Quantifiers with Split Scope

Quantifiers in intensional contexts cause difficulty in explaining their scope phenomena in the standard linguistics. A strong quantifier has narrower scope than an intensional operator, but the nominal predicate has the de re interpretation, and an indefinite can have wide scope over an intensional operator over a syntactic island. In the paper, I propose a new way of interpreting a strong quantifier, assuming that a strong quantifier triggers the presupposition that there is a non-empty set determined by the nominal predicate. A presupposition tends to be projected over an intensional operator. This gives the effect that the nominal predicate gets the de re interpretation. On the other hand, the nuclear scope of the quantifier, together with the nominal predicate, determines another set in the local context, and the quantificational force is determined by the relation of the two sets. A weak quantifier is ambiguous, and it can be interpreted as triggering a presupposition, as a strong quantifier does. A quantifier in this use can be analyzed in the same way, but it leads to the effect that the quantifier, not just the nominal predicate, has scope over an intensional operator because of a semantic property of the quantifier. This new way of interpreting a quantifier is independently motivated by the observation that the two sets are referred to at a later discourse.