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 in-
tertranslatability in general. In this paper, we show that their generalization
is not transitive and hence it is not an equivalence relation. Then we intro-
duce 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 generaliza-
tion 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.
and Halvorson introduce a straightforward generalization to non-disjoint
languages and they show that their generalization is not equivalent to in-
tertranslatability in general. In this paper, we show that their generalization
is not transitive and hence it is not an equivalence relation. Then we intro-
duce 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 generaliza-
tion 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.
Original language | English |
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Pages (from-to) | 709-729 |
Number of pages | 21 |
Journal | Journal of Philosophical Logic |
Volume | 48 |
Issue number | 4 |
Early online date | 24 Oct 2018 |
DOIs | |
Publication status | Published - 15 Aug 2019 |
Keywords
- Definability theory
- Definitional equivalence
- First-order logic
- Logical interpretation
- Logical translation