This is a survey of work on set-theoretical invariance criteria for logicality. It begins with a review of the Tarski-Sher thesis in terms, first, of permutation invariance over a given domain and then of isomorphism invariance across domains, both characterized by McGee in terms of definability in the language $L_{\omega ,\omega }$. It continues with a review of critiques of the Tarski-Sher thesis and a proposal in response to one of those critiques via homomorphism invariance. That has quite divergent characterization results depending on its formulation, one in terms of FOL, the other by Bonnay in terms of $L_{\omega ,\omega }$, both without equality. From that we move on to a survey of Bonnay's work on similarity relations between structures and his results that single out invariance with respect to potential isomorphism among all such. Turning to the critique that calls for sameness of meaning of a logical operation across domains, the paper continues with a result showing that the isomorphism invariant operations that are absolutely definable with respect to KPU–Inf are exactly those definable in full FOL; this makes use of an old theorem of Manders. The concluding section is devoted to a critical discussion of the arguments for set-theoretical criteria for logicality.

We present a way of classifying the logically possible ways out of Gärdenfors' inconsistency or triviality result on belief revision with conditionals. For one of these ways—conditionals which are not descriptive but which only have an inferential role as being given by the Ramsey test—we determine which of the assumptions in three different versions of Gärdenfors' theorem turn out to be false. This is done by constructing ranked models in which such Ramsey-test conditionals are evaluated and which are subject to natural postulates on belief revision and acceptability sets for conditionals. Along the way we show that in contrast with what Gärdenfors himself proposed, there is no dichotomy of the form: either the Ramsey test has to be given up or the Preservation condition. Instead, both of them follow from our postulates.

We study what the existence of a classical embedding between computable structures implies about the existence of computable embeddings. In particular, we consider the effect of fixing and varying the computable presentations of the computable structures.

In this paper, I start by showing that sorites paradoxes are inclosure paradoxes. That is, they fit the Inclosure Scheme which characterizes the paradoxes of self-reference. Given that sorites and self-referential paradoxes are of the same kind, they should have the same kind of solution. The rest of the paper investigates what a dialetheic solution to sorites paradoxes is like, connections with a dialetheic solution to the self-referential paradoxes, and related issues—especially so called "higher order" vagueness.

We study completely decomposable torsion-free abelian groups of the form ${\mathcal{G}}_S:={\oplus }_{n\in S}{\mathbb{Q}}_{p_n}$ for sets $S\subseteq \omega$. We show that ${\mathcal{G}}_S$has a decidable copy if and only if S is ${\mathrm{\Sigma }}_2^0$and has a computable copy if and only if S is ${\mathrm{\Sigma }}_3^0$.

I describe some paradoxes of ground and relate them to the semantic paradoxes.

Often where an axiomatization of an intensional logic using only finitely many axioms schemes and rules of the simplest kind is unknown, one has a choice between an axiomatization involving an infinite family of axiom schemes and one involving nonstandard "Gabbay-style" rules. The present note adds another example of this phenomenon, pertaining to the logic comparative probability ("p is no more likely than q"). Peter Gärdenfors has produced an axiomatization involving an infinite family of schemes, and here an alternative using a "Gabbay-style" rule is offered. Both axiomatizations depend on the Kraft-Pratt-Seidenberg theorem from measurement theory.

We present an overview of the topics in the title and of some of the key results pertaining to them. These have historically been topics of interest in computability theory and continue to be a rich source of problems and ideas. In particular, we draw attention to the links and connections between these topics and explore their significance to modern research in the field.