreciprocal.
Sigrid Beck and Arnim Von Stechow 217
We will introduce our background assumptions in section 2 and illustrate them with an analysis of ‘piece by piece’-type adverbials. Section 3 is devoted to the semantics of pseudo-reciprocals. In section 4, we address several issues closely related to our analysis, including a certain type of apparent reciprocal construction—inclusive alternative ordering (IAO) Reciprocals (Dalrymple et al. 1998)—as well as data like (8) below. We conclude the paper in section 5 with a summary, a discussion of related literature and some conclusions. (8) a. Sally ran and ran. b. Sally approached the horse step by step. c. Sally was sick and sick. d. The train arrived and arrived.
2 SIMPLE PLURACTIONALS
2.1 Background Besides individuals (type Æeæ), we use eventualities (type Ævæ). We assume that both De (the denotation domain of individuals—count and mass) and Dv (the denotation domain of eventualities) have a mereological structure. We assume a primitive (i.e. not formally defined) part of relation s(ie#: Katie enters the room in e#) b. ke. Min enters the room in e & Katie enters the room in pred(e) pred(e): the relevant predecessor of e (25$) ½½after Katie ¼ kRÆe, Æv, tææ.kx.ke.R(x)(e) & R(Katie)(pred(e)) (26) pred(e) ¼ ie#: s(e#) < s(e) & "e$[s(e$) < s(e) / e$ ¼ e# or s(e$) < s(e#)] A generalized version of this idea is given in (27) and (28). There is an ordering relation on events based on temporal precedence. We can identify the predecessor according to that order. (27) ordering relation on events: e is before e#: e : e# iff s(e) < s(e#) (28) the immediate predecessor of e: pred(e) ¼ ie#: e# : e & "e$[e$ : e / e$ ¼ e# or e$ : e#]
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(2) These three dogs entered the room one after the other. D3 / D2 / D1 (3) These three dogs entered the room, and the entering can be divided into a sequence of subevents in each of which one of the dogs enters, and the dogs can be divided into a sequence of individual dogs each of which entered in one of the subevents.
224 Pluractional Adverbials
3.2 The ‘other’ dog The instance of the ‘after’-modifier that we are confronted with is ‘after the other’. The key to our analysis of pluractional ‘one after the other’ lies in our understanding of the meaning of ‘the other’ in this construction. We suggest that for each dog, the relevant other dog is always the immediately preceding one. That is, we propose that there is an ordering on the individuals that is derived from the ordering of events, as in (29). The predecessor of an individual can be defined on the basis of that derived order. (29) ordering relation on individuals: x : y iff de[x is in e and "e#[y is in e# / e : e#]] x is before y iff x occurs in a relevant event before y does (30) the immediate predecessor of x: pred(x) ¼ iy: y : x & "z [z : x / z ¼ y or z : y] Finding the predecessor for each dog requires that the dogs can successfully be ordered into a sequence. (31) defines the notion of sequence; the cover has to have this property so that its members can be ordered. In our example, we would have (32). (31) Cov[x] is a sequence iff Cov[x] ¼ {x1, . . ., xn} and for any xi, xi+1: xi : xi+1 (32) Cov[e] ¼ {e1, . . ., en} such that for any ei, ei+1: ei : ei+1 Cov[these 3 dogs] ¼ {x1, . . ., xn} such that for any xi, xi+1 xi : xi+1d {D1, D2, D3}
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Two remarks are in order. There might be a worry how we identify the relevant predecessor of an event in the general case. Even if we limit our attention to enterings by Katie, Katie may have entered the room many times. Example (25) talks about the contextually relevant entering; hence the immediate predecessor ought to be defined relative to a contextually given set of events. We will not concern ourselves with this here, since our pluractional data come with an obvious way of resolving this problem: we will only need to look for the predecessor within one overall plural event of entering, the parts of which are ordered. Secondly, when we talk about the immediate predecessor, this is to be understood in terms of events, not time. In ‘Min entered the room two minutes after Katie’, the running time of Min’s entering is two minutes later than the running time of the immediate predecessor event.
Sigrid Beck and Arnim Von Stechow 225
If the appropriate sequence is given, then the rest of the truth conditions of our example (1) can be stated as in (33) below. From (33a) we get (33b). The overall truth conditions we propose are paraphrased in (34).
(34) e can be divided into a sequence of subevents, and the three dogs can be divided into a sequence of individual dogs, such that each dog entered the room in a relevant subevent, and its predecessor entered in the preceding subevent, and each subevent was one of one of the dogs entering, and the preceding event was one of the predecessor of that dog entering. These truth conditions can be derived straightforwardly from the Logical Form in (35). The subject is raised, with the movement binding an anaphor contained in the NP ‘the other’; the relevant pluralization operator is attached to the modified relation (the predicate created by the movement). We propose a version of our PL operator (henceforth PLseq) that incorporates the constraint on the cover that the cover of the relevant entity and event be a sequence. And we suggest a semantics for the modifier ‘one after the other’ that is essentially a combination of what we found out about ‘after NP’ in (25$) and the idea that the NP here contributes, for each dog, the predecessor of that dog (the full semantic details of the composition of the modifier will be discussed in section 3.5). With this, (35) will give rise to the truth conditions in (34). seq
(35) these 3 dogs [PLCov kx [vt x
[evt entered the room] [(evt)(evt) one after the other x]]] j__________QR_________j anaphor
(36) ½½one after the other R(pred(g(x)))(pred(e))
xg
¼
kR.ky.ke.
R(y)(e)
&
(37) ½½PLseq ¼ kCov.kR.kz.ke. Cov[e] is a sequence and Cov[z] is a sequence & [kz#.ke#.Cov(z#) & Cov(e#) & R(z#)(e#)](z)(e)]
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(33) a. Æ3D, eæ 2 [kx.ke#.Cov(x) & Cov(e#) & x enters the room in e# & pred(x) enters the room in pred(e#)] b. "x[x < 3D & Cov(x) / de#[e# < e & Cov(e#) & x enters the room in e# & pred(x) enters the room in pred(e#)]] & "e#[e# < e & Cov(e#) / dx[x < 3D & Cov(x) & x enters the room in e# & pred(x) enters the room in pred(e#)]]
226 Pluractional Adverbials (35#) ½½(35)g ¼ seq ke.Æ3D, eæ 2 ½½PLCov ]] (kx.ke#. x enter t.r. in e# & pred(x) enter t.r. in pred(e#)]) ¼ ke.Cov[3D] is a sequence & Cov[e] is a sequence & Æ3D, eæ 2 [kx.ke#.Cov(x) & Cov(e#) & x enter t.r. in e# & pred(x) enter t.r. in pred(e#)]
3.3 The first dog The observant reader will no doubt have noticed that the truth conditions in (34) suffer from a problem: We require that for each dog, that dog enter after its predecessor. But the first dog in the sequence does not have a predecessor. So (34) as such could never be true. We propose to embrace this prediction—so our compositional semantics will derive these truth conditions. There must then be a pragmatic process that allows us to ignore the first dog, and thus makes it possible for (34) to be true. We suggest that essentially the same process is at work in (38) and (39) below. In (39) for instance, we must subtract Arnim from the domain of quantification and understand ‘everyone’ to mean here ‘everyone but Arnim’; else the sentence could never be true. Likewise, we subtract the first sentry in the row from the domain that ‘each’ quantifies over. (38) 20 Wachposten sind so in einer Reihe aufgestellt, dass jeder den vorherigen sehen kann. 20 sentries are standing in a row such that each can see the one before him. (39) Everyone has a faster computer than Arnim. Thus, we think that it is generally possible to reinterpret a quantificational statement that could not come out true by
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We rely on the context to provide an ordering relation which is used both to form a sequence of the relevant parts of the cover and to identify the relevant predecessor of an event; in the example, the preposition ‘after’ expresses this ordering relation. The predecessor of an event must then be the predecessor in the sequence. We will see in section 3.5 that the individual predecessor is found through the same ordering relation. Context (which specifically includes the linguistic context of the sentence itself ) furnishes an order and a division into relevant sub-parts of our pluralities. Plural predication makes use of this information through the plural operator PLseq. PLseq and the adverbial modifier are jointly responsible for the assertion we get, requiring a sequence of enterings by individual dogs.
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subtracting the problematic individual from the domain of quantification. This process will also have to apply to our examples in (1). Let’s see how a pragmatic domain subtraction solution might work for (38) and (39). We assume with Westerstahl (1984), von Fintel (1994) and others that all natural language quantifiers come with their own domain restriction. The representation of (39) would be (39#). (39#) everyC one has a faster computer than Arnim every (C)(A)(B) ¼ 1 iff for every x such that C(x)&A(x):B(x)
(40) C ¼ {x: x is part of the situation talked about} C# ¼ C {Arnim} We refer to this as domain subtraction. The resource domain of a quantifier is determined by context, and we think that a pragmatic process applies after the usual interpretive mechanisms that could be described roughly as in (41). (41) Choose the value for the resource domain variable C in such a way that the sentence may come out true. We suggest that (38) is parallel. The first sentry does not have a predecessor, hence (38) could not be true as long as we consider the first sentry to be part of the domain of ‘every’. It is the presupposition triggered by the definite description ‘the one before him’ that is responsible for domain subtraction. This process has been discussed as presupposition accomodation (compare Kadmon (2001) and references therein). (38#) everyC one can see the one before him. C ¼ {x: x is part of the situation talked about} ¼ {x: x is a sentry} C# ¼ C {the 1st sentry} The parallel of (38) to our problem with the first dog is obvious. We would similarly want to exclude the individual for whom the predecessor does not exist from the quantification. It is not surprising that plural quantification is similar to other natural language quantification in this respect.
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The set C is the contextually determined resource domain for the quantifier. A normal value would be the set of individuals in the situation talked about with (39). While Arnim is clearly around, we must suppose that he is subtracted from C in order for (39) to receive reasonable truth conditions—that is, C is revised to C# to avoid necessary falsehood.
228 Pluractional Adverbials However, it must be acknowledged that plural quantification adds a small technical complication. The universal quantifier that needs to be affected by domain subtraction is not the quantifier that we find in the syntax. That is, domain subtraction cannot happen to PLseq itself, since we do not want to exclude the first dog once and for all (else the second dog would pose the same problem again, and so on!). We would like domain subtraction to apply at the level of cumulation, as shown below.
If the original value for C was the individuals in the context, the new value must be that set minus the first dog (and the first event) in the sequence. We propose to give the PL operator another parameter that is inherited by cumulation and serves as its domain restriction. Example (37#) gives the revised version of (37) and (14$) the revised version of the definition of cumulation (14#). We assume that C[z + e] 4 Cov[z + e] (i.e. nothing new suddenly becomes relevant); then (14$) results in (42). (37) Old definition of PL: ½½PLseq ¼ kCov.kR.kz.ke.Cov[z] is a sequence & Cov[e] is a sequence & [[kz#.ke#.Cov(z#) & Cov(e#) & R(z#)(e#)]](z)(e) (37#) New definition of PL: ½½PLseq ¼ kC.kCov.kR.kz.ke.Cov[z] is a sequence & Cov[e] is a sequence & [C[kz#.ke#.Cov(z#) & Cov(e#) & R(z#)(e#)]](z)(e) (14#) Old definition of of type Æe, Æv, tææ: Let R be a relation of type Æe, Æv, tææ. Then for any x, e: [R](x)(e) ¼ 1 iff "x# < x: de# < e:R(x#)(e#) & "e# < e: dx# < x:R(x#)(e#) (14$) New definition of of type Æe, Æv, tææ: Let R be a relation of type Æe, Æv, tææ. Then for any x, e: [C R](x)(e) ¼ 1 iff "x#[x# < x & C(x#) / de#[e# < e & C(e#) & R(x#)(e#)]]& "e#[e# < e & C(e#) / dx#[x# < x & C(x#) & R(x#)(e#)]]
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(42) "x[x < 3D & Cov(x) & C(x) / de#[e# < e & Cov(e#) & C(e#) & x enter in e# & pred(x) enter in pred(e#)]] & "e#[e# < e & Cov(e#) & C(e#) / dx[x < 3D & Cov(x) & C(x) & x enter in e# & pred(x) enter in pred(e#)]]
Sigrid Beck and Arnim Von Stechow 229
3.4 Similar cases: one above/upon/within the other In this subsection, we indicate how the analysis proposed for ‘one after the other’ extends to similar instances of pluractional adverbials with different prepositions. Some examples are given below. We will focus on (43a) with ‘above’. (43) a. These three children sleep one above/next to the other. b. She laid the books bundle beside/upon bundle on the porch. Our starting point is once more a regular occurrence of the modifier (44a). The semantics in (44b) leads to the meaning in (44c) for the modifier.2 Like our earlier example ‘after NP’, the PP modifies a relation. In this case, this is a relation between an individual and a place (not an event). It adds to the original relation the information
2
We leave out the event-argument slot in these data for convenience.
