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In this paper we analyze k-ary inclusion-exclusion logic, INEX[k], which isobtained by extending first order logic with k-ary inclusion and exclusionatoms. We show that every formula of INEX[k] can be expressed with a formula ofk-ary existential second order logic, ESO[k]. Conversely, every formula ofESO[k] with at most k-ary free relation variables can be expressed with aformula of INEX[k]. From this it follows that, on the level of sentences,INEX[k] captures the expressive power of ESO[k].  We also introduce several useful operators that can be expressed in INEX[k].In particular, we define inclusion and exclusion quantifiers and so-called termvalue preserving disjunction which is essential for the proofs of the mainresults in this paper. Furthermore, we present a novel method of relativizationfor team semantics and analyze the duality of inclusion and exclusion atoms.

Capturing k-ary Existential Second Order Logic with k-ary Inclusion-Exclusion Logic