# Logicality and Model Classes

## Juliette Kennedy, Helsinki

When is a property of a model a logical property? According to the so-called Tarski-Sher criterion this is the case when the property is preserved by isomorphisms. We relate this to the model-theoretic characteristics of abstract logics in which the model class is definable, resulting in a *graded* concept of logicality (in the terminology of Sagi’s paper “Logicality and meaning”). We consider which characteristics of logics, such as variants of the Löwenheim-Skolem Theorem, Completeness Theorem, and absoluteness, are relevant from the logicality point of view, continuing earlier work by Bonnay, Feferman, and Sagi. We suggest that a logic is the more logical the closer it is to first order logic, and offer a refinement of the result of McGee that logical properties of models can be expressed in *L*_{∞∞} if the expression is allowed to depend on the cardinality of the model, based on replacing *L*_{∞∞} by a “tamer” logic. This is joint work with Jouko Väänänen.