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Now, if R contains a presuppositional element, that element can be expected to trigger the same pragmatic effects in pluralization as it triggers in other quantification—in particular domain subtraction. This is how we propose to account for the first dog. We think that our Logical Forms and semantic representations already look complicated enough and will suppress the additional parameter C in what follows. The issue of the first dog, once resolved, does not seem to interact with the other topics we discuss below. Before we move on, let us use this opportunity to mention a different kind of exception to our sequence truth conditions. An anonymous referee points out to us that a sentence like ‘They shouted one man after the other’, could be considered true in a situation in which one individual, Boris (¼b), shouts twice, leading to the following sequence of shouting events (let a–d be the men and e1–e5 be the relevant events of shouting): Æa, e1æ, Æb, e2æ, Æc, e3æ, Æb, e4æ, Æd, e5æ. We concur that one might indeed use such a sentence in such a situation. We propose to view e4 as an exception in the sense of Brisson (1998). Brisson provides an analysis in terms of covers of the fact that a distributive plural predication may be used even though not all parts of a plurality contribute towards making the predication true. She proposes that in such cases one chooses an ‘ill-fitting’ cover of the plurality, which does not contain the exceptional part. See also Beck (2001) for more general use of covers to explain exceptions in plural predication.
230 Pluractional Adverbials that the relation also holds between the referent of the NP and the relevant preceding place, which is the place immediately below. (44) a. Hans sleeps above Fritz. b. kp. Hans sleeps at p & Fritz sleeps at bel(p) bel(p) ¼ the place immediately below p c. ½½above Fritz ¼ kR.kx.kp. R(x)(p) & R(Fritz)(bel(p)) Once more, then, we have an ordering relation, this time based on the meaning of the preposition ‘above’. A place is smaller than another one according to that ordering if it is below it. We then also have the notion of the immediately preceding place.
(47) ordering relation on individuals: x : y iff dp[x is in p and "p$[y is in p$ / p : p$]] x is below y iff x is in a place that is below any place that y is in. (48) the immediate predecessor of x bel(x) ¼ iy: y : x & "z [z : x / z ¼ y or z : y] The rest of the analysis is quite parallel to the analysis of the ‘after’ example. We must be able to divide both the place and the plural individual into a sequence. Given that, we propose the analysis in terms of the in (43#) which amounts to the truth conditional entailment in (43$). The resulting overall truth conditions are described roughly in (50). (49) Cov[p] ¼ {p1, . . ., pn} such that for any pi, pi+1: pi : pi+1 Cov[these 3 children] ¼ {x1, . . ., xn} such that for any xi, xi+1: xi : xi+1 (43#) Æ3C, pæ 2 [kx.kp#.Cov(x) & Cov(p#) & x sleeps in p# & bel(x) sleeps in bel(p#)] (43$) "x[ x < 3C & Cov(x) / dp#[p# < p & Cov(p#) & x sleeps in p# & bel(x) sleeps in bel(p#)]] &"p#[p# < p & Cov(p#) / dx[x < 3C & Cov(x) & x sleeps in p# & bel(x) sleeps in bel(p#)]] (50) The place p can be divided into a sequence of sub-places, and the three children can be divided into a sequence of individual
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(45) ordering relation on places: p : p# iff p is below p# (46) the immediate predecessor of p: bel(p) ¼ ip#: p# : p & "p$[p$ : p / p$ ¼ p# or p$ : p#] In order to find a denotation for the NP ‘the other’ in the pluractional adverbial ‘one above the other’, we again suppose that there is a derived ordering of individuals based on one of the places [as defined in (47)], which will permit us to define the predecessor of an individual according to the scale introduced by ‘above’ [cf. (48)].
Sigrid Beck and Arnim Von Stechow 231
children such that: each child sleeps above the one immediately below, and each place has a child sleeping in it (. . .). The compositonal derivation of these truth conditions is based on the Logical Form in (51) and uses the PL operator in (52)—the same one as before adapted to talk about places instead of events. seq kx [x [[sleep] [one above the other x]]] (51) these 3 children [PLCov seq ¼ kR.kz.kp.Cov[p] is a sequence and Cov[z] is (52) ½½PLCov a sequence & [kz#.kp#.Cov(z#) & Cov(p#) & R(z#)(p#)](z)(p)
3.5 Pseudo-reciprocal ‘one . . . the other’ The topic of this subsection is the composition inside the modifier. How do we derive the fact that the NP ‘the other’ contributes the relevant predecessor? The overt material in (53a) suggests an internal structure of the modifier as in (53b). We assume that in addition there is covert structure in the form of the anaphor x and a contextually given relation that will constrain us to the relevant other individual. A hidden anaphor in the expression ‘other’ has been suggested, for example, in Heim et al. (1991) on the basis of data like (54a): ‘another’ here means ‘a shirt different from this shirt’. The expression ‘another’ thus includes an anaphoric reference to ‘this shirt’. The difference between (54a) and our data (as well as the reciprocal pronouns investigated by Heim et al. 1991) is that the anaphor is bound in the latter case. More generally, hidden material in definites comparable to what we assume has been detected by Partee (1989) and Winter (2000) for examples like (54b) and underlies the analysis of E-type pronouns like (54c) spelt out in Heim & Kratzer (1998). (53) a. The dogs entered the room one after the other. b. [one [after [the [other]]]] c. [one [after [the [R other x]]]] (54) a. I don’t like this shirt, bring me another. another ¼ a shirt different from this shirt b. Every soldier hit the target. the target ¼ [the [target R x]] ¼ the target assigned to him
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Other prepositions occurring in the structure ‘one Preposition the other’ would give rise to different orderings, but be otherwise parallel to the examples discussed. Very roughly, all of these pluractional modifier constructions add to the relation modified that the members of the NP argument group can be ordered according to the preposition in the modifier.
232 Pluractional Adverbials c. Every host bought just one bottle of wine and served it with dessert. it ¼ the bottle of wine that he bought. In (55) we recall the desired semantics for ‘the other’, argued for in the previous section. We can achieve this result if the hidden relation variable is assigned by the context the value in (56a) (this must once more come from the preposition), and compositional interpretation proceeds as in (56b). We end up with the meaning ‘that y which is not x and immediately precedes x#—the predecessor of x according to the ’after’ relation.
(56) a. g(R) ¼ immediately precede b. ½½ [the [NP Æe,tæ [Æe,Æe,tææ R other] x]]] g ¼ iy: y 6¼ g(x) & g(R)(g(x))(y) ¼ pred(g(x)) The referential NP needs to combine with ‘after’ in the same way as the referential NP ‘Katie’ would in the simpler case, repeated in (57). The ‘after’ from (58b) is combined with the meaning of ‘the other’ in (59). The actual modifier we see also includes ‘one’. We propose that that provides an additional constraint on the individual argument of the relation, namely that that be a singular individual. The meaning of ‘one after the other’ is then as in (60). (57) a. Min entered the room after Katie. b. ke. Min enters the room in e & Katie enters the room in pred(e) (58) a. ½½after Katie ¼ kP.kx.ke.P(x)(e) & P(Katie)(pred(e)) b. ½½after ¼ kz.kP.kx.ke.P(x)(e) & P(z)(pred(e)) (59) ½½after the R other xg ¼ kP.ky.ke.P(y)(e) & P(pred(g(x)))(pred(e)) (60) ½½one after the R other xg ¼ kP.ky.ke.P(y)(e) & one(y) & P(pred(g(x)))(pred(e)) We believe that (61a,b) are equivalent. Hence, we suggest that the two modifiers make the same semantic contribution. One way to derive this would be to have an underlying form (62a) from which both are derived as different surface forms. (61) a. She washed them dog after dog.
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(55) ½½the R other xg ¼ pred(g(x)) ¼ iy: y immediately precedes g(x)
Sigrid Beck and Arnim Von Stechow 233
b. She washed them one (dog) after the other. (62) a. one dog after the other dog b. one dog after the other dog c. one dog after the other dog It is relatively obvious how to derive ‘one after the other’ from (62a), namely, through a process of N-deletion. This is not obligatory, at least not for the first N to be deleted, cf. (63). It is far less obvious how (62c) would be derived.3
The above considerations lead to a final revision for the internal semantics of the modifier which yields (64): we add the information that the relevant predecessor as well as the individual argument of the relation are Ns. (64) a. ½½ [the[[[R other]x]N]] g ¼ iy: y 6¼ g(x) & g(R)(g(x))(y) & ½½N (y) ¼ the dog immediately preceding g(x) pred(g(x)) b. ½½one N after [the R other x N] g ¼ kP.ky.ke.P(y)(e) & one(y) & ½½N (y) & P(pred(g(x)))(pred(e)) We call these modifiers pseudo-reciprocal. They are reminiscent of reciprocals formally in the use of ‘other’ and semantically in talking about a different member of the same group (in the example, the dogs). But they are not reciprocal pronouns formally—they are modifiers. Moreover, the NP in the modifier is a singular. By contrast, a reciprocal pronoun introduces a second plurality of individuals (Beck 2001).
3.6 Concluding remarks on Æe, Æv, tææ pluralization We have argued that ‘piece by piece’-type adverbials as well as ‘dog after dog’-type adverbials show us that natural language can pluralize relations of type Æe, Æv, tææ. The adverbial makes the pluralization visible by adding extra constraints on the simultaneous division into sub-parts (subevents and subindividuals) performed by the pluralization operation. In the case of ‘piece by piece’-type adverbials, this is the constraint 3
In view of data like (i), it may even be doubtful that one would want to make too close a connection between the two constructions. Thanks to David Pesetsky for pointing this out. (i)
I reviewed paper after paper after paper.
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(63) a. She put the books one bundle beside the other (bundle) on the porch. b. She examined the wine one bottle after the other (bottle).
234 Pluractional Adverbials that the division (the cover) of the plural entity in question be a partition into pieces. In the case of pseudo-reciprocal ‘dog after dog’ adverbials, the constraint is that the cover be a sequence of individual dogs. Now that we have an overview of the data, we think it makes sense to adopt a slightly more abstract analysis of ‘piece by piece’ and to subsume this kind of adverbial under plain Æe, Æv, tææ pluralization. The revised analysis is given in (66) and (67). The adverbial is just a modifier; the combination with the plain PL operator yields the same result as PLdiv earlier [repeated in (65)].
(66) plain Æe, Æv, tææ pluralization: ½½PL ¼ kCov.kRÆe,Æv,tææ.kx.ke: PART(Cov, e + x). [kx#. ke#. Cov(e#) & Cov(x#) & R(x#)(e#)](x)(e) (67) a. b. c. d.
John ate the cake piece by piece. [[the cake][PLCov [piece by piece [k2[John ate t2]]]]] ½½piece by piece ¼ kRÆe, Æv, tæækx.ke. x is a piece & R(x)(e) ke.Æe, the_cakeæ 2 [ky#.ke#.Cov(y#) & Cov(e#) & y# is a piece & John ate y# in e#]
We end up with just two pluralization operators: the plain PL operator and PLseq. This, as far as we can see, is irreducible. Sequence pluralization imposes a stronger requirement on the cover. (68) sequence Æe, Æv, tææ pluralization: ½½PLseq ¼ kCov.kR.kx.ke. Cov[e] is a sequence and Cov[x] is a sequence & [kx#. ke#. Cov(x#) & Cov(e#) & R(x#)(e#)] For both types of adverbials, the plural operator is not the adverbial itself. The adverbial is a modifier that merely seems to indicate the presence of the pluralization operator. This revision of the analysis raises a question. Pluractional adverbs like ‘piece by piece’ are predicates of singularities but come with a PL operator. There is nothing in the semantics of the pluractional adverb that triggers the semantic plural. Are these adverbs compatible with the singular, or do we need some mechanism that would force the adverbial to combine with a higher PL operator? (69) a. Ich habe die A¨pfel einzeln gekauft.
