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  1. The Craig Interpolation Theorem in abstract model theory.Jouko Väänänen - 2008 - Synthese 164 (3):401-420.
    The Craig Interpolation Theorem is intimately connected with the emergence of abstract logic and continues to be the driving force of the field. I will argue in this paper that the interpolation property is an important litmus test in abstract model theory for identifying “natural,” robust extensions of first order logic. My argument is supported by the observation that logics which satisfy the interpolation property usually also satisfy a Lindström type maximality theorem. Admittedly, the range of such logics is small.
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  • Compactness, interpolation and Friedman's third problem.Daniele Mundici - 1982 - Annals of Mathematical Logic 22 (2):197.
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  • δ-Logics and generalized quantifiers.J. A. Makowsky - 1976 - Annals of Mathematical Logic 10 (2):155-192.
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  • Algorithmic uses of the Feferman–Vaught Theorem.J. A. Makowsky - 2004 - Annals of Pure and Applied Logic 126 (1-3):159-213.
    The classical Feferman–Vaught Theorem for First Order Logic explains how to compute the truth value of a first order sentence in a generalized product of first order structures by reducing this computation to the computation of truth values of other first order sentences in the factors and evaluation of a monadic second order sentence in the index structure. This technique was later extended by Läuchli, Shelah and Gurevich to monadic second order logic. The technique has wide applications in decidability and (...)
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  • Countable approximations and Löwenheim-Skolem theorems.David W. Kueker - 1977 - Annals of Mathematical Logic 11 (1):57.
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  • Logicality and model classes.Juliette Kennedy & Jouko Väänänen - 2021 - Bulletin of Symbolic Logic 27 (4):385-414.
    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 (...)
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  • Infinitary Logic has No Expressive Efficiency Over Finitary Logic.Matthew Harrison-Trainor & Miles Kretschmer - forthcoming - Journal of Symbolic Logic:1-18.
    We can measure the complexity of a logical formula by counting the number of alternations between existential and universal quantifiers. Suppose that an elementary first-order formula $\varphi $ (in $\mathcal {L}_{\omega,\omega }$ ) is equivalent to a formula of the infinitary language $\mathcal {L}_{\infty,\omega }$ with n alternations of quantifiers. We prove that $\varphi $ is equivalent to a finitary formula with n alternations of quantifiers. Thus using infinitary logic does not allow us to express a finitary formula in a (...)
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  • Chain models, trees of singular cardinality and dynamic ef-games.Mirna Džamonja & Jouko Väänänen - 2011 - Journal of Mathematical Logic 11 (1):61-85.
    Let κ be a singular cardinal. Karp's notion of a chain model of size κ is defined to be an ordinary model of size κ along with a decomposition of it into an increasing union of length cf. With a notion of satisfaction and -isomorphism such models give an infinitary logic largely mimicking first order logic. In this paper we associate to this logic a notion of a dynamic EF-game which gauges when two chain models are chain-isomorphic. To this game (...)
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  • Definability and automorphisms in abstract logics.Xavier Caicedo - 2004 - Archive for Mathematical Logic 43 (8):937-945.
    In any model theoretic logic, Beth’s definability property together with Feferman-Vaught’s uniform reduction property for pairs imply recursive compactness, and the existence of models with infinitely many automorphisms for sentences having infinite models. The stronger Craig’s interpolation property plus the uniform reduction property for pairs yield a recursive version of Ehrenfeucht-Mostowski’s theorem. Adding compactness, we obtain the full version of this theorem. Various combinations of definability and uniform reduction relative to other logics yield corresponding results on the existence of non-rigid (...)
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  • Stationary logic.Jon Barwise - 1978 - Annals of Mathematical Logic 13 (2):171.
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