The interpretation of much/many has been argued to be regulated by Uniform Dimensionality (Hackl 2000; Solt 2009): much is underspecified but many encodes cardinality. However, given some data where many denotes ‘volume’, Snyder (2021) proposes the need for Multiform Dimensionality: both much and many are underspecifed. After reviewing the English data, and in light of novel cross-linguistic data, we argue that neither generalization is fully accurate. Instead, following Wellwood (2015, 2018), we argue for an alternative, Abstract Uniform Dimensionality, which we propose to be universal: MUCH always measures cardinality when it scopes over semantically interpretable plural. We derive the universal by proposing that MUCH can occupy different positions in the NP, only one of which has semantic plural in its scope. Variation is thus not semantic, but morpho-syntactic.
This paper presents a new uniform method for studying modal companions of superintuitionistic deductive systems and related notions, based on the machinery of stable canonical rules. Using our method, we obtain an alternative proof of the Blok-Esakia theorem both for logics and for rule systems, and prove an analogue of the Dummett-Lemmon conjecture for rule systems. Since stable canonical rules may be developed for any rule system admitting filtration, our method generalises smoothly to richer signatures. We illustrate this by applying our techniques to prove analogues of the Blok-Esakia theorem (for both logics and rule systems) and of the Dummett-Lemmon conjecture (for rule systems) in the setting of tense companions of bi-superintuitionistic deductive systems. We also use our techniques to prove that the lattice of rule systems (logics) extending the modal intuitionistic logic KM and the lattice of rule systems (logics) extending the provability logic GL are isomorphic.
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