I investigate whether Wittgenstein’s “weakly exclusive” Tractarian semantics (as reconstructed by Rogers and Wehmeier) is compositional. In both Tarskian and Wittgensteinian semantics, one has the choice of either working exclusively with total variable assignments or allowing partial assignments; the choice has no bearing on the compositionality of Tarskian semantics, but turns out to make a difference in the Wittgensteinian case. Some philosophical ramifications of this observation are discussed.

We propose a natural deduction calculus for the modal logic S4.2. The system is designed to match as much as possible the structure and the properties of the standard system of natural deduction for first-order classical logic, exploiting the formal analogy between modalities and quantifiers. The system is proved sound and complete with respect to (w.r.t.) the standard Hilbert-style formulation of S4.2. Normalization and its consequences are obtained in a natural way, with proofs that closely follow the analogous ones for first-order logic.

Within Bishop Set Theory, a reconstruction of Bishop’s theory of sets, we study the so-called completely separated sets, that is, sets equipped with a positive notion of an inequality, induced by a given set of real-valued functions. We introduce the notion of a global family of completely separated sets over an index-completely separated set, and we describe its Sigma- and Pi-set. The free completely separated set on a given set is also presented. Purely set-theoretic versions of the classical Stone–Čech theorem and the Tychonoff embedding theorem for completely regular spaces are given, replacing topological spaces with function spaces and completely regular spaces with completely separated sets.

Goodstein’s theorem states that certain sequences based on exponential notation for the natural numbers are always finite. The result is independent of Peano arithmetic and is a prototypical example of a proof of termination by transfinite induction. A variant based instead on the Ackermann function has more recently been proposed by Arai, Fernández-Duque, Wainer, and Weiermann, and instead is independent of the more powerful theory ${\mathsf{ATR}}_0$. However, this result is contingent on rather elaborate normal forms for natural numbers based on a “sandwiching” procedure. This leaves open both the question of whether the sandwiching procedure can be eliminated while retaining the full strength of the Ackermannian Goodstein principle, and whether other normal forms can lead to nontermination.

In this article we settle these questions by showing that any Goodstein process based on the Ackermann function is terminating, and indeed the sandwiching procedure gives rise to Goodstein principles of maximal length. We thus obtain an equivalent principle which does not involve normal forms at all and immediately implies all Ackermannian Goodstein principles that have been considered. Our techniques provide a new approach to termination proofs, where terms in a sequence do not necessarily decrease in complexity, but instead are majorized by some “master” process, already known to be terminating.

We explore Boolean-valued models of set theory with a class of urelements. In an existing construction, which we call $U^{\mathbb{B}}$, every urelement is its own $\mathbb{B}$-name. We prove the fundamental theorem of $U^{\mathbb{B}}$ in the context of ${\mathrm{ZFU}}_{\mathrm{R}}$ (i.e., ZF with urelements formulated with Replacement). In particular, $U^{\mathbb{B}}$ is shown to preserve Replacement and hence ${\mathrm{ZFU}}_{\mathrm{R}}$. Moreover, $U^{\mathbb{B}}$ can both destroy axioms, such as the DC${}_{{\mathit{\omega }}_1}$-scheme, and recover axioms, such as the Collection Principle. One drawback of $U^{\mathbb{B}}$ is that it does not permit mixing names, resulting in a lack of fullness. To address this, we introduce a new construction, $\overline{U^{\mathbb{B}}}$, which is closed under mixtures. We prove that there is an elementary embedding from $U^{\mathbb{B}}$ to $\overline{U^{\mathbb{B}}}$. Over ${\mathrm{ZFU}}_{\mathrm{R}}$ with the Axiom of Choice, $\overline{U^{\mathbb{B}}}$ is full for every complete Boolean algebra $\mathbb{B}$ just in case the Collection Principle holds.