We study theorems of ordered groups from the perspective of reverse mathematics. We show that $\mathit{RCA}_0$ suffices to prove Hölder's Theorem and give equivalences of both $\mathit{WKL}_0$ (the orderability of torsion free nilpotent groups and direct products, the classical semigroup conditions for orderability) and $\mathit{ACA}_0$ (the existence of induced partial orders in quotient groups, the existence of the center, and the existence of the strong divisible closure).

In this paper we deal with the logic ${\cal M\kern-1pt L}_{\omega_1}$ which is the infinitary extension of propositional modal logic that has conjunctions and disjunctions only for countable sets of formulas. After introducing some basic concepts and tools from modal logic, we modify Makkai's generalization of the notion of consistency property to make it fit for modal purposes. Using this construction as a universal instrument, we prove, among other things, interpolation for ${\cal M\kern-1pt L}_{\omega_1}$ as well as preservation results for universal, existential, and positive ${\cal M\kern-1pt L}_{\omega_1}$-formulas.

In "Intuitionistic validity in $T$-normal Kripke structures," Buss asked whether every intuitionistic theory is, for some classical theory $T$, that of all $T$-normal Kripke structures ${\cal H}(T)$ for which he gave an r.e. axiomatization. In the language of arithmetic $\mathit{Iop}$ and $\mathit{Lop}$ denote PA$^{-}$ plus Open Induction or Open LNP, $\mathit{iop}$ and $\mathit{lop}$ are their intuitionistic deductive closures. We show $\mathcal{H}\mathit{(Iop)}$ $=\mathit{lop}$ is recursively axiomatizable and $\mathit{lop}\vdash_{i\ c}\dashv \mathit{iop}$, while $i\forall_{1}\not\vdash \mathit{lop}$. If $iT$ proves PEM $_{\mathrm{atomic}}$ but not totality of a classically provably total Diophantine function of $T$, then $\mathcal{H}(T)\not\subseteq iT$ and so $iT\not\in \mathrm{range({\cal H})}$. A result due to Wehmeier then implies $i\Pi_{1}\not\in\mbox{range}({\cal H})$. We prove $\mathit{Iop}$ is not $\forall_{2}$-conservative over $i\forall_{1}$. If $\mathit{Iop}\subseteq T\subseteq I\forall_{1}$, then $iT$ is not closed under MR $_{\mathrm{open}}$ or Friedman's translation, so $iT\not\in$ range (${\cal H}$). Both $\mathit{Iop}$ and $I\forall_{1}$ are closed under the negative translation.

The laws of distribution follow at once from Lyndon's interpolation theorem and the fact that the fallacy of many terms is a fallacy.

The semantics of e-models for tense logics with binary operators for `until' and `since' (US-logics) was introduced by Bellissima and Bucalo in 1995. In this paper we show the adequacy of these semantics by proving a general Henkin-style completeness theorem. Moreover, we show that for these semantics there holds a Stone-like duality theorem with the algebraic structures that naturally arise from US-logics.

The idea of a property's being supervenient on a class of properties is familiar from much philosophical literature. We give this idea a linguistic turn by converting it into the idea of a predicate symbol's being supervenient on a set of predicate symbols relative to a (first order) theory. What this means is that according to the theory, any individuals differing in respect to whether the given predicate applies to them also differ in respect to the application of at least one of the predicates in the set. The latter relationship we show turns out to coincide with something antecedently familiar from work on definability: with what is called the piecewise (or modelwise) definability, in the theory in question, of the given predicate in terms of those in the set.

We investigate transfer of interpolation in such combinations of modal logic which lead to interaction of the modalities. Combining logics by taking products often blocks transfer of interpolation. The same holds for combinations by taking unions, a generalization of Humberstone's inaccessibility logic. Viewing first-order logic as a product of modal logics, we derive a strong counterexample for failure of interpolation in the finite variable fragments of first-order logic. We provide a simple condition stated only in terms of frames and bisimulations which implies failure of interpolation. Its use is exemplified in a wide range of cases.

Parsons has given a (nonconstructive) proof that the first-order fragment of the system of Frege's Grundgesetze is consistent. Here a constructive proof of the same result is presented.