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LOGICALITY AND MODEL CLASSES

Part of: Model theory Philosophical aspects of logic and foundations

Published online by Cambridge University Press:  26 July 2021

JULIETTE KENNEDY  and
JOUKO VÄÄNÄNEN
JULIETTE KENNEDY
Affiliation:
DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF HELSINKIHELSINKI, FINLANDE-mail: [email protected]
JOUKO VÄÄNÄNEN
Affiliation:
DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF HELSINKIHELSINKI, FINLAND and ILLC, UNIVERSITY OF AMSTERDAM AMSTERDAM, NETHERLANDS E-mail: [email protected]
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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 [46]. 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 $L_{\infty \infty }$ if the expression is allowed to depend on the cardinality of the model, based on replacing $L_{\infty \infty }$ by a “tamer” logic.

Keywords

logicalitymodel classgeneralized quantifierinfinitary logic

MSC classification

Primary: 03C07: Basic properties of first-order languages and structures
Secondary: 03A05: Philosophical and critical
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 27 , Issue 4 , December 2021 , pp. 385 - 414
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Copyright
The Author(s), 2021. Published by Cambridge University Press on behalf of Association for Symbolic Logic

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