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(65) divisive Æe, Æv, tææ pluralization: ½½PLdiv ¼ kPÆe, tæ.kCov.kRÆe, Æv, tæækx.ke: PART(Cov, e + x). [kx#. ke#. Cov(e#) & Cov(x#) & P(x#) & R(x#)(e#)](x)(e)
Sigrid Beck and Arnim Von Stechow 235
(70) ½½piece by piece ¼ kCov.kRÆe,Æv,tææ.kx.ke: Cov(x) & Cov(e) & R(x)(e) & x is a piece In order to rule out the ‘singular’ interpretation of sentences like (69a), we need a syntactic principle that states that a cover variable is only licensed if it is the variable of a PL operator, or if it is co-indexed with a licensed cover variable (this licensing condition would amount to the ‘plural polarity item’ effect). It follows that the following LF is not possible for sentence (69a): (69a#) [vt the cake [evt piece by pieceCov [evt 2 John ate t2]]] (ill-formed) In contrast, the following LF is a good one and it has the ‘plural’ reading, provided the plural predicates are only defined for pluralities proper. (69a$) [vt the cake PLCov1 [evt piece by pieceCov1 [evt 2 John ate t2]]] Both options—the pragmatic and the plural polarity item one—would derive the fact that pluractional adverbials are fully acceptable only in
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I have the apples individually bought b. ?Ich habe den Apfel einzeln gekauft I have the apple individually bought It seems to us to be arguable that (69b) is pragmatically odd (it may serve as a sort of joke), but is semantically well formed. Under this view, there could be an implicature brought about by the pluractional adverb, to the effect that the argument of the relation the adverb modifies be a genuine plurality. The implicature could be generated by the notion that division into parts is only relevant when there is more than one part of the kind mentioned. At any rate, the suggestion would be that the pluralization requirement is a weak pragmatic one, one can then leave the compositional semantics as is. Alternatively, if one wanted to enforce that pluralization accompanies the adverbial, one could view the adverbial as a plural polarity item in the style of Oh (2005). Oh proposes that elements like Korean -ssik mark elements that need to occur in the scope of a distributivity operator. Analogously, our pluractional adverbials would have to occur in the scope of a cumulative operator of the appropriate type. Here is a way to spell out the connection in more detail: Cover variables come with plural operators. So there is connection between them and pluralization. Let us suppose that pluractional adverbs select a cover variable as well. The variable of an adverb must be anaphoric to the variable of the local plural operator. So the entry for ‘piece by piece’ would be this:
236 Pluractional Adverbials
4 There is an interesting type of example brought to our attention by an anonymous referee that we cannot completely analyse with the tools at our disposal. An instance is given in (i).
(i)
John glued the book together page by page.
The interesting aspect of (i) is that we are talking about the creation of a complex whole (the book does not exist in the beginning of the event, but it does by the end); our pluractional adverb tells us about how the event proceeds in terms of the parts of the whole (its pages). We think that the analysis will have to rely on the detailed semantics of the creation verb, but have found this too complicated to develop here. 5
Malte Zimmermann (personal communication) raises the question of a cardinality requirement accompanying pluractional adverbials. He suggests that ‘piece by piece’-type adverbials require that there be at least two pieces, while ‘one Preposition the other’ adverbials require there to be more, say at least three. A cardinality of at least two follows from the requirement that there be a genuine plurality. A stronger cardinality requirement would have to be spelt out separately (e.g. as in (i) below). However, we are not sure that we want to do that. Example (ii) seems to us to be fine. Perhaps, a larger cardinality is just an assumption one tends to make, rather than part of the semantics of such sentences. Malte Zimmermann also observes that reduplicative adverbials seem to require an even larger cardinality of the group they are anaphorically related to. Thus, (iii) is odd. (i) ½½PLseq ¼ kCov.kR.kx.ke. Cov[e] is a sequence and Cov[x] is a sequence & card(Cov[x]) > 2 & card(Cov[e]) > 2 & [kx#.ke#.Cov(x#) & Cov(e#) & R(x#)(e#)] (ii) The two women entered the room one after the other. (iii) The two women/they entered the room woman after woman. The idea brought forth in the text above that adverbials themselves have a cover variable allows us to capture such effects. We could for example use the semantics in (iv). We leave the details of this empirically delicate issue for future research. (iv) ½½one after the R other xg ¼ (kCov.kP.ky.ke: card(Cov[y]) is large.P(y)(e) & one(y) & P(pred(g(x)))(pred(e)) 6 In examples such as ‘She placed the books on the porch bundle next to bundle’, we have to assume that an ordering relation is derived from ‘next to’ that mounts to either ‘to the right of ’ or ‘to the left of ’. That is, more goes into the ordering relation than just the meaning of the preposition.
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the context of a corresponding pluralization operator. We will leave the choice between these two options open. Under both of them, notice, the term ‘pluractional adverbial’ becomes a descriptive notion. What brings about pluractionality in the analysis is the invisible PL operator which is triggered (pragmatically or at the syntax/semantics interface) and restricted by the presence of the adverbial. The adverbial itself is not a plural operator.4,5 We have two types of pluractional adverbs, the ‘piece by piece’-type and the pseudo-reciprocals. How do we know which is which? Pseudo-reciprocals require an ordering relation such as ‘after’, ‘above’, ‘within’. The relation is provided by the preposition and is the basis for the sequence restriction used by PLseq. The ‘piece by piece’ adverbs are built around particles such as ‘by’, ‘for’, ‘and’ which mean nothing by themselves; in particular, they do not express an ordering relation. Hence, they are unsuitable for combination with PLseq and use plain PL instead.6
Sigrid Beck and Arnim Von Stechow 237
4 RELATED ISSUES This section explores some issues related directly to the analysis developed in section 3. These concern IAO reciprocals, the scope of our PL operators, pluractionals in argument position, the event distributivity reduplications mentioned in the introduction and finally other ‘Noun Preposition Noun’ modifiers.
4.1 Apparent reciprocals
(71) a. b. (71a#)
The children sleep above each other. The three dogs came into the room after one another/ The three dogs followed each other into the room. IAO: Each child sleeps above or below some other child.
(72) a. b.
Schema of an elementary reciprocal sentence: A R each other. antecedent relation reciprocal pronoun IAO: "x[x < A / dy[y < A & xRy or yRx]]
We suggest instead that the data in (71) (and IAO reciprocals in general) have a pseudo-reciprocal semantics. That is, (71a) really amounts to (73a). The semantics we assign to (73a), and by assumption then also to (71a), entails (73b). (73) a. The children sleep one above the other. b. Each child sleeps above some other child (namely, her ‘predecessor’ relative to the ‘below’ relation). Why do we pursue this idea? There are three kinds of facts that motivate us. The first is that the IAO truth conditions are very weak indeed and intuitively too weak for example for (71b). The IAO truth conditions for (71b) are given in (74a). These truth conditions predict the sentence to be true in the situation depicted in (74b). This does not accord with intuitions. By contrast, our truth conditions will render
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In this subsection, we explore the possibility of extending our analysis of pseudo-reciprocals to certain apparent reciprocals, namely those that have an IAO interpretation. Some examples of such reciprocals are given in (71). The interpretation of (71a) according to Dalrymple et al. (1998) is paraphrased in (71a#). The general schema of an IAO interpretation is given in (72). The data in (71) are all taken to have such a weak semantics.
238 Pluractional Adverbials (71b) equivalent to (74c) and correctly predict that the sentence is false in a situation like (74b). (74) a. Each dog came into the room after or before some other dog. b. D3 + D2 / D1 c. The dogs entered the room one after the other.
(75) a. The plates are stacked on top of each other. b. The Smiths and the Johnsons outnumber each other. A third and final problem with IAO is noted in Beck (2001): IAO reciprocals are restricted to local reciprocal relations while other reciprocals are not. To illustrate what is meant by a non-local reciprocal relation, consider (76). The sentence is judged true if (76#a) is the case. This can be derived from the truth conditions in (76#b): the reciprocal relation ‘want to kill’ holds between non-identical members of the antecedent group ‘Tracy and Joe’. Example (76) is an example of a regular reciprocal interpretation, weak reciprocity. The reciprocal relation ‘want to kill’ is non-local in that it is not a relation that exists as the meaning of a surface constituent. (76) Tracy and Joe want to kill each other. (76#) a. Tracy wants to kill Joe and Joe wants to kill Tracy. b. ÆT&J, T&Jæ 2 [kxky: x 6¼ y. x wants to kill y] We should contrast (76) with (77). The sentence can be understood as in (78)—Tracy and Joe agree that they want to sleep above each other rather than, say, beside each other. It cannot be understood as in (79), which would be made true by the fact that Tracy wants to sleep above Joe. Example (79) would be a non-local IAO interpretation with the reciprocal relation ‘want to sleep above’. Clearly, this is not possible. Only a local reading inside the embedded clause in (78) is acceptable. (77) Tracy and Joe want to sleep above each other.
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A second problem for IAO reciprocals is the fact that an IAO interpretation is only possible with a restricted set of relations. See Langendoen (1978), Sauerland (1998), Beck (2001) and Schein (2003) for discussion. As an illustration, notice that (75a) with the relation ‘on top of ’ is acceptable under an IAO interpretation while (75b) with ‘outnumber’ is unacceptable and cannot have an IAO reading (which would be made true by the fact that the Smiths are more numerous than the Johnsons, for instance). If IAO were a regular interpretation for reciprocal sentences, why should it not be generally available?
Sigrid Beck and Arnim Von Stechow 239
(78) Tracy and Joe both have the following desire: we sleep above each other (i.e. one of us sleeps above the other). (79) For each x, x one of Tracy and Joe: either x wants to sleep above the other one of Tracy and Joe, or the other one of Tracy and Joe wants to sleep above x. The pair in (80) makes the same point: in (80a), a non-local interpretation is possible in which the different members of the antecedent group ‘these people’ were introduced by different linguists. A similar interpretation is not available in (80b); the same apprentice magician has to line up the glasses.
The two constraints on the availability of IAO interpretations (limited set of relations and local interpretation only) are quite unexpected as long as one thinks of IAO as a regularly available interpretation of reciprocal pronouns. This is additional motivation then, besides the problem mentioned above with inappropriately weak truth conditions, for looking for an alternative analysis of the phenomenon of IAO. We propose that IAO reciprocals only appear to be reciprocals, and are really pseudo-reciprocals: (81) above each other 0 (one) above the other That is, the example in (82a) should really be interpreted as (82b). (82) a. Tracy and Joe want to sleep above each other. b. Tracy and Joe want to sleep one above the other. The truth conditions we predict are the ones of pseudo-reciprocals, which seem right to us. As for the unexpected constraints on the relations that participate in an IAO interpretation, we have nothing concrete to offer. One may suppose that whatever process relates (82a) and (82b) is somehow restricted and cannot apply to every relation. For all we know, the connection may be lexical. But no concrete predictions arise regarding which relations can participate. We do have something to say about the fact that apparent IAO reciprocals—now reanalysed as pseudo-reciprocals—only receive a local interpretation. In (77) ¼ (82), for instance, the whole ‘(one) above the other’ is an adverbial that can only modify the embedded predicate ‘sleep’ (whishes cannot plausibly be above each other). And since there is no further potentially scope-bearing element in this modifier (‘the
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(80) a. These people were introduced to each other by a linguist. b. The glasses were lined up behind each other by an apprentice magician.
