On the Expressive Power of Inquisitive Team Logic and Inquisitive First-Order Logic

Inquisitive team logic is a variant of inquisitive logic interpreted in team semantics, which has been argued to provide a natural setting for the regimentation of dependence claims.

With respect to sentences, this logic is known to be expressively equivalent with first-order logic.

In this article we show that, on the contrary, the expressive power of open formulas in this logic properly exceeds that of first-order logic.

On the way to this result, we show that if inquisitive team logic is extended with the range-generating universal quantifier adopted in dependence logic, the resulting logic can express finiteness; as a consequence, this logic is not compact and has non-arithmetic complexity.

We further extend our results to standard inquisitive first-order logic, showing that some sentences of this logic express non first-order properties of models, thus settling an open problem from the literature.