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On Generalization of Definitional Equivalence to Non-Disjoint Languages

Abstract

For simplicity, most of the literature introduces the concept of definitional equivalence only for disjoint languages. In a recent paper, Barrett and Halvorson introduce a straightforward generalization to non-disjoint languages and they show that their generalization is not equivalent to intertranslatability in general. In this paper, we show that their generalization is not transitive and hence it is not an equivalence relation. Then we introduce another formalization of definitional equivalence due to Andréka and Németi which is equivalent to the Barrett–Halvorson generalization in the case of disjoint languages. We show that the Andréka–Németi generalization is the smallest equivalence relation containing the Barrett–Halvorson generalization and it is equivalent to intertranslatability, which is another definition for definitional equivalence, even for non-disjoint languages. Finally, we investigate which definitions for definitional equivalences remain equivalent when we generalize them for theories in non-disjoint languages.

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Acknowledgements

The writing of the current paper was induced by questions by Marcoen Cabbolet and Sonja Smets during the public defence of [24]. We are also grateful to Hajnal Andréka, Michèle Friend, Mohamed Khaled, Amedé Lefever, István Németi and Jean Paul Van Bendegem for enjoyable discussions and feedback while writing this paper, as well as to the two anonymous referees who made valuable remarks which helped significantly improving our paper.

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Lefever, K., Székely, G. On Generalization of Definitional Equivalence to Non-Disjoint Languages. J Philos Logic 48, 709–729 (2019). https://doi.org/10.1007/s10992-018-9491-0

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Keywords

  • First-order logic
  • Definability theory
  • Definitional equivalence
  • Logical translation
  • Logical interpretation