In this paper, using a propositional modal language extended with the window modality, we capture the first-order properties of various mereological theories. In this setting, $\Box \varphi $ reads all the parts (of the current object) are $\varphi $, interpreted on the models with a whole-part binary relation under various constraints. We show that all the usual mereological theories can be captured by modal formulas in our language via frame correspondence. We also correct a mistake in the existing completeness proof for a basic system of mereology by providing a new construction of the canonical model.

Since Kripke, philosophers have distinguished a priori true statements from necessarily true ones. A statement is a priori true if its truth can be established before experience, and necessarily true if it could not have been false according to logical or metaphysical laws. This distinction can be captured formally using two-dimensional semantics.

There is a natural way to extend the notions of apriority and necessity so they can also apply to questions. Questions either can or cannot be resolved before experience, and either are or are not about necessary facts. Classical two-dimensionalism has no account of question meanings, so it has to be combined with a framework for question semantics in order to capture these observations. It is shown in [14] how two-dimensional semantics can be combined with inquisitive semantics, in which questions are analyzed in terms of information. The present paper investigates the logic of two-dimensional inquisitive semantics, and provides a complete proof system.

We present a new manifestation of Gödel’s second incompleteness theorem and discuss its foundational significance, in particular with respect to Hilbert’s program. Specifically, we consider a proper extension of Peano arithmetic ($\mathbf {PA}$) by a mathematically meaningful axiom scheme that consists of $\Sigma ^0_2$-sentences. These sentences assert that each computably enumerable ($\Sigma ^0_1$-definable without parameters) property of finite binary trees has a finite basis. Since this fact entails the existence of polynomial time algorithms, it is relevant for computer science. On a technical level, our axiom scheme is a variant of an independence result due to Harvey Friedman. At the same time, the meta-mathematical properties of our axiom scheme distinguish it from most known independence results: Due to its logical complexity, our axiom scheme does not add computational strength. The only known method to establish its independence relies on Gödel’s second incompleteness theorem. In contrast, Gödel’s theorem is not needed for typical examples of $\Pi ^0_2$-independence (such as the Paris–Harrington principle), since computational strength provides an extensional invariant on the level of $\Pi ^0_2$-sentences.

Standard Type Theory, ${\textrm {STT}}$, tells us that $b^n(a^m)$ is well-formed iff $n=m+1$. However, Linnebo and Rayo [23] have advocated the use of Cumulative Type Theory, $\textrm {CTT}$, which has more relaxed type-restrictions: according to $\textrm {CTT}$, $b^\beta (a^\alpha )$ is well-formed iff $\beta>\alpha $. In this paper, we set ourselves against $\textrm {CTT}$. We begin our case by arguing against Linnebo and Rayo’s claim that $\textrm {CTT}$ sheds new philosophical light on set theory. We then argue that, while $\textrm {CTT}$’s type-restrictions are unjustifiable, the type-restrictions imposed by ${\textrm {STT}}$ are justified by a Fregean semantics. What is more, this Fregean semantics provides us with a principled way to resist Linnebo and Rayo’s Semantic Argument for $\textrm {CTT}$. We end by examining an alternative approach to cumulative types due to Florio and Jones [10]; we argue that their theory is best seen as a misleadingly formulated version of ${\textrm {STT}}$.

This article provides an algebraic study of the propositional system $\mathtt {InqB}$ of inquisitive logic. We also investigate the wider class of $\mathtt {DNA}$-logics, which are negative variants of intermediate logics, and the corresponding algebraic structures, $\mathtt {DNA}$-varieties. We prove that the lattice of $\mathtt {DNA}$-logics is dually isomorphic to the lattice of $\mathtt {DNA}$-varieties. We characterise maximal and minimal intermediate logics with the same negative variant, and we prove a suitable version of Birkhoff’s classic variety theorems. We also introduce locally finite $\mathtt {DNA}$-varieties and show that these varieties are axiomatised by the analogues of Jankov formulas. Finally, we prove that the lattice of extensions of $\mathtt {InqB}$ is dually isomorphic to the ordinal $\omega +1$ and give an axiomatisation of these logics via Jankov $\mathtt {DNA}$-formulas. This shows that these extensions coincide with the so-called inquisitive hierarchy of [9].1

