Publications
Pre-Filtration, Pre-Stable Canonical Rules, and the Kuznetsov-Muravitsky Isomorphism (with Nick Bezhanishvili)
For The Legacy of A.V. Kuznetsov in Logic, Algebra and Foundations of Mathematics, forthcoming. Penultimate draft.
We introduce pre-filtrations and pre-stable canonical rules for the Kuznetsov–Muravitsky system of intuitionistic modal logic and provide a new proof of the Kuznetsov–Muravitsky isomorphism, along with several preservation results. The proofs employ these rules and a duality between modal (Heyting) algebras and their corresponding order-topological spaces.
Blok-Esakia Theorems via Stable Canonical Rules (with Nick Bezhanishvili)
The Journal of Symbolic Logic, 2025. Published version.
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. Since stable canonical rules can be developed for any rule system admitting filtration (in a fairly weak sense), our method can be applied uniformly across a wide range of signatures. We illustrate this by proving a version of the Blok-Esakia theorem for superintuitionistic rules systems, bi-superintuitionistic rule systems and modal superintuitionistic rule systems above KM. Furthermore, we prove a version of the Dummett-Lemmon conjecture for superintuitionistic and bi-superintuitionistic rule systems.
Debunking Multiform Dimensionality: many, Romance tant-PL, & morpho-syntactic opacity (with Luis Miguel Toquero Perez)
Proceedings of SALT 32, 2022. Published version.
The interpretation of ‘much/many’ has been argued to be regulated by Uniform Dimensionality: ‘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.
Work in progress
Quantificationalism
In preparation. Email for draft.
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 elucidates the content of Quantificationalism and argues for its philosophical fruitfulness, by outlining applications in metaphysics and the philosophy of logic.
The Logic of Quantificationalism, Part 1: Foundations.
Under review. Draft.
I formulate a higher-order logic in which Quantificationalism can be precisely formulated and shown to be consistent. This logic can be given a sound and complete semantics over quantificational substitution structures (QSS). Essentially, a QSS is a substitution structure in the sense of @Bacon2019SS, where each metaphysical substitution is determined by a domain of quantification. Intuitively, the effect of applying a substitution of this sort to an entity is that of forcing all quantification “involved” in that entity to range over the domain that determines the relevant substitution.
The algebra of Necessities
In preparation. Email for draft
Recently, modal metaphysics has seen a shift from the study of specific notions of necessity (nomic, counterfactual, metaphysical, logical, …) to the study necessities as a whole. Following @Bacon2018TBN, the idea is to define a necessity as the worldly analog of an operator governed by a normal modal logic, then investigate the theory of necessities in purely logical terms. @BaconZeng2022AToN showed that if reality is coarse-grained enough, necessities in this sense form a bounded lattice with respect to entailment.
This paper investigate the lattice of necessities in more detail. Solving an open problem from @BaconZeng2022AToN, I show that the lattice of necessities is in fact distributive. What is more, it admits the structure of a Heyting algebra. This allows us to define pseudocomplements of necessities: the pseudocomplement of a necessity is the weakest necessity whose meet with is the broadest necessity. I investigate the conditions under which pseudocomplements of necessities obey classical principles, like the law of excluded middle and double-negation elimination. These laws turn out to characterize different senses of the philosophically significant idea of an infinitely closed necessity.
A Harris Result for Quantificationalist Free Quantifiers
In preparation. Email for draft.
Classical higher-order logicians sometimes argue for the intelligibility of higher-order quantification by using an inferentialist strategy: we understand classical higher-order quantifiers because their inferential role uniquely pins down their meaning up to logical equivalence. The received view is that an inferentialist strategy is not available to those who take the correct logic of higher-order quantifiers to be free rather than classical. In standard higher-order free logics, multiple non-equivalent terms satisfy the axioms of free higher-order quantification.
I argue that higher-order free logicians in fact can articulate an inferentialist strategy if they are willing to embrace Quantificationalism. Quantificationalists have independent reasons to regard the correct inferential role of free quantifiers as richer than it is normally taken to be. This richer inferential role turns out to be strong enough to single out the meaning of free quantifiers uniquely.
The Logic of Quantificationalism, Part 2: Support and Predicativity
In preparation.
Quantificationalists can make sense of what it is for a non-linguistic entity like a property or a proposition to quantify over a domain, without assuming that reality has anything like quasi-syntactic structure. I introduce the notion of quantificational support for this purpose. As a corollary, Quantificaitonalists can make sense of a worldly or metaphysical notion of predicativity. Roughly, a proposition is metaphysically predicative when it quantifies (in the sense captured by quantificaitonal support) over a domain that includes that very proposition. This paper explores the logic of these notions.