240 Pluractional Adverbials other’ being a singular), there is no process that could generate a nonlocal interpretation. A final comment: there are cases of IAO reciprocals for which our pseudo-reciprocal truth conditions might be thought too strong. Example (83b) is a case in point. Dalrymple et al. (1998) point out that such a sentence can be considered true in a situation with two bunk beds each of which sleeps two children. This is different from (83a), our pseudo-reciprocal. We speculate that (83b) permits a partition of the children into two groups of two, on which its interpretation with the bunk beds is based. This is excluded by the overt element ‘one’ in (83a) which tells us that the partition of the children is into singletons.
4.2 The scope of PL Our theory of pluralization is essentially syntactic in the sense that the PL operator applies in the syntax to an LF constituent. It is thus in line with Sternefeld (1998), Heim et al. (1991) or Beck & Sauerland (2000), but not compatible with pure lexical theories of pluralization such as Krifka (1989) or Winter (2000). The syntactic nature of the PL operator is obvious from the fact that pluractional adverbs are in the scope of the PL operator. Recall, for example, our analysis for (84): (84) These three dogs entered the room one after the other. The LF was (85). Sec kx [vt x (85) these 3 dogs [PLCov
j__________QR________j
[evt entered the room] [(evt)(evt) one after the other x]]] anaphor
The lexicalist has to explain the PL operator away, which seems difficult but is perhaps not impossible in this case (by using an appropriate distributor). But there are harder examples: (86) a. Die Ma¨dchen schenkten den Jungen eine Blume. the girls gave the boys a flower b. Die Ma¨dchen schenkten den Jungen eines nach dem anderen eine Blume. the girls gave the boys one (Nominative) after the other a flower c. Die Ma¨dchen schenkten den Jungen einem nach dem anderen eine Blume.
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(83) a. These four children sleep one above the other. b. These four children sleep above each other.
Sigrid Beck and Arnim Von Stechow 241
the girls gave the boys one (Dative) after the other a flower. The examples are knitted after the pattern of the examples brought forward in Beck and Sauerland (2000) against Winter (2000). They seem hard nuts for the lexicalist but fall out easily in our theory. The formalizations of (86b) and (86c) are these:
The examples illustrate that the object variable of ‘other’ does a useful job. It disambiguates the two readings: in (86b) the antecedent is ‘the girls’ and in (86c) the antecedent is ‘the boys’. Two possible scenarios satisfying the LFs are the following ones: (87) a. e ¼ e1 + e2, G ¼ g1 + g2, B ¼ b1 + b2, ½½ flower ¼ {f1, f2} b. e1: g1 give f1 to b1 + b2 e2: g2 give f2 to b1 + b2 c. e1: g1 + g2 give f1 to b1 e2: g1 + g2 give f2 to b2 The general conclusion is that the analysis proposed supports the syntactic plural theory developed and defended in previous papers by Beck and others. It is worth noting that pluractional operators can be iterated. Example (88) provides an example as well as the sketch of an analysis. Suppose that the boys are b1, b2 and the girls are g1, g2, and that the sequence of events was e11 < e12 < e21 <e22. The LF in (88#a) is then predicted to be true in the situation (88#c), as desired. (88) Die Jungen haben einer nach dem anderen die Ma¨dchen eines nach dem anderen geku¨sst. the boys have one after the other the girls one after the other kissed (88#) a. ÆB, eæ 2 PLCovseq (kx.ke#. ÆG, e#æ 2 PLCov#(e#)seq (ky. ke$. [x kisse$ y] o.a.t.oy) o.a.t.ox) b. Cov[B + e] ¼ {b1, b2, e1, e2} Cov#(e1)[G + e1] ¼ {g1, g2, e11, e12} Cov#(e2)[G + e2] ¼ {g1, g2, e21, e22}
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(86#) a. ÆG, eæ 2 seq (kx.ke#. x give B a flower in e# & pred(x) give B PLCov a flower in pred(e#)) b. ÆB, eæ 2 seq PLCov (ky.ke. G give y a flower in e# & G give pred(y) a flower in pred(e#))
242 Pluractional Adverbials c.
(89) She bounced a ball from wall to wall. Only one ball can be involved in this multiple action. Since our theory assumes that pluralization takes place in the syntax, ‘a ball’ might have narrow scope with respect to the PL operator and we predict the bouncing of possibly different balls. For ‘from wall to wall’, see the discussion in section 4.5. We consider an example with a more familiar adverbial, (90). The question is whether pluralization can take scope over the indefinite. This would correspond to Logical Forms like (90#a). (90) a. The boys kicked a wall one after the other. b. One after the other, the boys opened a drawer. (90#a) ÆB, eæ 2 PLCovseq (kx.ke#.x kick a wall in e# and pred(x) kick a wall in pred(e#)) Perhaps, it is easiest to assume that one wall is involved—we are not really sure about the facts. We concur, at any rate, that pluralization often has a tendency to take narrow scope. We do not have a general analysis of this effect. One restriction relevant in the present case could be the fact that event arguments like to be bound before they encounter an aspectual operator—say, just above VP (e.g. as in Diesing 1990; von Stechow 2002). This may limit the scope taking possibilties of our pluractional pluralization operations. Importantly, we think that the clear examples in which a plural operator does take non-trivial scope show that a syntactic analysis of pluralization is called for. That such an analysis must be constrained to limit scope possibilities, we would certainly agree to.
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Cov# is a variable for functions from events into Covers. The value of Cov# depends on the bound variable e#. The scenario (88#c) shows why this must be so: g1 is before g2 in the sequence of kissing that e1 consists of, but after g2 in the sequence of kissing that e2 consists of. Note that examples where covers must be parametrized functions are also given in Schwarzschild (1996). His examples involve modality. Angelika Kratzer (personal communication) mentions the following objection to syntactic pluralization to us:
Sigrid Beck and Arnim Von Stechow 243
Example (90b) has the pluractional adverbial topicalized. Such a structure is incompatible with our assumption that the adverbial contains a variable bound by the subject. We must assume that at Logical Form, the adverbial is reconstructed to its position within the VP.
4.3 Nominal pluractionals The data we have analysed all involve pluractional adverbials like (91). However, there are many cases where our pluractionals occur in argument position. The semantics we think (92a,a#) have is given in (92#). (92) a. She washed dog after dog. a#. She washed one dog after the other. b. She wrote paper after paper. b#. She wrote one paper after another. (92#) ke. dX [Cov[e] is a sequence and Cov[X] is a sequence & [kx.ke#.Cov(x) & Cov(e#) & dog(x) & she washed x in e# & she washed pred(x) in pred(e#)](X)(e)]] It is interesting that these expressions may alternatively play the role of adverbials or arguments syntactically. As regards semantics, we can assume that the meaning of ‘dog after dog’ is as before. We then shift the pluralization operation to existentially bind the individual argument: (93) Regular PLseq: kCov.kR.kz.ke. Cov[e] is a sequence and Cov[z] is a sequence & [kz#.ke#.Cov(z#) & Cov(e#) & R(z#)(e#)](z)(e) (93#) existentially shifted PLseq: ½½ PLd ¼ kCov.kR.ke. dz[Cov[e] is a sequence and Cov[z] is a sequence & [kz#.ke#.Cov(z#) & Cov(e#) & R(z#)(e#)](z)(e) The way from PLseq to PLd involves an existential type shift: (94) ÆCov, Æe, Æv, tææ, Æe, Æv, tæææ 0 ÆCov, Æe, Æv, tææ, Æv, tææ k .kCov.kR.ke.dz[ (Cov)(R)(z)(e)] (95) She washed dog after dog. dCov [[dog after dog][Æe, Æv,tææ kx. she washed x]] [PL ½½dog after dog ¼ kR.ky.ke. R(y)(e) & dog(y) & R(pred(g(x)))(pred(e))
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(91) She washed them dog after dog.
244 Pluractional Adverbials We need to move ‘dog after dog’ for type reasons. We can suppose that an existential shift is generally available (see, e.g. Partee 1987 for another version); it has to apply in this case in order for us to get a reasonable result. In case of a nominal pluractional in subject position (example below), we would assume reconstruction, just as we proposed in the previous subsection. (95#) Dog after dog entered the room.
4.4 Reduplicative event distribution
(97) a. Sally ran and ran. b. Sally was sick and sick. c. The train arrived and arrived. Example (97a) means that Sally ran a long time. It would seem then that ‘and’ simply means ‘a long time’ in this context. It is tempting do derive this reading from a pluractional meaning of ‘and’. (98) Sally ANDCov ran ke.e 2 [ke#.Cov(e#) & ran(Sally)(e#)] AND is a plain PL operator that is restricted to cumulative and non-divisive events. (99) ½½AND ¼ kCov.kPvt: P is cumulative & P is not divisive. ke.[ke#.Cov(e#) & P(e#)](e) A predicate is cumulative iff, whenever it is true of two arguments, it is also true of their sum. A predicate is divisive (or not atomic) iff whenever it is true of an argument, it is also true of a part of that argument (see, e.g. Krifka 1998). The two presuppositions account for the ungrammaticality of (97b) and (97c). The requirement that covers have to be non-divisive presumably follows from general assumptions for covers: if we want to
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We already mentioned data in the introduction that look parallel to our pluractionals and are semantically obviously related. Some examples are given below. (96) a. Wir warteten Stunde um Stunde. we waited hour for hour ‘We waited for hours.’ b. She surprised us time after time. c. Sie hat uns immer wieder ueberrascht. she has us always again surprised ‘She surprised us again and again.’
Sigrid Beck and Arnim Von Stechow 245
split a plurality into smaller units, we need some contextually relevant standard of individuation, that is, a measure that gives us smallest units. In this particular case, the verb gives us the principle of individuation by itself. Since states are divisive, they are excluded as principles of individuation. Achievements and accomplishments are excluded because they are not cumulative. Since these are conceptual requirements, it is perhaps not necessary to write them into the semantics of AND. A simpler version is (99#). The semantics for AND is very similar to the semantics given to durative adverbials in Kratzer (2003).7 (99#) ½½AND ¼ kCov.kPvt: ke.[ke#.Cov(e#) & P(e#)](e)
(100) a. Wir stiegen ho¨her und ho¨her. we climbed higher and higher b. Es wurde ka¨lter und ka¨lter. it became colder and colder c. Es wurde immer ka¨lter. it became always colder d. Sally approached the horse step by step. (100#) a. It ANDCov became colder b. ke.e 2 [ke#.Cov(e#) & become_colder(e#)] c. become_colder(e) ¼1 iff it is colder at the end of e than at the beginning of e (von Stechow 1996) Pluralization of Æv, tæ predicates (i.e. event distribution) is relevant in the analysis of a type of example brought to our attention by an anonymous referee. A representative is (101) below. (101) As the wheel turned, the flags hit the water one after the other.
7
(i)
Kratzer’s duratives: ½½for twenty minutes ¼ kP.ke.P(e) & fminute(e) ¼ 20 & e ¼ re#[e# < e & P(e#)]
The right conjunct in the definition is the contribution of our AND.
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It belongs to the pragmatics of AND that the selected cover must have a rather large cardinality (compare section 3.6). This entails that the running must be quite long. The following examples are degree achievements, that is, predicates which have the property that the result is attained to a higher degree at the end of the action than it had been attained at the beginning. They are cumulative and are analysed as before. ‘immer’ is presumably a PL operator with the same semantics as AND.
246 Pluractional Adverbials We propose that this example combines a sequence interpretation (the flags as ordered along the rim of the wheel) with an iteration of pluralization operations as observed in section 4.2, viz. Æv, tæ pluralization above PLseq. (101#) One sequence reading (the wheel turns once): ke.Cov[F] is a sequence & Cov[e] is a sequence & ÆF, eæ 2 [kz.ke#: z hits the water in e# & pre(z) hits the water in pred(e#)]
The data discussed in this subsection illustrate the generality of our theory of plural predication and pluractionality.
