The main result of this paper, built on previous work by the author and T. Wilson, is the proof that the theory “${\mathsf{AD}}_{\mathbb{R}}+\mathsf{DC}$ $+$ there is an $\mathbb{R}$-complete measure on Θ” is equiconsistent with “$\mathsf{ZF}+\mathsf{DC}+{\mathsf{AD}}_{\mathbb{R}}$ $+$ there is a supercompact measure on ${\wp }_{{\mathit{\omega }}_1}(\wp (\mathbb{R}))+\mathrm{\Theta }$ is regular.” The result and techniques presented here contribute to the general program of descriptive inner model theory and in particular, to the general study of compactness phenomena in the context of $\mathsf{ZF}+\mathsf{DC}$.

The paper studies a cluster of systems for fully disquotational truth based on the restriction of initial sequents. Unlike well-known alternative approaches, such systems display both a simple and intuitive model theory and remarkable proof-theoretic properties. We start by showing that, due to a strong form of invertibility of the truth rules, cut is eliminable in the systems via a standard strategy supplemented by a suitable measure of the number of applications of truth rules to formulas in derivations. Next, we notice that cut remains eliminable when suitable arithmetical axioms are added to the system. Finally, we establish a direct link between cut-free derivability in infinitary formulations of the systems considered and fixed-point semantics. Noticeably, unlike what happens with other background logics, such links are established without imposing any restriction to the premises of the truth rules.

Belief logic is one of the most important branches of philosophical logic. The standard approach for characterizing belief is based on the so-called possible worlds model within which the problem of logical omniscience is considered a significant defect. We present an alternative semantics for belief logic based on a probabilistic model, which formalizes the so-called Lockean thesis in philosophical literature. The Lockean thesis states that a person believes a proposition whenever they see that the probability of the proposition has reached or exceeded a prespecified threshold. We give a complete deductive system for probabilistic belief logic as well as detailed comparisons to existing frameworks of belief logic.

In his own words, this historical note documents Robert Solovay’s argument in 2002 that a classical system BI with arithmetic comprehension and bar induction is equiconsistent, over primitive recursive arithmetic PRA, with Kleene’s formal system FIM for intuitionistic analysis plus Markov’s Principle.

In this paper, an incompleteness theorem for modal extensions of relevant logics is proved. The proof uses elementary methods and builds upon the work of Fuhrmann.

A von Neumann regular ring, named after John von Neumann for his work related to continuous geometry and operator algebras, has various equivalent characterizations in terms of popular properties of rings and modules. This paper mainly focuses on effective aspects of characterizations of von Neumann regular rings, and studies such characterizations by techniques of reverse mathematics. For not necessarily commutative rings, we obtain that $\mathrm{RC}{A}_0$ proves that a ring R is von Neumann regular iff every ${\mathrm{\Sigma }}_1^0$ finitely generated left ideal of R is generated by an idempotent iff every ${\mathrm{\Sigma }}_1^0$ cyclic left R-module is Lam-divisible. For commutative rings, we obtain that over $\mathrm{RC}{A}_0$, $\mathrm{WK}{L}_0$ (resp., ${\mathrm{ACA}}_0$) is equivalent to the statement that a commutative ring R is von Neumann regular iff the localization of R at any prime ideal (resp., maximal ideal) exists and it is a field. Lastly, we show that $\mathrm{AC}{A}_0$ proves that a commutative ring R is von Neumann regular iff every simple R-module is Lam-divisible.