Antonio Maria Cleani

Quantificationalism is the view that some true propositions are false at or relative to some domains of quantification, in much the same sense in which propositions can be true or false at times or possible worlds. This paper seeks to give solid formal foundations to quantificationalism, comparable to what modal and tense logics are to the notions of contingent and temporary truth. It has the twofold purpose of understanding what sort of problems arise when we try and make precise sense of Quantificationalism, and of solving these problems by outlining a comprehensive higher-order logical framework in which Quantificationalism can be precisely formulated and shown to be consistent.

We present a new uniform method for studying modal companions of superintuitionistic rule systems and related notions, based on the machinery of stable canonical rules. Using this method, we obtain alternative proofs of the Blok-Esakia theorem and of the Dummett-Lemmon conjecture for rule systems. Since stable canonical rules may be developed for any rule system admitting filtration, our method generalizes smoothly to richer signatures. Using essentially the same argument, we obtain a proof of an analogue of the Blok-Esakia theorem for bi-superintuitionistic and tense rule systems, and of the Kuznetsov-Muravitsky isomorphism between rule systems extending the modal intuitionistic logic KM and modal rule systems extending the provability logic GL. In addition, our proof of the Dummett-Lemmon conjecture also generalizes to the bi-superintuitionistic and tense cases.

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.