In this paper, we axiomatize the deontic logic in Fusco (2015), which uses a Stalnaker-inspired account of diagonal acceptance and a two-dimensional account of disjunction to treat Ross’s Paradox and the Puzzle of Free Choice Permission. On this account, disjunction-involving validities are a priori rather than necessary. We show how to axiomatize two-dimensional disjunction so that the introduction/elimination rules for boolean disjunction can be viewed as one-dimensional projections of more general two-dimensional rules. These completeness results help make explicit the restrictions Fusco’s account must place on free-choice inferences. They are also of independent interest, as they raise difficult questions about how to “lift” a Kripke frame for a one-dimensional modal logic into two dimensions.

This paper introduces three model-theoretic constructions for generalized Epstein semantics: reducts, ultramodels and $\textsf {S}$-sets. We apply these notions to obtain metatheoretical results. We prove connective inexpressibility by means of a reduct, compactness by an ultramodel and definability theorem which states that a set of generalized Epstein models is definable iff it is closed under ultramodels and $\textsf {S}$-sets. Furthermore, a corollary concerning definability of a set of models by a single formula is given on the basis of the main theorem and the compactness theorem. We also provide an example of a natural set of generalized Epstein models which is undefinable. Its undefinability is proven by means of an $\textsf {S}$-set.

We develop an untyped framework for the multiverse of set theory. $\mathsf {ZF}$ is extended with semantically motivated axioms utilizing the new symbols $\mathsf {Uni}(\mathcal {U})$ and $\mathsf {Mod}(\mathcal {U, \sigma })$, expressing that $\mathcal {U}$ is a universe and that $\sigma $ is true in the universe $\mathcal {U}$, respectively. Here $\sigma $ ranges over the augmented language, leading to liar-style phenomena that are analyzed. The framework is both compatible with a broad range of multiverse conceptions and suggests its own philosophically and semantically motivated multiverse principles. In particular, the framework is closely linked with a deductive rule of Necessitation expressing that the multiverse theory can only prove statements that it also proves to hold in all universes. We argue that this may be philosophically thought of as a Copernican principle that the background theory does not hold a privileged position over the theories of its internal universes. Our main mathematical result is a lemma encapsulating a technique for locally interpreting a wide variety of extensions of our basic framework in more familiar theories. We apply this to show, for a range of such semantically motivated extensions, that their consistency strength is at most slightly above that of the base theory $\mathsf {ZF}$, and thus not seriously limiting to the diversity of the set-theoretic multiverse. We end with case studies applying the framework to two multiverse conceptions of set theory: arithmetic absoluteness and Joel D. Hamkins’ multiverse theory.

In this essay we introduce a new tool for studying the patterns of sentential reference within the framework introduced in [2] and known as the language of paradox $\mathcal {L}_{\mathsf {P}}$: strong $\mathcal {L}_{\mathsf {P}}$-homomorphisms. In particular, we show that (i) strong $\mathcal {L}_{\mathsf {P}}$-homomorphisms between $\mathcal {L}_{\mathsf {P}}$ constructions preserve paradoxicality, (ii) many (but not all) earlier results regarding the paradoxicality of $\mathcal {L}_{\mathsf {P}}$ constructions can be recast as special cases of our central result regarding strong $\mathcal {L}_{\mathsf {P}}$-homomorphisms, and (iii) that we can use strong $\mathcal {L}_{\mathsf { P}}$-homomorphisms to provide a simple demonstration of the paradoxical nature of a well-known paradox that has not received much attention in this context: the McGee paradox. In addition, along the way we will highlight how strong $\mathcal {L}_{\mathsf {P}}$-homomorphisms highlight novel connections between the graph-theoretic analyses of paradoxes mobilized in the $\mathcal {L}_{\mathsf {P}}$ framework and the methods and tools of category theory.