4.5 Other ‘noun preposition noun’ adverbials We have investigated the semantics of three kinds of modifiers of the form ‘Noun Preposition Noun’. They are exemplified by ‘piece by piece’, ‘dog after dog’ and ‘Stunde um Stunde’. These three types of modifiers do not have the same semantics, although their semantics is similar in that all three can be called pluractionals. We think that the form ‘Noun Preposition Noun’ may be used by other kinds of modifiers not all of which may be pluractional. To illustrate, consider the data in (101): (101) a. The houses stand side by side. b. The lovers walked hand in hand to the river. c. The squirrel jumped from tree to tree. The formal similarity to the data investigated in this paper may lead one to suppose that ‘side by side’, ‘hand in hand’ and perhaps also ‘from tree to tree’ should share the semantics of pluractionals like ‘piece by pice’ or ‘dog after dog’. But we do not think that this is the case. Consider the uses in (102): (102) a. Anna walked hand in hand with Lena. b. Haus A steht Seite an Seite mit/neben Haus B. House A is standing side by side with/next to House B We propose the semantics in (103) for example (102b).
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(101$) Iterative reading (the wheel continues to turn): ke$.e$ 2 ke: PART(Cov#, e). Cov[F] is a sequence & Cov[e] is a sequence & ÆF, eæ 2 [kz.ke#: z hits the water in e# & pre(z) hits the water in pred(e#)]
Sigrid Beck and Arnim Von Stechow 247
(103) ÆA, Bæ 2 [kx.ky. x is next to y and one side of x is adjacent to one side of y] The contribution of ‘side by side’ would be (104): (104) [kx.ky. one side of x is adjacent to one side of y] There is no evidence of a plural operation according to this analysis of (102b). The original examples (101) offer more justification of a connection to plural predication through the use of a plural subject. Their semantics could be as in (105):
(105#) ÆH, Hæ 2 [kx.ky: x 6¼ y. x is next to y and one side of x is adjacent to one side of y] That is, their semantics could plausibly be seen in terms of hidden reciprocity, the relevant reciprocal reading being weak reciprocity. Hence the appearance of a plural semantics in (101). But, we claim that this is not triggered by the ‘Noun Preposition Noun’ modifier—the modifier is equally happy without any plurality in the semantics of the sentence it occurs in. The plural semantics in (101) comes about through independent means (the plural subject plus covert reciprocity). We think that the semantics of (106a) is similarly weak (thanks to Cecile Meier for the example): (106) a. The squirrel jumped from tree to tree. b. The squirrel jumped from one tree to another tree. It seems clear that ‘from tree to tree’ does not involve a pseudoreciprocal semantics, that is, require a scenario in which the trees can be arranged in a sequence according to the jumping. The meaning of ‘from tree to tree’ seems to be something like ‘from one tree to another tree’. Ignoring possibly serious issues of compositionality, the prepositions generating the adverb mean perhaps the following: (107) ½½ from-to ¼ [kx.ky.kPevt.kz.ke.P(z)(e) & e starts at x & e ends at y] The meaning of sentence (106a) could then be analysed as (108) Æe, T, Tæ 2 ky.kx.ke#: x 6¼ y.Cov(x) & Cov(y) & Cov(e#) & the squirrel jumps in e# & e# starts at x & e# ends at y
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(105) a. Die Haeuser stehen Seite an Seite nebeneinander. The houses stand side by side next to each other b. The lovers walked hand in hand with each other.
248 Pluractional Adverbials The LF is generated by means of a three-place PL operator and means approximately: ‘The trees are divided in a way such that each tree is involved in a jumping to and from some other tree.’ If something along these lines is correct, weak reciprocity would be involved here as well. To be sure, this has to be worked out and might turn out to be wrong. For our purposes, the important point is that not all ‘Noun Preposition Noun’ sequences share the semantics of pluractional modifiers, and hence one should not take our proposals to extend to all data with a similar surface appearance without careful semantic consideration.
5.1 Summary To summarize, we subscribe to the view that all pluralization is sensitive to a division of pluralities into appropriate sub-parts. Pluractionals make this visible; in our cases with ‘piece by piece’ and ‘dog after dog’, they tell us which units are contained in the cover. They also show that natural language has pluralization of Æe, Æv, tææ predicates, that is, simultaneous pluralization of an event- and an individual-argument slot. We have integrated the two types of pluractional adverbials into a theory of pluralization which combines Sternefeld’s plural operators and one-place covers. As has been argued in Beck (1999), the partition of pluralities into parts is interdependent with the partition of events into parts. Pluractional adverbials restrict the partition induced by a cover by imposing restrictions on individuals: the adverbs say which criterion an individual must meet in order to be in a cover. Suppose we start from a plural fact R(e, x) and we know that a part y of x is in the relevant Cov, then that part e# for which R(e#,y) is true must be in Cov as well. So pluractional adverbs indirectly restrict covers of events. We have introduced two types of Æe, Æv, tææ PL operators, plain PL and PLseq. The former is plain pluralization and needed for ‘N by N’type adverbs. The second type is needed for pseudo-reciprocal adverbs. The two operators are very similar in structure but cannot be reduced to one operator in an obvious way. The PL operators apply in the syntax. Adverbials with ‘N Preposition the other’ are pluractionals that give rise to a sequence interpretation, which we have called pseudoreciprocal. Our theory of pseudo-reciprocals brings these together closely with reciprocals. The common core is the expression ‘other x’. In the case of reciprocals, it contributes towards ‘the others among
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5 CONCLUSIONS
Sigrid Beck and Arnim Von Stechow 249
5.2 Related literature There have of course been earlier approaches to these or related phenomena. Our goal in this paper has not so much been to develop a compositional semantics of pluractional adverbials, but rather to develop such an analysis in the framework of plural predication developed in Beck (2001). The other proposals do not have that aim. The point of our choice of framework is generality: we believe that embedding our semantic analysis in a general theoretical framework for the analysis of plural predication has allowed us to develop an analysis of pluractionality that is also very general, empirically and theoretically. Other approaches that, to our knowledge, our analysis still owes a significant theoretical debt to are the following: Lasersohn (1995) on pluralities of events; Moltmann (1997), who proposes an analysis of ‘one at a time’ adverbials and related data and some papers that add to our stock of pluractional phenomena, for example, Matthewson (2000) and Oh (2005). There are just a couple of works that propose an analysis of pseudo-reciprocals: Stockall (2001), who analyses ‘dog after dog’-type adverbials and Zimmermann (2002), who proposes a refinement of Stockall’s analysis. Also, Schein (2003) offers an analysis of reciprocals as adverbials. We will discuss the former literature first. Lasersohn (1995) introduces pluralities of events. The primary applications of this are analyses of ‘together’, verb conjunctions (plus
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them’. In the case of pseudo-reciprocals ‘the R other x# means ’the R-predecessor of x#, where R denotes an ordering. Pseudo-reciprocals are not reciprocals because they do not introduce a group. If apparent reciprocals with an IAO interpretation are reanalysed as pseudo-reciprocals (i.e. pluractional ‘one . . . the other’), this may explain some peculiarities that otherwise set apart IAO reciprocals from better behaved reciprocals. Pseudo-reciprocals are different from regular reciprocals in not introducing a plurality of type Æeæ. Rather, they are a modifier containing a singular ‘the other’ NP. Our truth conditions for IAO reciprocals are stronger than in the previous literature and empirically more adequate, we think. They give rise to the problem of the first dog viz. the problem of the bottom child. We solved these by assuming a pragmatic restriction for quantification. We have extended the analysis to event iteration as in ‘ran and ran’ and to occurrences of pluractionals in argument position. Finally, we have pointed out that not all ‘Noun Preposition Noun’ sequences share the semantics we propose.
250 Pluractional Adverbials
(109) John sold the apples one at a time. [112a] If we understand her correctly, the syntax she proposes in her definition [118] amounts to this: (110)
The node BIN-TIME is an ad hoc grammatical relation, whose interpretation is a semantic operation that makes sure that precisely the desired meaning of (109) is obtained when the value of BIN-TIME is applied first to ‘the apples’ and then to ‘John’. Something similar would have to happen in the case of ‘piece by piece’. By contrast, we suggest that natural language systematically makes available a family of pluralization operations for various types of predicates, all of which are restricted to contextually relevant part-whole structures (covers). Pluralization effects as such depend on those operators. Adverbials only add very particular information (for example, what goes in the cover). This amounts to a totally different architecture. Thus, we are indebted to Moltmann’s (1997) insights on simultaneous division of indivdual and event arguments of relations. We have a different theory on how this comes about in plural predication. There are several other works on adverbials like ‘one at a time’, including Matthewson (2000) on Salish and Oh (2005). Oh’s (2005)
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‘alternately’) and pluractional markers, which we need not go into here. His work is, we believe, the crucial reference for explicitly plural event predication. Moltmann (1997) includes an analysis of ‘one at a time’ (under the heading ‘binary distributive event quantifiers’ in section 6.6.4). The nature of that adverbial’s semantic contribution makes necessary a simultaneous division of individuals and events into sub-parts in the predication. This property emerges clearly from Moltmann’s analysis. It is shared by ‘piece by piece’ and ‘dog after dog’. She thus anticipates this aspect of our analysis. It is, however, embedded into a different architecture, in that her views of the syntax–semantics interface and pluralization operations in particular are incompatible with our own. We illustrate her proposal briefly by discussing her analysis of the pluractional adverb ‘one at a time’ in the following sentence:
Sigrid Beck and Arnim Von Stechow 251
work on Korean pluractional adverbials ought to be mentioned because her theoretical outlook is similar to ours, in attributing significant aspects of the semantics of the construction to standard plural operators, and only parts of it to the adverbial itself. The resulting semantics is similar to our ‘piece by piece’-type adverbials. In contrast to the authors mentioned above, Stockall and Zimmermann are concerned with the phenomenon whose analysis is the central purpose of the present paper. Stockall (2001) analyses the sentence in (111). She assumes the S-structure in (112). (111) Girl after girl arrived.
In this construction, ‘after’ has a complicated meaning. It applies to a pair of individuals and an intransitive verb, which is a relation between an individual and an event: (113) after#(x)(y)(arrive#) ¼ ke.de#,e$[e# < e & e$ < e & arrive(x)(e#) & arrive(y)(e$) & e# after e$] REDC is a reduplication operator, which applies to the common noun ‘girl’ and the two-place relation ky.kx.after(x)(y)(arrive). (114) REDC(girl)(ky.kx.after(x)(y)(arrive)) ¼ ke.C is a large subset of girls & "x,y[(C(x) & C(y) / x arrive after y in e or y arrive after x in e] The LF yielding the desired interpretation is the following: (115)
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(112) [IP[PPRED [after girl]] arrived]
252 Pluractional Adverbials The movement index of RED is 1 and that of girl is 2. So we have crossing k-abstraction. The structure gives us those events that are the arrival of a group of girls at different times. The approach gets this particular example right, but it does not generalize non-transitive prepositions such as in the following examples: (116) Sie schlug Nagel neben Nagel in die Wand. she hit nail next to nail into the wall
Acknowledgements We would like to thank audiences at Universita¨t Tu¨bingen, Sinn und Bedeutung 10 and MIT and moreover Philippe Schlenker, Malte Zimmermann and an anonymous reviewer for important comments.
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Zimmermann (2002: 287f.) criticizes Stockall’s (2001) proposal on syntactic grounds. He proposes a somewhat different syntax, which, however, yields the same truth conditions. Stockall/Zimmermann’s analysis of ‘dog after dog’ is also different from ours in theoretical outlook. Like Moltmann, they hold the adverbial itself and/or its composition within its local structure responsible for all of the specific semantics of the construction. Our analysis has been guided by the idea that we have a system of plural predication in place independently which includes plural operators of various types plus a restriction on relevant part-whole structures. Thus, the adverbial has a very slim semantics, with much of the burden to be carried by the pluralization operation. We believe that our approach offers a more general theoretical perspective. Finally, there is a connection between Schein (2003) and our paper in that Schein proposes that reciprocals are adverbial, just like we have suggested for IAO reciprocals above. The difference between Schein’s suggestions and ours is that he proposes this for reciprocals in general. That makes his suggestion unsuitable for the ground we want to cover in this paper: we want to analyse what distinguishes IAO reciprocals from other reciprocals. For this purpose, we attribute a pseudo-reciprocal (adverbial) semantics to IAO reciprocals (and derive their truth conditions and their scopal constrainedness). This cannot be extended to other reciprocals. There is a further connection in that Schein observes for IAO reciprocals the role of sequencing, which he attributes to temporal flow. We concur for the case of ‘after’, but use non-temporal sequencing for other cases like ‘above’, ‘within’, etc.
