# Each vs. Jeweils: A Cover-Based View on Distance Distributivity

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5 sentence like The shoes cost \$50 can be interpreted as saying that each pair of shoes costs \$50, as opposed to each shoe or all the shoes together. To model this kind of example, Schwarzschild modifies D and makes it anaphoric to a salient cover (a partition of a plural individual that allows overlap). C, the cover variable, is free and anaphoric on the context. Schwarzschild assumes that C is a cover of the entire universe of discourse, but for most purposes one can instead think of C as a cover or a partition of the sum individual in question into salient parts, which may be plural sums. In this case, C partitions the sum of shoes into pairs. Schwarzschild refers to his own version of the D operator as Part. (10) [Part C ]= λp et λx y[y x C(y) P (y)] (Schwarzschild, 1996) This operator optionally applies to a VP and shifts it to a nonatomic distributive reading. For example, Lasersohn s sentence is modeled as follows: (11) The shoes [Part [cost \$50]]. Each salient plurality of shoes costs \$50. nonatomic distributive It is of course possible to think of D as a special case of Part, namely the one that results when the variable C is resolved to the predicate Atom. However, I assume that both D and Part are present in the grammar. This assumption will allow us to capture the distinction between each and jeweils. The former corresponds to D and the latter corresponds to Part. This accounts for the fact that jeweils and its relatives across languages have a wider range of readings than each and its relatives do. In count domains, distributivity over atoms is expected to be salient in almost all contexts and to obscure the presence of nonatomic distributive readings (Schwarzschild, 1996). It is therefore useful to look for nonatomic VP-level distributivity in a noncount domain, such as time. Here we find once again that the readings in question are available given appropriate contextual information or world knowledge. Example (12) is based on observations in Moltmann (1991). It is odd out of the blue because pills cannot be taken repeatedly, but it is acceptable in a context where the patient s daily intake is discussed. Example (13) is from Deo and Piñango (2011), and is acceptable because it is clear that snowmen are typically built in winter. (12) The patient took two pills for a month and then went back to one pill. (13) We built a huge snowman in our front yard for several years. Since for-adverbials are otherwise not able to cause indefinites to covary (Zucchi and White, 2001), and since Part is dependent on a salient level of granularity just like (12) and (13) are, it is plausible to assume that a temporal version of Part is responsible for the distributive interpretation of these sentences. See Champollion (2010) for more discussion of this point. The contribution of this temporal version of Part can be paraphrased as daily in (12) and yearly in (13). The original formulations of the operators in (8) and (10) can only target (that is, distribute over parts of) the subject. Examples like (12) and (13)

6 motivate a reformulation of the operators that allows them to target different thematic roles, including time. I will represent the relationship between D and the thematic role it targets through coindexation. For evidence that this relationship can be nonlocal, which justifies the use of coindexation, see Champollion (2010). Coindexation also allows us to capture the fact that DD items can also target different thematic roles (Zimmermann, 2002). For example, (14) can either involve two stories per boy or two stories per girl, depending on which thematic role is targeted by each. (14) The boys told the girls two stories each. In the following, I assume a Neo-Davidsonian algebraic semantic system loosely based on Krifka (1989) and Champollion (2010). Events, verbs and thematic roles are each assumed to be closed under sum formation. Verbs and their projections are all of type vt (event predicates). Here is a sample entry of a verb. (15) [see]] = λe [ see(e)] This entry includes the star operator from Link (1983) as a reminder that the predicate is closed under sum formation. The star operator maps a set P to the predicate that applies to any sum of things each of which is in P. It can be easily generalized to functions such as thematic roles (Champollion, 2010). Noun phrases are interpreted in situ (I do not consider quantifier raising in this paper). Silent theta role heads, which denote functions of type ve (event to individual), are located between noun phrases and verbal projections. I will often omit them in the LFs for clarity. The precise nature of the compositional process is not essential, but it affects the types of the lexical entries of DD items so let me make it concrete. I assume that the following type shifters apply first to the theta role head, then to the noun phrase, and finally to the verbal projection. (16) a. Type shifter for indefinites: λθ ve λp et λv vt λe[v (e) P (θ(e))] b. Type shifter for definites: λθ ve λxλv vt λe[v (e) θ(e) = x] Each of these type shifters combines a noun phrase with its theta role head to build an event predicate modifier of type vt, vt. For example, after the noun phrases the children (definite) and two monkeys (indefinite) combine with the theta role heads agent and theme respectively, their denotations are as follows. Here, child stands for the sum of all children, a plural individual of type e. (17) [[agent [the children]]]] = λv λe[v (e) ag(e) = child] (18) [[theme [two monkeys]]] = λv λe[v (e) th(e) = 2 monkey( th(e))] After the verb has combined with all its arguments, the event variable is existentially bound if the sentence is uttered out of the blue. If the sentence is understood as referring to a specific event, the event variable is instead resolved to that event. If the noun phrases combine directly with the verb, we get a scopeless reading as in (19). Here and below, I write 2M as a shorthand for λe[ th(e) = 2 monkey( th(e))].

