Logicality and Model Classes

TL;DR

This paper explores when properties of models are considered logical, relating model-theoretic features of logics to the concept of logicality, and refines existing results on expressing logical properties within certain logical frameworks.

Contribution

It introduces a graded notion of logicality based on model-theoretic characteristics and refines McGee's result on expressing model properties in extended logics.

Findings

Logical properties are preserved under isomorphisms.

Closer similarity to first-order logic increases logicality.

A tamer logic can express model properties depending on cardinality.

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 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.