Craige Roberts 11/28/05; rev. 11/30 English Partitive NPs Partitive NPs are those of the form Det of DP. E.g.: all of the boxes some of the twenty toys you got for Christmas a lot of them very few of Marie’s friends only twenty of the poems two of those the first five of the people who come to the opening It has long been observed that there are constraints on what DPs can serve as partitive complements. E.g., Jackendoff (1972) noticed that partitives like the following are unacceptable, and proposed the partitive constraint: *most of four girls *all of friends *only a little of many quarts *none of some men *very few of sixty diplomats etc. Partitive Constraint (preliminary): the complement NP in a partitive must be definite. Barwise & Cooper (1981) argued that what characterized a definite NP was that it denoted a sieve: Definite: A determiner D is definite if for every model M = <E,|| ||> and every A for which ||D||(A) is defined, there is a non-empty set B, such that ||D||(A) is the sieve {X∈E | B ⊆ X} (i.e. ||D||(A) is the principal filter generated by B). It would not suffice that they require definites to be principal filters, because that would not rule out universally quantified complements; hence, the requirement that there be a non-empty set B that generates the principal filter. Accordingly, they redefined the partitive constraint as a requirement that a partitive complement be interpreted as a sieve. But Ladusaw (1982) noted that there is one remaining counterexample to Barwise & Cooper's version of the constraint: both CN presupposes that the cardinality of the denotation of CN is two, so it is only defined when this is the case. But where defined, it always denotes the principal filter generated by the denotation of CN. And yet such NPs are not generally acceptable as partitive complements: *none of both teams *each of both keys Ladusaw (1982) offers a suggestion about how to rule out both the universal quantifiers and both. (In what follows, I am abstracting away from Ladusaw's particular formulation.) He observes that if we adopt the idea that a group may be semantically an individual, then, roughly, group NPs denote individuals, or, more precisely, they denote principal filters generated by an individual. Though he didn’t have the benefit of Link’s (1983) analysis of groups as individuals, we can recast his observations in those terms. Then we can state the partitive constraint as: Partitive Constraint: the complement in a partitive must denote a principal filter generated by an individual. We can make the Partitive Constraint follow from the meaning of the partitive construction, by giving it something like the following meaning:1 |Det of DP| = Det′({y| Atomic(y) & y ≤i ιx.x = ∩|DP|}) This says that the head Det of the partitive takes as its domain the set of individuals who are atomic individual parts of the (existent, unique/maximal) individual which generates the principal filter denoted by the complement. In case the generator of the filter is a group, then the domain is the set of atomic individuals in the non-atomic individual denoted by the complement. Now suppose that obligatorily quantificational DPs like every CN and both CN must undergo QI/QR, so that the partitive complement is a variable bound by the raised DP. This “denotes” an individual, but it is not the individual that generates the principal filter corresponding to the complement generalized quantifier in Barwise & Cooper’s sense. Whatever the value of this variable under some assignment function, it is an atomic count individual; so in that case the domain of the head Det will be just the singleton set consisting of the value of that individual. But the partitive construction with count head Det seems to presuppose that the domain is non-singleton—in keeping with its name, it is about parts of some greater whole. So, singular complements are anomalous (on the non- mass sense; see below) when the head Det is a count Det: #two of the man #a lot of the box #very few of Marie’s friend #some of John #all of the box If we take the partitive to presuppose that the denotation of the complement has more than one atomic part, this will account for the anomaly of count partitives (where Det 1 This is quite closely related to Barwise & Cooper’s semantics for partitives, except that it requires an atomic individual as generator of the filter. It isn’t what Ladusaw has in mind, because it doesn’t rule out both. But, as we will see, it predicts the occurrence of mass partitives with quantificational complements, to be discussed below, which I don’t believe have been observed before. takes a count domain) both with these singular complements and with quantificational complements like every CN or both CN. But among the anomalous examples with singular complements, there are some that are acceptable if we give the Det a non-count interpretation. Ladusaw also notes the existence of examples with singular complement NPs, such as the following: some of the book most of the oatmeal very little of the rain Proper names and pronouns also occur as complement NPs: all of John all of me These are mass partitives. The complements in these examples are all definite, individual denoting NPs, and thus obey Ladusaw's partitive constraint—they each denote the principal filter generated by an individual. But here they are singular. They demonstrate an unexpected bonus of couching Ladusaw's insight in the context of a theory such as Link's. Since the class of individuals in Link's structured domain includes not only objects, but also individual portions of matter, we can make sense of mass partitives by giving the complement NP its marked, but readily available mass interpretation. Link (1983) points out the need for the existence of a mass denotation of CNs and NPs which are generally count, citing examples such as: There is apple in this salad. The apple in this salad is mealy. where the portions of apple denoted by the underlined NPs in these examples need not come from or constitute one piece of fruit. In the mass partitives above, the book may be viewed as its contents, the oatmeal as a mass, me as my physical mass or, more abstractly, the mass of my consciousness. Roberts (1987) gave an explanation of how the proportional quantifiers work in the mass partitives, an explanation that depends on the fact that in Link’s theory the elements of the atomic subdomain D of individual portions of matter are also ordered by a relation of material part/whole. The lattice which is formed by this relationship is non-atomic, but in other respects it is like the lattice on the entire domain E. As with plural quantification examples, we cannot claim that the quantification here is over all parts of the denotation of the oatmeal: In a non-atomic lattice, there may be infinitely many parts of a given individual; how could we determine whether most of those infinitely many parts have some property? Instead, we need to quantify over parts homogeneously characterized in some appropriate fashion. Suppose that in most of the oatmeal we consider the material correlate of the oatmeal, say the individual portion of matter am in the model. We then find a way of characterizing a level of homogeneous individual parts of am. For example, in most of the oatmeal got wet, we might consider a partition of the mass of oatmeal into equal portions of matter. These will also be atomic individual portions of matter on the count lattice which are related to the entire mass of the oatmeal by the material part relation on the non-atomic mass lattice. Then we compare the proportion of those portions of oatmeal which are wet to those which are dry. The mass partitives support Ladusaw's characterization of the constraint on the partitive construction in terms of groups as individuals, especially in a framework such as Link's which has a built-in relationship between individuals in the count and mass domains. The individual denoted by a partitive complement need not be non-atomic. In the cases examined above where the individual is atomic on the count domain, we simply shift to its counterpart in the mass domain, and the distributivity over parts of the individual proceeds in analogous fashion in the two domains. Given the possibility of interpreting a count individual as its mass counterpart, we can even derive an acceptable mass interpretation of partitives with non-individual-denoting, universally quantified complements. The interpretations are those we would derive if the complement was quantified in: A large portion of each book was boring. ‘for each of the familiar set of books, its mass counterpart (the abstract content) is such that a large portion of it was boring’ Most of every quart of milk we bought last summer went sour before we finished it. ‘for every quart of milk, its mass counterpart (the supremum of the portions of milk in the quart) is such that most of it went sour before we finished it’ This may even work for both NPs and for only-NPs. What do you think of these? Most of both books was ghost-written. Some of only three books was ghost-written. Though it doesn’t seem to work for downward entailing (right monotone decreasing) DPs: Most of few books was entertaining. Some of none of the books was ghost-written.