We study expressive power of continuous logic in classes of metric groups defined by properties of their actions. We concentrate on unbounded continuous actions on metric spaces. For example, we consider the properties non‐OB, non‐FH and non‐FR.

We give a general framework for the treatment of perturbations of types and structures in continuous logic, allowing to specify which parts of the logic may be perturbed. We prove that separable, elementarily equivalent structures which are approximately $aleph_0$-saturated up to arbitrarily small perturbations are isomorphic up to arbitrarily small perturbations. As a corollary, we obtain a Ryll-Nardzewski style characterisation of complete theories all of whose separable models are isomorphic up to arbitrarily small perturbations.

We give a formalism for approximate isomorphism in continuous logic simultaneously generalizing those of two papers by Ben Yaacov [2] and by Ben Yaacov, Doucha, Nies, and Tsankov [6], which are largely incompatible. With this we explicitly exhibit Scott sentences for the perturbation systems of the former paper, such as the Banach‐Mazur distance and the Lipschitz distance between metric spaces. Our formalism is simultaneously characterized syntactically by a mild generalization of perturbation systems and semantically by certain elementary classes of two‐sorted (...) structures that witness approximate isomorphism. As an application, we show that the theory of any ‐tree or ultrametric space of finite radius is stable, improving a result of Carlisle and Henson [8]. (shrink)

We study a real valued propositional logic with unbounded positive and negative truth values that we call ‐valued logic. Such a logic is semantically equivalent to continuous propositional logic, with a different choice of connectives. After presenting the deduction machinery and the semantics of ‐valued logic, we prove a completeness theorem for finite theories. Then we define unital and Archimedean theories, in accordance with the theory of Riesz spaces. In the unital setting, we prove the equivalence of consistency and satisfiability (...) and an approximated completeness theorem similar to the one that holds for continuous propositional logic. Eventually, among unital theories, we characterize Archimedean theories as those for which strong completeness holds. We also point out that ‐valued logic provides alternative calculi for Łukasiewicz logic and for propositional continuous logic. (shrink)