Graham Priest has influentially claimed that the Sorites paradox is an Inclosure paradox, concluding that his favored dialetheic solution to the Inclosure paradoxes should be extended to the Sorites paradox. We argue that, given Priest’s dialetheic solution to the Sorites paradox, the argument purporting to show that that paradox is an Inclosure is unsound, and discuss some issues surrounding this fact.

This paper provides a study of sequent calculi for intuitionistic Gödel–Löb logic (iGL), which is the intuitionistic version of the classical modal logic GL, known as Gödel–Löb logic. We present two different sequent calculi, one of which we prove to be the terminating version of the other. We study those systems from a proof-theoretic point of view. One of our main results is a syntactic proof for the cut-admissibility result for those systems. Finally, we apply this to prove Craig interpolation for intuitionistic Gödel–Löb logic.

In this note, we show that the first-order logic ${\mathrm{IK}}^{\mathit{\omega }}$ is sound with regard to the models obtained from continuum-valued Łukasiewicz-models for first-order languages by treating the quantifiers as infinitary strong disjunction/conjunction rather than infinitary weak disjunction/conjunction. Moreover, we show that these models cannot be used to provide a new consistency proof for the theory of truth ${\mathrm{IKT}}^{\mathit{\omega }}$ obtained by expanding ${\mathrm{IK}}^{\mathit{\omega }}$ with transparent truth, because the models are inconsistent with transparent truth. Finally, we show that whether or not this inconsistency can be reproduced in the sequent calculus for ${\mathrm{IKT}}^{\mathit{\omega }}$ depends on how vacuous quantification is treated.

Let $\mathcal{K}=⟨\mathcal{R},\mathit{\delta }⟩$ be a closed ordered differential field, in the sense of Singer, and let C be its field of constants. In this note, we prove that, for sets definable in the pair $\mathcal{M}=⟨\mathcal{R},C⟩$, the δ-dimension of Brihaye, Michaux, and Rivière and the large dimension from of Eleftheriou, Günaydin, and Hieronymi coincide. As an application, we characterize the definable sets in $\mathcal{K}$ that are internal to C as those sets that are definable in $\mathcal{M}$ and have δ-dimension 0. We further show that, for sets definable in $\mathcal{K}$, having δ-dimension 0 does not generally imply co-analyzability in C (in contrast to the case of transseries). We also point out that the coincidence of dimensions also holds in the context of differentially closed fields and in the context of transseries.

In this paper we analyze the connection between reflexive-insensitive modal logics, logics of provability, and the modal logic of forcing. Because of the inter-definability of the ∘-operator that characterizes the reflexive-insensitive logics and Boolos’s ⊡-operator, characterization results for the reflexive-insensitive logics can be used to give a partial solution of the boxdot conjecture, and to illuminate some closely related problems. In turn, this will facilitate the application of reflexive-insensitive logics to the study of the modal logic of forcing and truth and the logic of forcing persistent sentences. We then show that the modal logic of forcing, the modal logic of forcing and truth, and the modal logic of forcing persistency give rise to the same characterizability results. We conclude by arguing that the modal logic of forcing should be better understood as the modal logic of generic absoluteness.

This paper develops some modal metatheory for quantified modal logic. In such a theory, the logic of a first-order modal object-language is made sensitive to the modal facts, stated in the metalanguage. This is radically different from possible worlds semantics, which reduces questions of validity to questions of nonmodal set theory. We consider theories which characterize a notion of truth under a second-order interpretation, where an operator for metaphysical necessity is treated homophonically. The form they take is crucially influenced by whether the Barcan formulas are part of the metalogic. The main result is that, with the Barcan formulas, formulating a modal metatheory is easy; principles of classical semantics transfer over smoothly with natural adaptations to the modal setting. Without the Barcan formulas, it seems to be much harder. Several options are considered, but each involves commitments plausibly in tension with the underlying philosophical motivations for rejecting them.

We propose a new direct presentation of full (propositional) classical logic by means of proof nets, for which strong normalization and confluence hold true. Our proposal is based on the breaking of symmetry between propositional connectives, combined with a principle of focusing/defocusing.

In this paper, we study a generalization of a question, raised by C. Fefferman and J. Kollár, on the existence of solutions of linear functional equations. Suppose that $\mathfrak{R}$ is a definably complete expansion of a real closed field $(R;+,\cdot )$. Let $f,g_1,\dots ,g_k:R^n\to R$ be continuous functions that are definable in $\mathfrak{R}$. We prove that if there exist continuous functions $y_1,\dots ,y_k:R^n\to R$ such that $f=g_1{y}_1+\cdots +g_k{y}_k$, then there exist continuous functions $y_1,\dots ,y_k$ such that $y_1,\dots ,y_k$ are definable in $\mathfrak{R}$ and $f=g_1{y}_1+\cdots +g_k{y}_k$.

We construct a computable module $\mathcal{M}$ over a computable commutative ring R such that the radical of $\mathcal{M}$, $\mathrm{rad}(\mathcal{M})$, defined as the intersection of all proper maximal submodules, is ${\mathrm{\Pi }}_1^1$-complete. This shows that in general such radicals are as (logically) complicated as possible and, unlike many other kinds of ring-theoretic radicals, admit no arithmetical definition.

Recently, several authors have pointed out that substructural logics are adequate for developing naive theories that represent semantic concepts such as truth. Among them, three proposals have been explored: dropping cut, dropping contraction and dropping reflexivity. However, nowhere in the substructural literature has anyone proposed rejecting the structural rule of weakening, while accepting the other rules. Some theorists have even argued that this task was not possible, since weakening plays no role in the derivation of semantic paradoxes. In this article, I introduce a theory for naive truth based on the logic resulting from dropping the rule of weakening from classical logic, and maintaining the other structural rules.