Abstract
We prove that the expressive power of first-order logic with team semantics plus contradictory negation does not rise beyond that of first-order logic (with respect to sentences), and that the totality atoms of arity k + 1 are not definable in terms of the totality atoms of arity k. We furthermore prove that all first-order nullary and unary dependencies are strongly first-order, in the sense that they do not increase the expressive power of first-order logic if added to it.
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Notes
- 1.
To be more precise, this results holds if we are allowing models over all signatures. The case in which only models over the empty signature are considered is yet open.
- 2.
Since \(Z \subseteq X[F/v]\), such a s always exists. Of course, there may be multiple ones; in that case, we just pick arbitrarily one.
- 3.
On the other hand, if θ were a first-order sentence over the nonempty vocabulary, then it would not be a dependency.
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Acknowledgements
This research was supported by the Deutsche Forschungsgemeinschaft (project number DI 561/6-1). The author thanks an anonymous reviewer for a number of useful corrections and suggestions.
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Galliani, P. (2016). On Strongly First-Order Dependencies. In: Abramsky, S., Kontinen, J., Väänänen, J., Vollmer, H. (eds) Dependence Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-31803-5_4
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