the week of november 12

quantifier phrases in object positions

Following Frege, we have been assuming that quantifier phrases denote second order properties, that is, they have denotations of type <<et>t>. Given our current assumptions about semantic composition, which only allow a small set of composition operations, this means that quantifier phrases need sister constituents of type <et>. Surface syntax does not always provide such sister constituents. Semanticists have not yet reached a consensus about how natural languages solve potential type mismatches that might result from quantifier phrases sitting in the wrong positions - in the direct object positions of transitive verbs, for example.

The syntactic repair strategy called "Quantifier Raising" (QR) moves the quantifier phrase from its original position and adjoins it to a constituent of type t, thereby creating a binding configuration. A trace of type e is left in the original position and a matching binder is introduced between the moved QP and the constituent it adjoins to. Many questions came up at this point: what determines the type of the trace? A possible answer was that we pick the lowest type that guarantees interpretability. Another question concerned the origin of the binder. Is the binder an index on the moved QP that is reparsed as a binder index, as suggested by Heim & Kratzer? Or is the quantifier phrase "attracted" by a binder carried by an already existing functional head, as in the case of relative pronouns? What exactly determines the adjunction site? Is it the closest constituent that leads to interpretability, that is, the closest constituent of type t? How plausible is a syntactic solution to our type mismatch problem? In the end, the answer will depend on whether the operations created by QR can be shown to have independent motivation. Do independently attested constraints on movement constrain quantifier scope in just the right way, for example?

After investigating the properties of Quantifier Raising, we looked at one possible semantic strategy for dealing with the potential type mismatch created by quantifier phrases in object position. You can read about other conceivable semantic strategies in the Heim & Kratzer book. What if we allowed verbs to have argument positions of type <<et>t>? Exploring this proposal provided the welcome opportunity for us to get a thorough workout with our lambda notation. If all of a verb's argument positions are of type <<et>t>, we have to shift the denotations of proper names and pronouns of type e. That is a straightforward shift, however. We can easily map an individual into (the characteristic function of) the set of its properties. We could translate the proper name John as λP P(john), then.

We did not come up with conclusive arguments for or against one or the other approach to the type mismatch problem. The combined handout for Wednesday and Friday is here.

At the very end of Friday's class, we reminded ourselves that we had already heard arguments about severing the external argument from its verb within a Davidsonian event semantics. In that case, no problems with quantifier phrases in direct object positions are expected to come up in the first place. The same is true for quantifier phrases in indirect object positions, if indirect objects are introduced by separate heads as well. We can now interpret quantifier phrases in situ, provided we give them a semantic type that allows them to operate over relations between individuals and events. If s is the type of events, and R a variable ranging over functions of type <e<st>>, the translation of everybody into a suitable type logic would look as in (i).

(i) λRλe ∀x (person(x) → R(x)(e) )

The expression in (i) could then combine with a predicate of type <e<st>>, which might be the translation of a transitive verb, as in (ii).

(ii) λxλe offend(x)(e)

After beta-conversion, we get (iii):

(iii) λe ∀x (person(x) → offend(x)(e) )

After adding the external argument, the result would be (iv), which is a suitable argument for the translation of a quantifier phrase sitting in subject position. If there are no predicates that have more than one non-event argument, we do no longer run into a type mismatch with quantifier phrases in object positions.

(iv) λeλy (agent(y)(e) & ∀x (person(x) → offend(x)(e) ) )

The predictions made by this approach to quantification is that we should find quantifiers in natural languages that show event sensitivity. This prediction is borne out. English each versus every is a case in question. This topic is discussed in Susanne Tunstall's dissertation. As argued by Kimiko Nakanishi in her dissertation and related articles, floated quantifiers in Japanese show event sensitivity, too. In his forthcoming UMass dissertation, Jan Anderssen demonstrates that the German quantifier lauter shows event sensitivity. Interestingly, quantifiers that show event sensitivity often occur low within the extended verbal projection, and this fits with our earlier observation that a verb's event argument must be quantified off at some point, definitely in the scope of negation. Negative quantifiers (and other downward-entailing) quantifiers should then tend to show up higher in the tree. Chapter 4 of my Event Argument discusses examples illustrating that this is indeed so in German, where scope relations are represented more transparently than in English.