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Addendum to ‘The semantics of possessives’: Barker on quantified possessives

From: Language
Volume 89, Number 4, December 2013
pp. s1-s4 | 10.1353/lan.2013.0078

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Addendum to ‘The semantics of possessives’: Barker on quantified possessives Stanley Peters Stanford University Dag Westerst˚ ahl Stockholm University and University of Gothenburg As far as we know, the phenomenon of narrowing for possessives was discussed for the first time in Barker (1995), with examples such as 1. (1) a. Most planets’ rings are made of ice. b. Not every school’s linguistics program is as good as that one. Barker also proposed a general scheme for semantic interpretation of quantified (non-expanded) prenominal possessor DPs. His idea was to use a generalized quantifier that simultaneously binds two variables; one variable for possessors and one for possessions. Variable-binding was effected with the mechanism of unselective binding from Lewis (1975). The semantics enforces narrowing. How does it relate to our semantics with Poss and two separate quantifiers over possessors and possessions, respectively? In this note we make a brief comparison between his treatment and the one in Peters and Westerst˚ ahl (2013). Lewis took adverbial quantifiers to be resumptive, i.e. ordinary quantification over pairs (or triples, etc.) of individuals, but it has been noted that in other linguistic contexts (and perhaps sometimes for the adverbs too) where quantification over pairs seems natural, counting pairs rather than individuals may not give correct truth conditions. This is known as the proportion problem . A well-known case, which Barker takes as his point of departure, is donkey sentences. At one time resumption was thought to provide a smooth account of the semantics of these sentences, but it soon became clear that this only gives the right result for donkey sentences with every, some, and no, and fails for practically all other quantifiers.1 For example, sentence 2 gets completely wrong truth conditions if one tries to use the resumption of at least two. (2) At least two farmers who own a donkey beat it. So perhaps the relevant quantification over pairs is sometimes not resumptive, but of a different kind. Here is an illustrative (if artificial) example. 1For details, see Peters and Westerst˚ ahl (2006), ch. 10.2.1–2. 1 (3) a. Usually, when a farmer owns a donkey, he beats it. b. Usually, when a farmer owns a donkey, he is happy. c. Usually, when a farmer owns a donkey, it is unhappy. In 3b and 3c, it is natural to think that the quantification over pairs is reducible to ordinary quantification over individuals, so that 3b means 4. (4) a. ‘most farmers that own a donkey are happy’ b. most x(Fx ∧ ∃y(Dy ∧ Oxy) , Hx) Likewise, 3c means 5. (5) a. ‘most donkeys that a farmer owns are unhappy’ b. most y(Dx ∧ ∃x(Fx ∧ Oxy) , ¬Hy) However, neither reduction captures the meaning of 3a. Could 3a perhaps be taken to be the resumption of most to pairs, as in 6?2 (6) a. ‘most farmer-donkey pairs in the ownership relation are such that the farmer beats the donkey’ b. most2 xy(Fx ∧ Dy ∧ Oxy , BTxy) Barker reasoned similarly about possessives, as follows.3 Consider a sentence like 1a, that is, one of the form 7. (7) Q C’s As are B With R as the possessor relation, 7 is interpreted as in 8, where the narrowing requirement is made explicit. (8) Q∗ xy(Cx ∧ Ay ∧ Rxy , By) Here Q∗ is a quantifier over pairs somehow derived from the ordinary Q (over individuals). Which such quantifier is it? In what Barker calls the possessordominant reading, which is analogous to 5b, we have 9 (where ϕ(x, y) and ψ(y) contain at most the variables shown free). (9) Q∗ xy(ϕ(x, y), ψ(y)) ↔ Qy(∃xϕ(x, y), ψ(y)) This is a modifying reading of 7; in the case of 1a it says (somewhat implausibly ) that most rings-of-the-kind-planets-have are made of ice. The crux is to obtain the most likely reading, on which most seems to quantify over planets. reading. In analogy with 4b one might try 10. (10) Q∗ xy(ϕ(x, y), ψ(y)) ↔ Qx(∃yϕ(x, y), ψ(y)) 2So most 2(R, S) holds iff more than half of the ordered pairs in R belong to S. This quantifier is not logically definable...