Sigrid Beck and Arnim Von Stechow 253 SIGRID BECK Englisches Seminar Universita¨t Tu¨bingen Wilhelmstrasse 50 72074 Tu¨bingen Germany
[email protected] REFERENCES Krifka, Manfred (1989), Nominalreferenz und Zeitkonstitution: Studien zur Theoretischen Linguistik. Wilhelm Fink. Mu¨nchen. Krifka, Manfred (1998), ‘The origins of telicity’. In Susan Rothstein (ed.), Events and Grammar. Kluwer. Dordrecht. 197–235. Langendoen, D. Terence (1978), ‘The logic of reciprocity’. Linguistic Inquiry 9:177–97. Lasersohn, Peter (1995), Plurality, Conjunction and Events. Kluwer Academic Publishers. Boston. Lewis, David. (1991), Parts of Classes. Basil Blackwell. Oxford. Link, Godehart (1983), ‘The logical analysis of plurals and mass terms: a lattice-theoretical approach’. In R. Ba¨uerle, C. Schwarze, and A. v. Stechow (eds.), Meaning, Use, and Interpretation of Language. de Gruyter. Berlin. 302–23. Matthewson, Lisa (2000), ‘On distributivity and pluractionality’. In B. Jackson and T. Matthews, Proceedings of SALT X. CLC Publications. Ithaca, NY. Moltmann, Friederike (1997), Parts and Wholes in Semantics. Oxford University Press. Oxford. Oh, Sei-Rang (2005), Plurality Markers across Languages. Ph.D. thesis, University of Connecticut, Storrs. Partee, Barbara Hall (1987), ‘Noun phrase interpretation and type-shifting
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Beck, Sigrid 1999. ‘Plural predication and partitional discourses’. In Paul Decker (ed.), Proceedings of the 12th Amsterdam Colloquium. 67–72. Beck, Sigrid (2001), ‘Reciprocals are definites’. Natural Language Semantics 9:69–138. Beck, Sigrid, & Uli Sauerland (2000), ‘Cumulation is needed: a reply to Winter (2000)’. Natural Language Semantics 8:349–71. Brisson, Christine (1998), Distributivity, Maximality and Floating Quantfiers. Ph.D. thesis, Rutgers University, New Brunswick. Dalrymple, Mary, Makoto Kanazawa, Yookyung Kim, Sam Mchombo, & Stanley Peters (1998), ‘Reciprocal expressions and the concept of reciprocity’. Linguistics and Philosophy 21:159–210. Diesing, Molly (1990), The Syntactic Roots of Semantic Partition. Ph.D. thesis, University of Massachusetts, Amherst. Heim, Irene & Anglika Kratzer (1998), Semantics in Generative Grammar. Blackwell. Oxford. Heim, Irene, Lasnik, Howard, & May, Robert (1991), ‘Reciprocity and plurality’. Linguistic Inquiry 22:63–101. Kadmon, Nirit (2001), Formal Pragmatics. Blackwell. Oxford. Kratzer, Angelika (2003), The Event Argument. Chapter 4. Unpublished MS, University of Massachusetts, Amherst.
254 Pluractional Adverbials principles. In J. Groenendijk et al. (eds.), Studies in Discourse Representation Theory and the Theory of Generalized Quantifiers. Foris. Dordrecht. Partee, Barbara Hall (1989), ‘Binding implicit variables in quantified contexts. In CLS 25. Chicago Linguistic Society. Chicago. 342–65.
Schein, Barry (2003), ‘Adverbial, descriptive reciprocals’. In John Hawthorne (ed.), Language and Philosophical Linguistics, Philosophical Perspectives 17.1:333–67. Schwarzschild, Roger (1996), Pluralities: Studies in Linguistics and Philosophy. Kluwer. Dordrecht. Sternefeld, Wolfgang (1998), ‘Reciprocity and cumulative predication’. Natural Language Semantics 6:303–37. Stockall, Linnea (2001), ‘Pluractionality and prepositions in Germanic. The
First version received: 28.03.2006 Second version received: 30.10.2006 Accepted: 12.12.2006
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Sauerland, Uli (1998), ‘Plurals, derived predicates and reciprocals’. In Uli Sauerland and Orin Percus (eds.), The Interpretive Tract. MIT Working Papers in Linguistics, Vol. 25. 177–204.
syntax and semantics of [NP-P-NP]s’. In ConSole X. Leyden. von Fintel, Kai (1994), Restrictions on Quantifier Domains, Ph.D. thesis, University of Massachusetts, Amherst. von Stechow, Arnim (1996), ‘The different readings of Wieder ‘again’: a structural account. Journal of Semantics 13:87–138. von Stechow, Arnim (2002), ‘Temporal prepositional phrases with quantifiers. Some additions to Pratt & Francez (2001)’. Linguistics and Philosophy 25:1–40. Westerstahl, Dag (1984), ‘Some results on quantifiers’. Notre Dame Journal of Formal Logic 25:152–70. Winter, Yoad (2000), ‘Distributivity and dependency’. Natural Language Semantics 8:27–69. Zimmermann, Malte (2002), Boys Buying Two Sausages Each. On the Syntax and Semantics of Distance-Distributivity, Ph.D. thesis, Universiteit van Amsterdam, The Netherlands.
Journal of Semantics 24: 255–288 doi:10.1093/jos/ffm004
The Ingredients of Reciprocity in Cuzco Quechua MARTINA FALLER The University of Manchester
Abstract
1 INTRODUCTION This paper develops a compositional analysis of reciprocity in Cuzco Quechua (CQ) with the aim of elucidating the question of how languages may differ in the compositional derivation of this complex semantic notion. In a wider sense, it is a contribution to the growing literature on the possible space of cross-linguistic semantic variation. Consider the following Quechua reciprocal sentence and its English translation.1 (1) Hayt’a-na-ku-n-ku. kick-PA-REFL-3-PL ‘They kick each other’. 1
Where not indicated otherwise, examples are elicited, that is either translated from Spanish or constructed by the author and accepted/confirmed by native speaker consultants. The labels for morphemes used in the glosses are largely based on Cusihuaman’s (2001) study. The main exception is the label PA (‘pluractional’) for the suffix -na, which is based on my own analysis. Abbreviations used in glosses: 1O: first-person object, 1S2O: first-person subject second-person object, 1EXCL: firstperson exclusive, 1INCL: first-person inclusive, 3: third person, ABL: ablative, ACC: accusative, ADD: additive, BEN: benefactive, CAUS: causative, CIS: cislocative, COLL: collective, COM: comitative, CONT: continuative, DIM: diminutive, DIR: direct evidence, EMO: emotive, ILLA: illative, DIST: distributive, IMP: imperative, INC: inchoative, INT: intensifier, LIM: limitative, LOC: locative, NEG: negation, NMLZ: nominalizer, NMLZ.SS: same subject nominalizer, NX.PST: non-experienced past, PA: pluractional, PL: plural, PROG: progressive, PRTC: participle, PST: past, REFL: reflexive and TOP: topic. The Author 2007. Published by Oxford University Press. All rights reserved. For Permissions, please email:
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In Cuzco Quechua, reciprocity is marked by means of two verbal suffixes, one of which is a marker of reflexivity, the other of which is a marker of pluractionality. The paper develops an analysis that composes reciprocity from these more basic notions. Two further ingredients that are needed will be argued to derive from independent principles: universal quantification over parts of the reciprocal plural agent derives from plural predication, as has been argued by other researchers for English reciprocity, and distinctness of the participants in the reciprocal subevents derives from a semantic version of Condition B. This way of composing reciprocity is not universal, and other languages have dedicated reciprocal markers or make other reciprocal ingredients overt. The compositional derivation of reciprocity is therefore a clear candidate for cross-linguistic semantic variation.
256 The Ingredients of Reciprocity in Cuzco Quechua
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Both entail that there are at least two kicking events, that the agent and the theme of each kicking event are distinct but drawn from the same group consisting of at least two members (namely, the group denoted by they, which serves as the antecedent for the pronominal each other) and that each of the members of this group is an agent of at least one of these events and a theme of at least another one. In general, reciprocity is a complex concept that encompasses the more primitive notions of plurality, distinctness of co-arguments, reflexivity and universal quantification over parts of the reciprocal agent. While reciprocal sentences in the two languages share these semantic features, they are very different morphosyntactically: English employs the nominal bipartite anaphor each other, CQ employs two verbal suffixes: -na, which I will argue is a pluractional marker, and -ku, a reflexive. The main question for typological semantics that is raised by these observations is how the cross-linguistically uniform semantics is derived from the rather different morphosyntactic realizations, a question first raised in the formal literature by Dalrymple et al. (1994). The CQ data are interesting in this respect because the reciprocal construction in this language is not like other verbal reciprocals discussed in the literature (Mchombo 1993; Bruening 2006; Siloni forthcoming) in that it involves two morphemes rather than just one and thus raises the possibility that reciprocity is derived compositionally in this language. Dalrymple et al. (1994) studied the Chichew ˆ a monomorphemic verbal reciprocal and found that it shares the same cluster of semantic features as English reciprocal sentences; in addition to the features mentioned above, Chichew ˆ a and English reciprocals also have in common that they give rise to scope ambiguities when embedded. Dalrymple et al. (1994) concluded that the best way to deal with this semantic uniformity in the face of morphosyntactic diversity is to assume that any reciprocal construction is lexically specified as having reciprocal semantics. More precisely, they argued that reciprocal constructions denote RECIP, a polyadic quantifier, the semantics of which is such that it entails all the above features. The language-specific morphosyntactic realization of this quantifier is therefore largely semantically irrelevant. In particular, they argued that an account such as that of Heim et al. (1991) which composes reciprocity from a universal quantifier (each) and a so-called reciprocator (other) is not viable cross-linguistically as it cannot account for the fact that Chichew ˆ a reciprocal sentences have the same properties as their English counterparts despite not employing two morphemes that could
Martina Faller 257
2 In order to account for the morphosyntactic diversity of reciprocal constructions, Dalrymple et al. (1994: 157) hypothesize ‘that linguistic universals governing the interpretive relation between morphosyntax and meaning explain the naturalness of the quite different forms of expression English and Chichew ˆ a choose for reciprocal propositions’.
