On the expressive power of inquisitive team logic and inquisitive first-order logic

TL;DR

This paper demonstrates that inquisitive team logic's open formulas are more expressive than first-order logic, and extending it with certain quantifiers leads to non-compactness and non-recursive axiomatizability.

Contribution

It proves that inquisitive team logic's open formulas surpass first-order logic in expressive power and explores the implications of extending the logic with range-generating quantifiers.

Findings

Open formulas in inquisitive team logic exceed first-order logic in expressiveness.

Adding range-generating quantifiers allows expressing finiteness, leading to non-compactness.

Some sentences in inquisitive first-order logic express properties beyond first-order logic.

Abstract

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, and as a consequence, it is neither compact nor recursively axiomatizable. 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.