Dalrymple and Kaplan (2000) (DK) propose a treatment of indeterminacy in morphosyntactic features which uses sets as values for such features. DK propose that case requirements placed by governing verbs are checked by set membership. This treatment gives a straightforward account of the fact that an indeterminate noun can satisfy inconsistent case requirements, as, for example, an argument of coordinated verbs independently imposing conflicting requirements. However, the set-based account has several shortcomings. First, it does not make the right predictions when modifiers and governing predicates both place agreement requirements on the same noun. In this case, contrary to the predictions of the set-based analysis, the requirements must be compatible (see Levy 2001). This is the transitivity problem. Second, the set-based account does not extend easily to cases where a predicate places indeterminate requirements on an indeterminate feature such as case. This is the second-order indeterminacy problem. Our analysis covers the cases of indeterminacy that the set-based account was designed to handle, while also providing a simple and straightforward account of the transitivity problem and the second-order indeterminacy problem.
Consider the simple cases that motivate the set-based analysis. Our approach treats grammatical case not as a single atomic feature but as an f-structure, with attributes corresponding to each case distinguished in the language. Nouns and their modiers specify negative values for the cases which they do not give expression to.
Now consider the transitivity problem. A noun that is indeterminately accusative or dative must take a dative modifier if the predicate requires dative, and an accusative modifier for an accusative predicate; other patterns are disallowed. For example, a dative determiner is required with a noun in construction with a dative verb, even when the noun is indeterminately specified for case. The set-based analysis does not predict this pattern. On the set-based account, a noun that is indeterminately dative and accusative may satisfy a dative requirement imposed by the predicate as well as an accusative requirement imposed by the modifier. On our account, modifiers like determiners and adjectives simply provide additional specication of the noun's case value. They state negative values for case features, placing further constraints on the case features of the noun they modify.
The second-order indeterminacy problem arises when indeterminate verbs interact with indeterminate nouns. For example, some Russian verbs require objects that are either genitive or accusative. Coordinated objects with one genitive and one accusative conjunct are also possible. Such examples are problematic for the set-based account, since they require a non-null intersection between the set of case values specified by the noun and the set required by the verb, a requirement that is not possible to impose within the standard formal assumptions of LFG. Our analysis requires an underspecified case requirement for the verb. This allows the verb to govern coordinated objects with different case features, as long as each conjunct is compatible with a positive specication for either acc or gen.
In sum, our approach uses underspecication rather than set values to handle indeterminacy. It is formally simple, and correctly allows for incremental and monotonic renement of case requirements in particular contexts.