Generalized quantifiers and discontinuous type constructors

ion property for the main clause is greater than or equal to the cardinality of the than clause property. The latter is obtained by taking the boolean meet of the property corresponding to the common noun argument of the hypothetical determiner in the than clause and the noun-phrase abstraction property of the than clause. So our example would be paraphraseable as: ‘the number of books bought by John is greater than the number of records sold by Mary’. ‖more-than‖(CN) = {〈X,Y 〉 ∈ E | |X ∩ ‖CN‖| ≥ |Y |} (39) Now let us see how we can capture the syntactic fine-tuning in terms of a type assignment to the head of the construction — the determiner more in the main clause — so that it can reach out discontinuously to the than clause. The essential point is that the result of np binding in the main clause s domain is not a complete sentence, bu a sentence still looking for a than complement. (Compare the type assignment to an ordinary s level determiner: q(np, s, s)/n.) The than clause is headed by the complementizer than which combines with a clause missing a determiner: s↑(q(np, s, s)/n). more ∈ q(np, s, s/sthan)/n than ∈ sthan/(s↑(q(np, s, s)/n)) To make the semantics explicit, we can assign the complementizer than a lexical lambda term which takes the boolean meet of the property corresponding to the common noun argument of the hypothetical determiner and the noun-phrase abstraction property of the than clause. than = λQ((et)(et)t)tλxe.Q(λPetλQet.P (x) ∧Q(x)) (40) Below the reader can find the semantic recipes computed for the main clause and for the than clause. John bought more books 7→ λQet.more(books)(λxe.bought(x)(john))(Q) (41) than Mary sold records 7→ than(λD(et)(et)t.D(records)(λye.sold(y)(mary))) (42) Working out the lexical recipe for than, they combine to the term given below. more(books)(λxe.bought(x)(john))(λy.(records(y) ∧ sold(y)(mary))) (43) 4.2 Focus, gapping Work by Oehrle (1991) and Van der Linden (1991) on the semantic impact of prosodic organisation shows that one can analyse focus assignment in terms of a focus binder