8 the other one is a determiner). In both cases, the result is a phrase of VP modifier type vt, vt, which is also the type of D θ. Some intermediate steps of the derivations of (1) are shown in (25) and (26). (25) [[[[two monkeys] each ag ] theme]]] = λv vt λe[[d ag ](λe [V (e ) 2M(e )])(e)] = λv vt λe[e λe [ see(e ) 2M(e ) Atom(ag(e )]] (26) [[[Each child] agent]]] = λv vt λe[ ag(e) = child [D ag ](V )(e)] = λv vt λe[ ag(e) = child e λe [V (e ) Atom(ag(e ))]] The result of these derivations is always the same, which reflects their synonymy: (27) [The children each ag saw two monkeys] = [[The children saw two monkeys each ag ] = [[Each child saw two monkeys]] = (21) = [The children D ag saw two monkeys] We now come to the event-based reformulation of Part. We obtain it by replacing Atom in (20) with a free variable C, which is assumed to be anaphoric on the context. This minimal change reflects the close connection between D and Part. (28) [Part θ,c ] = λp vt λe[e λe [P (e ) C(θ(e ))]] Part takes an event predicate P and returns a predicate that holds of any event e which can be divided into events that are in P and whose θs satisfy the contextually salient predicate C. Note that the definition of (28) entails that C is a cover of θ(e). The operator (28) is also the lexical entry of adverbial jeweils. The same type shift as in (23) brings us from (28) to adnominal jeweils: (29) [jeweils θ,c ] adverbial = [Part θ,c ] = (28) (30) [jeweils θ,c ] adnominal = λp λθλv λe[[part θ,c ](λe [V (e ) P (θ(e ))])(e)] As in the case of the Part operator, the C parameter of jeweils can be set to Atom so long as θ is set to a function which points into a count domain, such as ag. In that case, jeweils distributes over individuals and is equivalent to each. The following example illustrates this with sentence (2a); sentence (2b) is equivalent. (31) Die Kinder haben jeweils ag,atom zwei Affen gesehen. The children have Dist two monkeys seen. The children have each seen two monkeys. If and only if there is a supporting context, the anaphoric predicate C can be set to a salient antecedent other than Atom, and in that case θ is free to adopt values like τ (runtime). This leads to occasion readings. Suppose for example that it is in the common ground that the children have been to the zoo to see animals last Monday, last Wednesday and last Friday, and that (2a) is uttered with reference to that state of affairs, or sum event. It is interpreted as follows.

9 (32) [Die Kinder haben jeweils τ,zoovisit zwei Affen gesehen.] = ag(e 0 ) = child e 0 λe [ see(e ) 2M(e ) zoovisit(τ(e ))] The children have seen two monkeys on each occasion. Since the sentence refers specifically to the sum e 0 of the three events in question, the event variable in (32) is resolved to e 0 rather than being existentially bound. The predicate that is true of any time interval at which a zoo visit takes place, call it zoovisit, is also salient in this context. So C can be resolved to zoovisit rather than to Atom. Since there are no atoms in time, it is only now that θ can be set to τ, rather than to ag as in (31). What (32) asserts is that e 0 has the children as its agents; that it can be divided into subevents, each of whose runtimes is the time of a zoo visit; and that each of these subevents is a seeingtwo-monkeys event. Runtime is closed under sum just like other thematic roles (τ = τ), or in other words, it is a sum homomorphism (Krifka, 1989). This means that any way of dividing e 0 must result in parts whose runtimes sum up to τ(e 0 ). Assuming that τ(e 0 ) is the (discontinuous) sum of the times of the three zoo visits in question, this entails that each of these zoo visits is the runtime of one of the seeing-two-monkeys events. This is the occasion reading. 5 Summary and Discussion This analysis has captured the semantic similarities between DD items across languages, as well as their variation, by relating them to distributivity operators. DD items can be given the same lexical entry up to type shifting and parameter settings. The parameters provided by the reformulation of the D and Part operators capture the semantic variation: DD items like English each are hard-coded for distribution over atoms, which blocks distributivity over a noncount domain like time. DD items like German jeweils can distribute over noncount domains, but only if they can pick up salient nonatomic covers from context. The remaining question is how to capture the correlation expressed in Zimmermann s Generalization (7). That is to say, why does a DD item which can also be used as a distributive determiner lack the occasion reading? Zimmermann himself proposes a syntactic explanation: Determiners must agree with their complement; DD each also has a complement, a proform that must acquire its agreement features from its antecedent, the target of each; only overt targets have agreement features. Alternatively, a semantic explanation seems plausible: Distributive determiners like English each are only compatible with count nominals (each boy, *each mud). Formally, this amounts to an atomicity requirement of the kind the D operator provides. This requirement can be seen as independent evidence of the atomic distributivity hard-coded in the entry (24) via the D operator (20). In other words, the DD item inherits the atomicity requirement of the determiner. Both explanations are compatible with the present framework. Acknowledgments. Thanks to Anna Szabolcsi, Robert Henderson and to the reviewers for helpful comments. I am also grateful to audiences at the 2011 Stuttgart workshop on quantification and at the 2011 Amsterdam Colloquium.

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