In earlier work by the first and second authors, the equivalence of a finite square principle ${\square }_{\lambda ,D}^{\mathrm{fin}}$ with various model-theoretic properties of structures of size $\lambda$ and regular ultrafilters was established. In this paper we investigate the principle ${\square }_{\lambda ,D}^{\mathrm{fin}}$—and thereby the above model-theoretic properties—at a regular cardinal. By Chang’s two-cardinal theorem, ${\square }_{\lambda ,D}^{\mathrm{fin}}$ holds at regular cardinals for all regular filters $D$ if we assume the generalized continuum hypothesis (GCH). In this paper we prove in ZFC that, for certain regular filters that we call ${\mathit{doubly}}^{+}$ regular, ${\square }_{\lambda ,D}^{\mathrm{fin}}$ holds at regular cardinals, with no assumption about GCH. Thus we get new positive answers in ZFC to Open Problems 18 and 19 in Chang and Keisler’s book Model Theory.

The author has previously shown that for a certain class of structures $\mathcal{I}$, $\mathcal{I}$-indexed indiscernible sets have the modeling property just in case the age of $\mathcal{I}$ is a Ramsey class. We expand this known class of structures from ordered structures in a finite relational language to ordered, locally finite structures which isolate quantifier-free types by way of quantifier-free formulas. This result is applied to give new proofs that certain classes of trees are Ramsey. To aid this project we develop the logic of EM-types.

The “high school algebra” laws of exponentiation fail in the ordinal arithmetic of sets that generalizes the arithmetic of the von Neumann ordinals. The situation can be remedied by using an alternative arithmetic of sets, based on the Zermelo ordinals, where the high school laws hold. In fact the Zermelo arithmetic of sets is uniquely characterized by its satisfying the high school laws together with basic properties of addition and multiplication. We also show how in both arithmetics the behavior of exponentiation depends on whether the empty set is an element of the base.

We characterize when the elementary diagram of a mutually algebraic structure has a model complete theory, and give an explicit description of a set of existential formulas to which every formula is equivalent. This characterization yields a new, more constructive proof that the elementary diagram of any model of a strongly minimal, trivial theory is model complete.

This paper proposes a new topic in substructural logic for use in research joining the fields of relevance and fuzzy logics. For this, we consider old and new relevance principles. We first introduce fuzzy systems satisfying an old relevance principle, that is, Dunn’s weak relevance principle. We present ways to obtain relevant companions of the weakening-free uninorm (based) systems introduced by Metcalfe and Montagna and fuzzy companions of the system R of relevant implication (without distributivity) and its neighbors. The algebraic structures corresponding to the systems are then defined, and completeness results are provided. We next consider fuzzy systems satisfying new relevance principles introduced by Yang. We show that the weakening-free uninorm (based) systems and some extensions and neighbors of R satisfy the new relevance principles.

The definition of a pseudofinite structure can be translated verbatim into continuous logic, but it also gives rise to a stronger notion and to two parallel concepts of pseudocompactness. Our purpose is to investigate the relationship between these four concepts and establish or refute each of them for several basic theories in continuous logic. Pseudofiniteness and pseudocompactness turn out to be equivalent for relational languages with constant symbols, and the four notions coincide with the standard pseudofiniteness in the case of classical structures, but the details appear to be slightly more important here than in the usual translation of definitions from classical logic. We also prove that injective “formula-definable” endofunctions are surjective, and conversely, in strongly pseudofinite omega-saturated structures.