Any theory of truth must find a way around Curry’s paradox, and there are well-known ways to do so. This paper concerns an apparently analogous paradox, about validity rather than truth, which JC Beall and Julien Murzi (“Two flavors of Curry’s paradox”) call the v-Curry. They argue that there are reasons to want a common solution to it and the standard Curry paradox, and that this rules out the solutions to the latter offered by most “naive truth theorists.” To this end they recommend a radical solution to both paradoxes, involving a substructural logic, in particular, one without structural contraction.

In this paper I argue that substructuralism is unnecessary. Diagnosing the “v-Curry” is complicated because of a multiplicity of readings of the principles it relies on. But these principles are not analogous to the principles of naive truth, and taken together, there is no reading of them that should have much appeal to anyone who has absorbed the morals of both the ordinary Curry paradox and the second incompleteness theorem.

The variety of Heyting algebras has two well-behaved locally finite reducts, the variety of bounded distributive lattices and the variety of implicative semilattices. The variety of bounded distributive lattices is generated by the $\to$-free reducts of Heyting algebras, while the variety of implicative semilattices is generated by the $\vee$-free reducts. Each of these reducts gives rise to canonical formulas that generalize Jankov formulas and provide an axiomatization of all superintuitionistic logics (si-logics for short).

The $\vee$-free reducts of Heyting algebras give rise to the $(\wedge ,\to )$-canonical formulas that we studied in an earlier work. Here we introduce the $(\wedge ,\vee )$-canonical formulas, which are obtained from the study of the $\to$-free reducts of Heyting algebras. We prove that every si-logic is axiomatizable by $(\wedge ,\vee )$-canonical formulas. We also discuss the similarities and differences between these two kinds of canonical formulas.

One of the main ingredients of these formulas is a designated subset $D$ of pairs of elements of a finite subdirectly irreducible Heyting algebra $A$. When $D=A^2$, we show that the $(\wedge ,\vee )$-canonical formula of $A$ is equivalent to the Jankov formula of $A$. On the other hand, when $D=\varnothing$, the $(\wedge ,\vee )$-canonical formulas produce a new class of si-logics we term stable si-logics. We prove that there are continuum many stable si-logics and that all stable si-logics have the finite model property. We also compare stable si-logics to splitting and subframe si-logics.

Kreisel’s set-theoretic problem is the problem as to whether any logical consequence of ZFC is ensured to be true. Kreisel and Boolos both proposed an answer, taking truth to mean truth in the background set-theoretic universe. This article advocates another answer, which lies at the level of models of set theory, so that truth remains the usual semantic notion. The article is divided into three parts. It first analyzes Kreisel’s set-theoretic problem and proposes one way in which any model of set theory can be compared to a background universe and shown to contain internal models. It then defines logical consequence with respect to a model of ZFC, solves the model-scaled version of Kreisel’s set-theoretic problem, and presents various further results bearing on internal models. Finally, internal models are presented as accessible worlds, leading to an internal modal logic in which internal reflection corresponds to modal reflexivity, and resplendency corresponds to modal axiom 4.

If $\mathcal{F}$ is a filter on $\omega$, we say that $\mathcal{F}$ is Canjar if the corresponding Mathias forcing does not add a dominating real. We prove that any Borel Canjar filter is $F_{\sigma }$, solving a problem of Hrušák and Minami. We give several examples of Canjar and non-Canjar filters; in particular, we construct a $\mathsf{MAD}$ family such that the corresponding Mathias forcing adds a dominating real. This answers a question of Brendle. Then we prove that in all the “classical” models of $\mathsf{ZFC}$ there are $\mathsf{MAD}$ families whose Mathias forcing does not add a dominating real. We also study ideals generated by branches, and we uncover a close relation between Canjar ideals and the selection principle $S_{\mathrm{fin}}(\Omega ,\Omega )$ on subsets of the Cantor space.

Let $L$ be a countable language, let $\mathcal{I}$ be an isomorphism-type of countable $L$-structures, and let $a\in 2^{\omega }$. We say that $\mathcal{I}$ is $a$-strange if it contains a computable-from-$a$ structure and its Scott rank is exactly ${\omega }_1^a$. For all $a$, $a$-strange structures exist. Theorem (AD): If $\mathcal{C}$ is a collection of ${\aleph }_1$ isomorphism-types of countable structures, then for a Turing cone of $a$’s, no member of $\mathcal{C}$ is $a$-strange.

Logical inferentialists contend that the meanings of the logical constants are given by their inference rules. Not just any rules are acceptable, however: inferentialists should demand that inference rules must reflect reasoning in natural language. By this standard, I argue, the inferentialist treatment of quantification fails. In particular, the inference rules for the universal quantifier contain free variables, which find no answer in natural language. I consider the most plausible natural language correlate to free variables—the use of variables in the language of informal mathematics—and argue that it lends inferentialism no support.

Hindman’s theorem says that every finite coloring of the natural numbers has a monochromatic set of finite sums. A Ramsey algebra is a structure that satisfies an analogue of Hindman’s theorem. In this paper, we present the basic notions of Ramsey algebras by using terminology from mathematical logic. We also present some results regarding classification of Ramsey algebras.

We present a new proof of the existence of Morley sequences in simple theories. We avoid using the Erdős–Rado theorem and instead use only Ramsey’s theorem and compactness. The proof shows that the basic theory of forking in simple theories can be developed using only principles from “ordinary mathematics,” answering a question of Grossberg, Iovino, and Lessmann, as well as a question of Baldwin.

We show that the number of types of sequences of tuples of a fixed length can be calculated from the number of $1$-types and the length of the sequences. Specifically, if $\kappa \le \lambda$, then

$${sup\hspace{0.17em}}_{\Vert M\Vert =\lambda }\left|S^{\kappa }\right(M\left)\right|=\left({sup\hspace{0.17em}}_{\Vert M\Vert =\lambda }\right|S^1\left(M\right)|{)}^{\kappa }.$$ We show that this holds for any abstract elementary class with $\lambda$-amalgamation. No such calculation is possible for nonalgebraic types. However, we introduce a subclass of nonalgebraic types for which the same upper bound holds.