  We ask, 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 model-theoretic characteristics of abstract
logics in which the model class is definable. This results in a graded concept
of logicality in the terminology of Sagi. We investigate which characteristics
of logics, such as variants of the L\"owenheim-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. We also offer a
refinement of the result of McGee that logical properties of models can be
expressed in $L_{\infty\infty}$ if the expression is allowed to depend on the
cardinality of the model, based on replacing $L_{\infty\infty}$ by a ``tamer"
logic.
