Logicality and Model Classes
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by
Juliette Kennedy, Jouko Väänänen
2021
Abstract
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 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. We also 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.
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