One of the most fruitful applications of substructural logics stems from their capacity to deal with self-referential paradoxes, especially truth-theoretic paradoxes. Both the structural rules of contraction and the rule of cut play a crucial role in typical paradoxical arguments. In this paper I address a number of difficulties affecting noncontractive approaches to paradox that have been discussed in the recent literature. The situation was roughly this: if you decide to go substructural, the nontransitive approach to truth offers a lot of benefits that are not available in the noncontractive account. I sketch a noncontractive theory of truth that has these benefits. In particular, it has both a proof- and a model-theoretic presentation, it can be extended to a first-order language, and it retains every classically valid inference.

We construct creature forcings with strong Axiom A that specialize a given Aronszajn tree. We work with tree creature forcing. The creatures that live on the Aronszajn tree are normed and have the halving property. We show that our models fulfill $${\aleph }_1=\mathfrak{d}<unif(\mathcal{M})={\aleph }_2=2^{\omega }.$$

This paper aims to answer the question of whether or not Frege’s solution limited to value-ranges and truth-values proposed to resolve the “problem of indeterminacy of reference” in Section 10 of Grundgesetze is a violation of his principle of complete determination, which states that a predicate must be defined to apply for all objects in general. Closely related to this doubt is the common allegation that Frege was unable to solve a persistent version of the Caesar problem for value-ranges. It is argued that, in Frege’s standards of reducing arithmetic to logic, his solution to the indeterminacy does not give rise to any sort of Caesar problem in the book.

Nous isolons des propriétés valables dans certaines théories de purs corps ou de corps munis d’opérateurs afin de montrer qu’une théorie est simple lorsque les clôtures définissables et algébriques sont contrôlées par une théorie stable associée.

In this article, we mimic the proof of the simplicity of the theory ACFA of generic difference fields in order to provide a criterion, valid for certain theories of pure fields and fields equipped with operators, which shows that a complete theory is simple whenever its definable and algebraic closures are controlled by an underlying stable theory.

Working under large cardinal assumptions such as supercompactness, we study the Borel reducibility between equivalence relations modulo restrictions of the nonstationary ideal on some fixed cardinal $\kappa$. We show the consistency of $E_{\lambda \text{-}\mathrm{club}}^{{\lambda }^{++},{\lambda }^{++}}$, the relation of equivalence modulo the nonstationary ideal restricted to $S_{\lambda }^{{\lambda }^{++}}$ in the space $({\lambda }^{++}{)}^{{\lambda }^{++}}$, being continuously reducible to $E_{{\lambda }^{+}\text{-}\mathrm{club}}^{2,{\lambda }^{++}}$, the relation of equivalence modulo the nonstationary ideal restricted to $S_{{\lambda }^{+}}^{{\lambda }^{++}}$ in the space $2^{{\lambda }^{++}}$. Then we show that for $\kappa$ ineffable $E_{reg}^{2,\kappa }$, the relation of equivalence modulo the nonstationary ideal restricted to regular cardinals in the space $2^{\kappa }$ is ${\Sigma }_1^1$-complete. We finish by showing that, for ${\Pi }_2^1$-indescribable $\kappa$, the isomorphism relation between dense linear orders of cardinality $\kappa$ is ${\Sigma }_1^1$-complete.

This article proposes the axiomatizations of contingency logics of various natural classes of neighborhood frames. In particular, by defining a suitable canonical neighborhood function, we give sound and complete axiomatizations of monotone contingency logic and regular contingency logic, thereby answering two open questions raised by Bakhtiari, van Ditmarsch, and Hansen. The canonical function is inspired by a function proposed by Kuhn in 1995. We show that Kuhn’s function is actually equal to a related function originally given by Humberstone.

We give a short proof of the Marker–Steinhorn theorem for o-minimal expansions of ordered groups. The key tool is Ramakrishnan’s classification of definable linear orders in such structures.

We study classes of right-angled Coxeter groups with respect to the strong submodel relation of a parabolic subgroup. We show that the class of all right-angled Coxeter groups is not smooth and establish some general combinatorial criteria for such classes to be abstract elementary classes (AECs), for them to be finitary, and for them to be tame. We further prove two combinatorial conditions ensuring the strong rigidity of a right-angled Coxeter group of arbitrary rank. The combination of these results translates into a machinery to build concrete examples of AECs satisfying given model-theoretic properties. We exhibit the power of our method by constructing three concrete examples of finitary classes. We show that the first and third classes are nonhomogeneous and that the last two are tame, uncountably categorical, and axiomatizable by a single $L_{{\omega }_1,\omega }$-sentence. We also observe that the isomorphism relation of any countable complete first-order theory is $\kappa$-Borel reducible (in the sense of generalized descriptive set theory) to the isomorphism relation of the theory of right-angled Coxeter groups whose Coxeter graph is an infinite random graph.

We study the degree structure of the $\omega$-c.e., $n$-c.e., and ${\Pi }_1^0$ equivalence relations under the computable many-one reducibility. In particular, we investigate for each of these classes of degrees the most basic questions about the structure of the partial order. We prove the existence of the greatest element for the $\omega$-c.e. and $n$-computably enumerable equivalence relations. We provide computable enumerations of the degrees of $\omega$-c.e., $n$-c.e., and ${\Pi }_1^0$ equivalence relations. We prove that for all the degree classes considered, upward density holds and downward density fails.