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be analysed as a universal quantifier and a reciprocator.2 While I agree with the conclusion that Heim et. al’s (1991) or any other compositional approach to the semantics of each other will not necessarily be applicable to reciprocals in other languages, not even other bipartite reciprocals, this does not necessitate an approach that associates reciprocal semantics directly with a morphosyntactically complex construction. In this paper, I will explore a different approach. It starts from the observation that the semantics of reciprocity is inherently complex. Even when one assigns a universal meaning such as RECIP to a reciprocal construction, one nevertheless has to require that this meaning has the components of plurality, universal quantification and distinctness of coarguments somehow bundled into it. It therefore should be equally possible to derive the universal features of reciprocity by composing two or more markers which have these more basic notions as their meaning. Thus, the observation that reciprocal constructions across languages share a number of features does not exclude the possibility that these features are derived compositionally in some languages. I will argue that CQ is such a language. Given languages like Chichew ˆ a with dedicated reciprocals, it follows that there is cross-linguistic variation in the way the complex semantic notion of reciprocity is composed (or not) from more basic building blocks. I will not enter the debate about what the best analysis of English each other is. But I will take my cue for developing an analysis of CQ reciprocal sentences from a third approach to English reciprocals, which neither tries to derive reciprocity compositionally from the meaning of each and other nor claims that each other directly denotes all features of reciprocity. Instead, this approach treats reciprocal sentences as special instances of relational plurals and argues that most semantic properties of reciprocal sentences follow from their plural properties (Langendoen 1978; Sternefeld 1998; Beck 2001). In particular, universal quantification over parts of the subject group is a direct consequence of plural predication and the definition of plural thematic roles. The overtly marked ingredients for reciprocity in CQ will be shown to be pluractionality (-na) and reflexivity (-ku). Most, if not all, previous accounts of reciprocity assume that the requirement that the arguments of the reciprocal subevents are distinct is encoded by the reciprocal construction itself. This paper departs from
258 The Ingredients of Reciprocity in Cuzco Quechua
1.1 A note on reciprocal truth conditions Before proceeding with the main purpose of this paper, I should clarify that it is not my intention to contribute to the ongoing debate on how many truth-conditionally distinct reciprocal readings should be recognized. This question has still not been settled for English. For example, Langendoen (1978) considers six truth-conditionally distinct readings for elementary reciprocal sentences (ERS, those in which each other occupies the second argument position of a two-place relation R), but rejects five of these as generally adequate in favour of just one, Weak Reciprocity. An ERS with relation xRy and A as the denotation of the plural subject is true according to weak reciprocity if (2) holds: (2) Weak Reciprocity ("x 2 A)(dy, z 2 A)(x ¼ 6 y ^ x ¼ 6 z ^ xRy ^ zRx) (Langendoen 1978: 179) Under this reading, (1) is true if each member of the group denoted by ‘they’ kicks at least one other member of that group and is kicked by at least one other member. This is weaker than strong reciprocity which would require every pair of individuals in the subject group to be in the kicking relation, but this is in accordance with the intuitions of most speakers, who would not consider (1) false in a situation in which not every member kicks every other member, especially if the group is fairly large. Subsequently, other researchers have argued for distinguishing three (e.g. Bruening 2006), four (e.g. Beck 2001) or six readings (e.g. Dalrymple et al. 1998), and I refer the reader to these studies for the data and arguments on which these diverse counts of readings are based. All agree, however, that weak reciprocity is at least a possible
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this assumption and instead explores the possibility that the distinctness requirement corresponds to a general condition on semantic predicates requiring co-arguments to be distinct (Reinhart & Reuland 1993). The paper is organized as follows. Section 2 presents the CQ data to be accounted for and motivates an analysis of -na as a pluractional marker. Section 3 introduces the notion of cumulativity and how it has been used to account for reciprocity in English. Section 4 develops a formal account of simple reciprocal sentences in CQ based on the notions of pluractionality, reflexivity and cumulativity and suggests that the distinctness condition is a direct consequence of a semantic formulation of Condition B. Section 5 concludes and raises some issues to be investigated in the future.
Martina Faller 259
interpretation, and Bruening (2006) moreover argues that it is sufficient for most reciprocal sentences with eventive predicates.3 I will therefore only derive the weak reading for CQ reciprocal sentences. Future work will have to show whether it is empirically justified to recognize other readings in this language and how they can be derived compositionally. 2 THE CQ DATA
(3) a. Kunan-qa chay-lla-ta-raq tapu-na-yu-ku-nchik. now-TOP this-LIM-ACC-CONT ask-PA-INT-REFL-1INCL ‘For now we only ask each other this’. (‘Eso noma´s por ahora nos preguntaremos’.) (Vengoa Zu´n˜iga 1998: 15) b. Pay-kuna pura qu-na-ku-sha-n-ku. (s)he-PL amongst give-PA-REFL-PROG-3-PL ‘They are giving each other (things)’. (Description of video clip) c. Qusqu kay-man-qa ham-pu-ra-ni, chicu-cha-y-pa Cuzco this-ILLA-TOP come-DEF-PST-1 boy-DIM-1-GEN papa-n-wan t’aqa-na-ku-spa. father-3-COM separate-PA-REFL-NMLZ.SS ‘I came here to Cuzco after the father of my boy and I separated from each other’. (Conversation) The agent of the reciprocal event may be encoded by only the person suffixes on the verb (3a), by an overt plural subject NP (3b) or by 3 Following Fiengo & Lasnik (1972), Bruening (2006) claims that strong reciprocity is only required for stative predicates.
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CQ is an agglutinative language which employs a large number of derivational and inflectional verbal suffixes encoding a variety of semantic notions (see Cusihuaman 2001 for an overview). Some of the derivational suffixes are valence changing such as the causative suffix -chi, the benefactive -pu and the reflexive -ku. The basic word order in CQ is (S)(O)V, though this is quite variable. Person suffixes on the verb are sufficient for marking the grammatical relations of subject and object, that is independent pronouns can be, and often are omitted. In this section, I will provide the data to be accounted for and describe informally the meanings of the suffixes -na and -ku, which together derive reciprocity. The textual examples in (3) further illustrate the reciprocal construction.
260 The Ingredients of Reciprocity in Cuzco Quechua a (potentially dropped) singular or plural subject NP plus a comitative NP (3c).4 The reciprocal relation may hold between the agent and theme, (3a), or, with a ditransitive verb, between the agent and recipient, (3b).5 In the presence of derivational suffixes that introduce additional arguments such as the causative -chi the reciprocal relationship may also hold between the agent and this additional argument, as shown in (4).6
4 In CQ, using the comitative suffix -(pu)wan is a regular strategy of forming conjoined NPs, as shown in (i). (i) Tayta-y-wan mama-y-puwan-mi llaqta-ta-qa ri-n-ku. father-1-COM mother-1-COM-DIR town-ACC-TOP go-3-PL ‘My father and mother have gone to town’. (‘Al pueblo fueron mi papa´ y mi mama´’.) (Cusihuaman 2001: 140) In (i), the conjoined subject NP is plural as indicated by the plural agreement on the verb. In (3c), however, the use of the comitative NP does not result in a syntactically plural subject NP. The use of the same subject nominalizer -spa indicates that the subject of the subordinate clause is first-person singular, that is the comitative NP does not seem to form a single NP with the subject here. I thank an anonymous reviewer for pointing this out. The same reviewer suggests that there is a semantic implication of this construction that the singular subject assumes responsibility for the action, so that a more adequate translation might be ‘. . . after I separated from the father of my boy’. I will leave it to future research to determine the precise semantics of comitative NPs in reciprocal constructions. See Dimitriadis (2004) for a discussion of this strategy from a cross-linguistic perspective. 5 In CQ, it appears to be impossible to establish a reciprocal relation between the agent and theme of ditransitive verbs or between the theme and recipient. The former is the case for verbal reciprocals in other languages as well, and the latter appears to be impossible in any language (Bruening 2006). 6 Arguably, (i) and (ii) are examples in which the reciprocal relation holds between the agent and a beneficiary argument introduced by the benefactive suffix -pu. (i) Chay-pis waqa-pu-na-ku-n-ku. this-ADD cry-BEN-PA-REFL-3-PL ‘They cried with each other’. (‘Lloraron en los brazos el uno del otro’.) (Itier 1999: 196)
(ii)
Asi-pu-na-ku-n-ku pay-kuna pura laugh-BEN-PA-REFL-3-PL (s)he-PL amongst ‘They are laughing together/with each other’. (elicited description of a video clip in which two people are laughing) However, it is not clear whether -pu in these examples really is the benefactive suffix -pu. According to Cusihuaman (2001: 193), -puna in such examples is a single suffix which is used instead of -na with verbs that express that two things are put in contact with each other or that two people are looking at each other. An argument for considering -pu the benefactive in this construction is the fact that with intransitive verbs its presence appears to be necessary for introducing a second argument. While some speakers accept just -naku by itself with intransitive verbs (Faller 2005, 2007), fieldwork in 2006 has revealed that most do not, and those that do, strongly prefer the forms with -pu. More research is necessary to understand this construction properly.
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(4) Chay-pi tawa runa-kuna malli-chi-na-ku-sha-n-ku person-PL taste-CAUS-PA-REFL-PROG-3-PL this-LOC four imaymana-ta. whatever-ACC ‘There, four people make each other taste different things’. (Description of video clip)
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2.1 The meaning of -na The suffix -na has a variety of uses.8 This section discusses its use as a pluractional marker. As such, it is most productive within the reciprocal construction as illustrated above. When -na precedes the causative suffix -chi (5a), or when used in a passive participle (5b) (cf. van de Kerke 1996) it alone can convey reciprocity.9 (5) a. Hayt’a-na-chi-rqa-ni. kick-PA-CAUS-PST-1 ‘I made them kick each other’. b. Pin˜a-chi-na-sqa puri-ri-n-ku. anger-CAUS-PA-PRTC walk-INC-3-PL ‘They walk being in a state of anger with each other’. 7 See van de Kerke (1996) for a first analysis of the effect of the relevant suffixes in different Quechua varieties on argument structure, as well as Bruening (2006) for a cross-linguistic study of which arguments can be reciprocalised. 8 One is nominalizing -na which is used for different kinds of nominalizations, including not only the formation of nominalized clauses as, for example the second -na in (7a), but also the derivation of nouns from verbs such as pun˜u- ‘to sleep’ / pun˜una ‘bed’. Conversely, -na can verbalize certain types of nouns. With nouns denoting locations, it derives verbs meaning ‘to put in that location’, for example k’uchu ‘corner’ / k’uchuna- ‘to put in a corner’ (Cusihuaman 2001: 184). With nouns that denote sets of often small objects attached to a bigger one, for example feathers or seeds, the resulting verbs mean to separate the numerous smaller objects from the big one (Cusihuaman 2001: 185). For example, phuru ‘feather’ / phuruna- ‘to pluck’, ruru ‘seed’ / ruruna- ‘to deseed’. In Faller (2007) I had hypothesized that this use might be related to the suffix -na that occurs in the reciprocal construction, possibly providing a way of accounting for the distinctness of co-arguments, as these verbs tend to denote separating events, but I have in the mean time come to the conclusion that this cannot be the correct analysis. 9 In CQ the sequence -nakuchi is ungrammatical. However, in the Ancash and Huanca and possibly other varieties, this is the regular way of expressing the causation of a reciprocal event (Muysken 1981: 458; Cerro´n Palomino 1976).
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Further research is required to determine exactly which arguments can participate in reciprocity marked by -na-ku,7 and I will in the following only develop a formal analysis of reciprocity holding between the agent and theme of simple transitive verbs. While some grammar writers treat -naku as a single suffix, Muysken (1981) showed that it consists of the two suffixes -na and -ku. One argument for the bipartite analysis is the fact that -na and -ku can be split by intervening suffixes. For example, in (3a), they are separated by -yu. Furthermore, both -ku and -na can occur without the other: -ku on its own marks reflexivity and, as argued below, -na marks pluractionality. In the following two sections, the meaning of these two suffixes will be discussed informally.
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10 In the descriptive literature, pluractional markers are sometimes called distributive. This is somewhat misleading in comparison with the use of the term ‘distributive’ in the formal semantics literature, where distributivity is usually taken to entail universal quantification. That distributivity should not be identified with pluractionality has been argued, for example by Matthewson (2000), who has shown that an apparently distributive element in St’a´t’imcets is a pluractional marker precisely because it does not entail universal quantification. 11 Glosses are the current author’s. van de Kerke glosses -na as REC ‘reciprocal’.
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Besides showing that -na can occur on its own, the examples in (5) also raise the possibility that -na itself is a reciprocal marker. In fact, Muysken (1981) calls -na ‘the reciprocal’. However, he assumes that -ku is necessary to establish an anaphoric relationship with the subject, that is to ensure that the participants of the subevents are drawn from the plural individual denoted by the subject. Muysken’s ‘reciprocal’ does therefore not encode all essential components of reciprocity. If -na were a true reciprocal, we would expect it to be able to mark reciprocity on its own in any circumstance. But this is not the case, with the exception of the two environments just mentioned, reciprocity always requires both -na and -ku. -na can therefore not be analysed as a reciprocal marker. Most previous studies have focussed on the role of -na as part of the reciprocal construction. However, -na can also be used to mark plurality of events without implications of reciprocity. I will therefore analyse it as a pluractional marker, a term coined by Newman (1990) in order to distinguish plural agreement markers on verbs (well known from Indo-European languages) from verbal markers that express ‘a broad range of notions typically including action by more than one individual, temporally iterated action, and spatially scattered action (among others)’ (Lasersohn 1995: 238). The pluractional analysis of -na is weaker than the analysis presented in the predecessors of this article, Faller (2005, 2007), which had followed van de Kerke’s (1996) analysis of Bolivian Quechua -na as marking distributivity over participants, times or locations. Distributivity involves universal quantification, but pluractionality not necessarily.10 Since pluractionality subsumes distributivity as a special instance, the pluractional analysis of -na can account for the same set of data as the distributive analysis, and moreover predicts that -na should be possible in situations which do not involve universal quantification. van de Kerke presents the following two examples in support of analysing -na as marking distributivity and against analysing it as a reciprocal, both observed in naturally spoken Bolivian Quechua.11 Note that these also contain the reflexive -ku, demonstrating that this combination does not necessarily express reciprocity in this variety.
