Logicality and Model Classes

Kennedy, Juliette; Väänánen, Jouko

doi:10.1017/bsl.2021.42

Cited by 5 publications

(5 citation statements)

References 45 publications

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“…The best known of these combinations are: Löwenheim-Skolem theorem + Compactness Löwenheim-Skolem theorem + Recursively enumerable set of validities These are by no means exhaustive though (the reader can consult the encyclopaedic monograph [3] for a thorough treatment of this topic). Philosophically, these results have been interpreted as providing a case for first-order logic being the "right" logic in contrast to higher-order, infinitary, or logics with generalized quantifiers, which can be argued to be more mathematical beasts (see [19,25]). An implicit assumption of Lindström's work is that identity (=) is a most basic notion and belongs in the base logic.…”

Section: Introductionmentioning

confidence: 99%

Maximality of logic without identity

Badía¹,

Caicedo²,

Noguera³

2022

Preprint

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Lindström theorem obviously fails as a characterization of L − ωω , first-order logic without identity. In this note we provide a fix: we show that L − ωω is maximal among abstract logics satisfying a weak form of the isomorphism property (suitable for identityfree languages and studied in [10]), the Löwenheim-Skolem property, and compactness. Furthermore, we show that compactness can be replaced by being recursively enumerable for validity under certain conditions. In the proofs we use a form of strong upwards Löwenheim-Skolem theorem not available in the framework with identity.

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Section: Introductionmentioning

confidence: 99%

Maximality of logic without identity

Badía¹,

Caicedo²,

Noguera³

2022

Preprint

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“…See(Kennedy and Väänänen, 2021) for refinements of this result and(Bonnay and Engström, 2018) for a more systematic investigation.…”

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confidence: 93%

Logical Constants and Arithmetical Forms

Speitel

2023

LLP

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This paper reflects on the limits of logical form set by a novel criterion of logicality proposed in (Bonnay and Speitel, 2021). The interest stems from the fact that the delineation of logical terms according to the criterion exceeds the boundaries of standard first-order logic. Among ‘novel’ logical terms is the quantifier “there are infinitely many”. Since the structure of the natural numbers is categorically characterisable in a language including this quantifier we ask: does this imply that arithmetical forms have been reduced to logical forms? And, in general, what other conditions need to be satisfied for a form to qualify as “fully logical”? We survey answers to these questions.

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“…The best known of these combinations are: This list is by no means exhaustive though (the reader can consult the encyclopaedic monograph [3] for a thorough treatment of this topic). Philosophically, these results have been interpreted as providing a case for first-order logic being the “right” logic in contrast to higher-order, infinitary, or logics with generalized quantifiers, which can be argued to be more mathematical beasts (see [21, 29]). An implicit assumption of Lindström’s work is that identity ( ) belongs in the base logic.…”

Section: Introductionmentioning

confidence: 99%

Maximality of Logic Without Identity

Badía¹,

Caicedo²,

Noguera

2023

J. symb. log.

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Lindström’s theorem obviously fails as a characterization of first-order logic without identity ( $\mathcal {L}_{\omega \omega }^{-} $ ). In this note, we provide a fix: we show that $\mathcal {L}_{\omega \omega }^{-} $ is a maximal abstract logic satisfying a weak form of the isomorphism property (suitable for identity-free languages and studied in [11]), the Löwenheim–Skolem property, and compactness. Furthermore, we show that compactness can be replaced by being recursively enumerable for validity under certain conditions. In the proofs, we use a form of strong upwards Löwenheim–Skolem theorem not available in the framework with identity.

show abstract

Logicality and Model Classes

Cited by 5 publications

References 45 publications

Maximality of logic without identity

Maximality of logic without identity

Logical Constants and Arithmetical Forms

Maximality of Logic Without Identity

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