Abstract
Displaced scope is a hallmark of natural language, and Quantifier Raising (QR) has long been the standard tool for analyzing scope. Yet despite the foundational importance of QR to theoretical linguistics, as far as I know, there has never been a study of its formal properties. For instance, consider the decidability problem: given an initial syntactic structure, is there an algorithm that will determine whether a semantically coherent QR derivation exists? If at least one such derivation exists, is the number of semantically different analyses always finite? How do we know when we have found them all? Do the answers to these questions depend on imposing scope islands or other constraints on QR, such as forbidding vacuous movement, re-raising, remnant raising, raising of names, repeated type lifting, and so on? I settle these issues by defining QRT (Quantifier Raising with Types), a substructural logic that is a faithful model of QR in the following respect: every semantically coherent QR derivation corresponds to a semantically equivalent proof in QRT, and vice-versa. Since QRT is decidable and has the finite readings property, it follows that a broad class of theories that rely on QR also have these properties, without needing to place any formal constraints on QR. I go on to study the special relationship between type lifting and QR, drawing an analogy with eta reduction in the lambda calculus. Allowing unrestricted type lifting does not compromise decidability. In addition, it turns out that QR with type lifting validates the core type shifting principles of Flexible Montague Grammar, a paradigm example of an in-situ type-shifting approach to scope taking. This suggests that QR is compatible with a local, directly compositional view of scope taking. These results put Quantifier Raising on a reassuringly firm formal footing. EARLY ACCESS
Highlights
Scope-taking is one of the most dramatic, distinctive, and ubiquitous phenomena in natural language, and Quantifier Raising has long been the standard technique for investigating scope
If we examine either the Quantifier Raising (QR) derivation of Ann saw everyone in (10) or the corresponding QRT proof in (17), there is no stage at which the structure corresponding to saw everyone is established as a constituent: there is no type associated with that particular substructure either in the QR derivation or in the corresponding QRT proof, and no Curry-Howard labeling that contains the semantic contribution of saw and everyone and nothing else
We can reconstruct the same type-shifted interpretations delivered by Flexible Montague Grammar whenever desired, so we can consider QRST to provide in-situ scope taking on demand
Summary
Scope-taking is one of the most dramatic, distinctive, and ubiquitous phenomena in natural language, and Quantifier Raising has long been the standard technique for investigating scope. Given a syntactic structure before Quantifier Raising, is there a procedure that is guaranteed to terminate and that will decide whether there is a series of QR operations that will result in a semantically coherent analyses? If QR were not decidable, any grammar that included QR would be committed to the existence of unanalyzable sentences For such a sentence, it would be impossible to say whether it had any coherent semantic interpretation: no matter how many derivations you had already tried, the mere fact that you hadn’t found one yet that works wouldn’t guarantee that there isn’t one. It turns out that QR with type lifting validates the core type shifting principles of Hendriks’ 1993 Flexible Montague Grammar, which is a paradigm example of a non-movement, in-situ, directly compositional approach to scope taking. The project reported here is both foundational — seeking a deeper understanding of Quantifier Raising — and crossdisciplinary, bringing to bear the metatheoretical techniques of formal logic
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