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(6) a. Pay-kuna runa-ta maqa-na-ku-n-ku. man-ACC hit-PA-REFL-3-PL (s)he-PL ‘They hit the man in turn’. (van de Kerke 1996: 149) b. Pay turril-is-man taqha-na-ku-sqa. (s)he vessel-PL-ILLA bump-PA-REFL-NX.PST ‘He bumped into the vessels’. (van de Kerke 1996: 150)
(7) a. . . .mana-n saru-na-wa-na-nchis-chu ka-sqa-nchis-wan. . . .not-DIR step.on-PA-1O-NMLZ-1INCL-NEG be-NMLZ-1INCL-COM ‘. . .(that) they must not discriminate against us (lit.: ‘trample on us’) for what we are’. (‘que no nos discriminen por lo que somos’.) (FARTAC 2002: 46) b. Maskha-na-ri-ku-spa puri-sha-n alqu. search-PA-INC-REFL-NMLZ.SS walk-PROG-3 dog ‘The dog walks, searching (for food) all over the place’. (spontaneous utterance in casual conversation)
12 In Faller (2005, 2007), I also presented the example in (i) as an illustration of the distributive use of -na: (i) Asnu-man chaqna-na-y kay kustal-kuna-ta. donkey-ILLA load-PA-IMP this sack-PL-ACC ‘Load all the sacks onto the donkeys’. This was judged acceptable by one of my consultants under the interpretation that each sack should be loaded onto the donkey. However, this same consultant does not accept this example anymore under this reading, and there is therefore some doubt that -na can be used to indicate distribution over objects. Another consultant accepts the example under a repeated action interpretation; her comment ‘amarrarlos a cada rato’, ‘to tie them up again and again’, reinforces the analysis of -na as a pluractional marker with temporal interpretations.
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The verbs of both these examples normally receive reciprocal interpretations, but in the contexts in which they were uttered, they are better translated in the indicated way. The beating in (6a) involved several men beating on another one with whips without that man reciprocating, and given that the vessels described in (6b) were stationary, a reciprocal interpretation could not have been intended. Instead, (6b) means that he bumped into each of the vessels or, perhaps weaker, that there were several events of bumping into vessels. van de Kerke notes that such purely distributive uses of -na are not very common, and I have not been able to get CQ versions of these examples accepted by my consultants. However, I, too, have been lucky to find some naturally occurring examples in which -na marks plurality of events. These are given in (7).12
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13 However, as pointed out by an anonymous reviewer, one could also analyse (7a) as involving distribution over the members of the plural object such that each of us is being stepped on (potentially only once). I have not been able to obtain clear judgments on whether this reading is to be preferred over the repeated stepping reading, though it seems to me that this example or any other use of saruna does not entail that each member of the object group is being stepped on. I have also not been able to obtain examples with this or other verbs containing -na that unequivocally involve distributivity over the object group (see also footnote 28). To really show that we are dealing with pluractionality, one would need an example with a singular object. Unfortunately, I have also not been able to obtain examples of this kind. However, the fact that consultants volunteered as a translation of saruna- into Spanish pisotear ‘to trample’ suggests to me a pluractional interpretation. Moreover, (7b) cannot be analysed as distribution over the object. 14 As an anonymous reviewer has pointed out, (7b) could out of context also mean that the dog is engaged in a mutual searching activity with other dogs (‘searching for each other’), and this would in fact be the more common interpretation. However, this example was uttered in a context in which there was only one dog and he was searching for food in every corner of the kitchen. In this context, no reciprocal interpretation could have been intended. 15 Given the relative unproductivity of -na as a purely pluractional marker, taking this meaning as basic and deriving reciprocity from it is a bit like ‘turning the world upside down’ in the words of an anonymous reviewer. This is true, but leads in my opinion to a simpler analysis than taking the reciprocal meaning as basic and trying to derive the non-reciprocal uses from it.
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Saru- means to step on something, and, according to the consultants who accept (7a), saruna- is best translated as pisotear ‘to trample’, that is an action that involves repeated stepping on something. (The Spanish translation as discriminar ‘to discriminate’, provided in the publication from which this example is taken, is a rather free translation.) Example (7a) is a relatively clear case in which -na marks plurality of events, but not necessarily distributivity. That is, there is no sense in which -na here introduces universal quantification over times.13 In contrast, (7b) is interpretable as distribution over locations, though it is also compatible with the weaker notion of plurality of events.14 Note that the contribution of -ku in this example is to indicate that the subject is a beneficiary of the described event. This use of -ku will be discussed in the next section. While such purely pluractional examples of -na occur, they are quite rare. There is also considerable variation between speakers and not everyone consulted accepts the examples in (7) as grammatical. Nevertheless, the existence of these examples indicates that marking plurality of events is at least one possible meaning of -na, and I will argue below that assuming this meaning for -na is sufficient for deriving reciprocity.15 The rarity of the examples calls for an explanation, of course, but it is not within the scope of this paper to fully investigate this. My hypothesis is that -na is restricted in its use because of the existence of other pluractional markers in the language. For example, the suffix -paya adds the meaning of repeated or repetitive action (wisq’a- ‘to close’, wisq’a-paya- ‘to close again and again’) (Cusihuaman 2001: 189), and the suffix -(y)kacha adds the meaning that the described action is scattered in
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space (apa- ‘to take’, apa-ykacha- ‘to take everywhere’) or repeated (kumpa-ku- ‘to fall’, kumpa-kacha-ku- ‘to fall again and again’) (Cusihuaman 2001: 188).16 In fact, the example in (7a) becomes acceptable to everyone if -na is replaced with -paya. There are other verbal suffixes (Cusihuaman 2001) which appear amenable to a pluractional analysis, and the study of the precise semantics of these markers will be an interesting topic for further investigation.
2.2 The meaning of -ku
(8) Asnu-n hayt’a-ku-n. donkey-DIR kick-REFL-3 ‘The donkey kicks itself ’. A related use of -ku is as a benefactive (cf. van de Kerke 1996: 144), requiring the beneficiary to be co-referential with the subject, as in (9). (9) Asnu-ta-n ranti-ku-rqa-ni. donkey-ACC-DIR buy-REFL-PST-1 ‘I bought myself a donkey’. Arguably, -ku is still reflexive in this use, indicating that the reflexive relation holds between a benefactive argument and the subject, and it can therefore be considered an extension of its basic reflexive use. In addition, as is typologically common, -ku also has uses as a middle and anticausative marker (van de Kerke 1996: 161). With the pluractional meaning for -na and the reflexive meaning of -ku having been given empirical support, we are now in a position to see how the reciprocal interpretation can arise when they are combined. Reciprocity involves plurality of events and this is marked 16 I would like to thank an anonymous reviewer for drawing my attention to the suffix -(y)kacha and its spatial meaning. 17 Note that this has to be understood as semantic reflexivity in the sense that the two arguments of a transitive verb are co-referential, but the resulting verb still can combine with a syntactic complement. This position can be occupied, for example by a body part noun, as shown in (i). (i) Juan uya-n-ta maqlli-ku-n. Juan face-3-ACC wash-REFL-3 ‘Juan washes his face’. The possessor of the body part noun is restricted to be the subject. van de Kerke (1996: 160) suggests that this is explained if one assumes that in Quechua ‘body parts are identified by means of the referential index of the subject possessor on the whole NP’. Thus, the body part noun is in fact coreferential with the sentence subject.
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The verbal suffix -ku has several functions, the one most relevant for current purposes is the marking of reflexivity illustrated in (8).17
266 The Ingredients of Reciprocity in Cuzco Quechua
3 CUMULATIVITY AND RECIPROCITY Langendoen (1978) observed that reciprocal sentences are interpreted in a similar fashion to relational plurals. A sentence like (10) is true if (i) each woman released some prisoner and (ii) each prisoner was released by some woman. This is captured by the semantic representation in (10), where A ¼ the set denoted by the women, B ¼ the set denoted by the prisoners and R ¼ the relation denotated by release. (10) The women released the prisoners. ð"x 2 AÞðdy 2 BÞðxRyÞ ^ ð"w 2 BÞðdz 2 AÞðzRwÞ (Langendoen 1978: 185) Compare this with the weak interpretation of the reciprocal sentence in (11) (an unpacked version of (2)). (11) The women released each other. ("x 2 A)(dy 2 A)(xRy ^ x 6¼ y) ^ ("w 2 A)(dz 2A)(zRw ^ w 6¼ z)
The two semantic representations differ only minimally. We can see that replacing the second plural NP with the reciprocal pronominal 18
Note that the analysis of pluractionality does not require events as ontological primitives, as is shown, for example by van Geenhoven’s (2004) successful analysis of pluractionality in West Greenlandic.
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by -na. The function of -ku is to identify the agent and the theme of the sum event, ensuring that the participants of the subevents are drawn from the same group. This alone does not constitute reciprocity yet, but if we can find the other ingredients, universal quantification and distinctness of the participants in the subevents, somewhere else, then the combination of -na and -ku provides a very good skeleton. In the following sections I argue that universal quantification comes for free with plural predication and the notion of plural agents and themes and that the distinctness condition is in fact a general condition on coarguments of a predicate. The account to be developed in the following sections is cast in a neo-Davidsonian event semantics. Having events in the representation language and as ontological primitives makes certain aspects in the analysis of reciprocity straightforward. For example, it is easy to distinguish the reciprocal sum event and the subevents of which it is constituted. Event semantics also is usually the framework of choice for work on pluractionality, as again, it captures straightforwardly the idea that a pluractional marker pluralizes events, not objects.18
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(12) For any set P, P is the smallest set such that a. P 4 P; and b. if a 2 P and b 2 P; then a4b 2 P For example, if dog# ¼ {Fido, Rover, Buddy}, then dog# ¼ {Fido, Rover, Buddy, Fido4Rover, Fido4Buddy, Rover4Buddy, Fido4Rover4Buddy}. Note that P contains atomic individuals, that is a starred predicate is true of single individuals as well. To derive a plural proper denotation, the atoms have to be subtracted from P: Rullmann & You (2003) argue that Mandarin Chinese unmodified nouns have ‘general number’, a term coined by Corbett (2000), that is they are P predicates. In this language, an optional plural marker PL, the semantics of which is given in (13), can be used to derive a properly plural interpretation. As will be shown below, CQ is like Mandarin in this respect. (13) PLðNÞ ¼ NAt 19
(Rullmann & You 2003)
I assume an ontology along the lines of Link (1998), where the domain of individuals De contains atomic/singular individuals as well as plural individuals. De is closed under the sum operation: if a and b are individuals in De, then a4b is an individual in De. Sums and their constituent parts form a join semi-lattice, structured by the ordering relation n
(Lasersohn 1995: 256)
First, as pointed out to me by Philippe Schlenker (personal communication), the biconditional in (i) will never be true, because the non-overlap condition :½f ðeÞ+f ðe#Þ is false whenever e ¼ e#. The condition e# 6¼ e$ has therefore been added in (25). Second, for simplicity’s sake, (25) only requires e to be non-atomic, without requiring there to be a (pragmatically determined) number n of subevents. Third, Lasersohn’s definition contains the free variable P ranging over properties of the atomic subevents. P may be different from V to allow for subevents that do not fit the verb’s description. This is needed in particular for repetitive event types such as nibbling, where the constituent atomic events are not nibbling events but biting events. However, by making this allowance, we open the door wide for all kinds of subevents. For example, at some level of a kicking event, we find an event of a leg moving without that movement counting as a kicking. This event would, however, not involve a theme, and we would therefore have to allow subevents to differ from not only their superevent in P but also in their argument structure. Since these kinds of complications are not directly relevant to the purposes of this paper, I will simplify matters and restrict the type of subevents to which the non-overlap condition applies to those events which have the property denoted by the verb. To do this, I introduce the notation e#