This paper investigates a set of issues connected with the so-called conservativeness argument against deflationism. Although I do not defend that argument, I think the discussion of it has raised some interesting questions about whether what I call “compositional principles,” such as “a conjunction is true iff its conjuncts are true,” have substantial content or are in some sense logically trivial. The paper presents a series of results that purport to show that the compositional principles for a first-order language, taken together, have substantial logical strength, amounting to a kind of abstract consistency statement.

Let $\mathcal{K}$ be a family of structures, closed under isomorphism, in a fixed computable language. We consider effective lists of structures from $\mathcal{K}$ such that every structure in $\mathcal{K}$ is isomorphic to exactly one structure on the list. Such a list is called a computable classification of $\mathcal{K}$, up to isomorphism. Using the technique of Friedberg enumeration, we show that there is a computable classification of the family of computable algebraic fields and that with a $\mathbf{0\text{'}}$-oracle, we can obtain similar classifications of the families of computable equivalence structures and of computable finite-branching trees. However, there is no computable classification of the latter, nor of the family of computable torsion-free abelian groups of rank $1$, even though these families are both closely allied with computable algebraic fields.

It is widely thought that the acceptability of an abstraction principle is a feature of the cardinalities at which it is satisfiable. This view is called into question by a recent observation by Richard Heck. We show that a fix proposed by Heck fails but we analyze the interesting idea on which it is based, namely that an acceptable abstraction has to “generate” the objects that it requires. We also correct and complete the classification of proposed criteria for acceptable abstraction.

By rephrasing quantifier-free axioms as rules of derivation in sequent calculus, we show that the generalized Steiner–Lehmus theorem admits a direct proof in classical logic. This provides a partial answer to a question raised by Sylvester in 1852. We also present some comments on possible intuitionistic approaches.

We consider the set of jumps below a Turing degree, given by $\mathsf{JB}\left(\mathbf{a}\right)=\{\mathbf{x}\text{'}:\mathbf{x}\le \mathbf{a}\}$, with a focus on the problem: Which recursively enumerable (r.e.) degrees $\mathbf{a}$ are uniquely determined by $\mathsf{JB}\left(\mathbf{a}\right)$? Initially, this is motivated as a strategy to solve the rigidity problem for the partial order $\mathcal{R}$ of r.e. degrees. Namely, we show that if every high${}_2$ r.e. degree $\mathbf{a}$ is determined by $\mathsf{JB}\left(\mathbf{a}\right)$, then $\mathcal{R}$ cannot have a nontrivial automorphism. We then defeat the strategy—at least in the form presented—by constructing pairs ${\mathbf{a}}_0$, ${\mathbf{a}}_1$ of distinct r.e. degrees such that $\mathsf{JB}\left({\mathbf{a}}_0\right)=\mathsf{JB}\left({\mathbf{a}}_1\right)$ within any possible jump class $\{\mathbf{x}:\mathbf{x}\text{'}=\mathbf{c}\}$. We give some extensions of the construction and suggest ways to salvage the attack on rigidity.

The dual character of invariance under transformations and definability by some operations has been used in classical works by, for example, Galois and Klein. Following Tarski, philosophers of logic have claimed that logical notions themselves could be characterized in terms of invariance. In this article, we generalize a correspondence due to Krasner between invariance under groups of permutations and definability in ${\mathcal{L}}_{\infty \infty }$ so as to cover the cases (quantifiers, logics without equality) that are of interest in the logicality debates, getting McGee’s theorem about quantifiers invariant under all permutations and definability in pure ${\mathcal{L}}_{\infty \infty }$ as a particular case. We also prove some optimality results along the way, regarding the kinds of relations which are needed so that every subgroup of the full permutation group is characterizable as a group of automorphisms.