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We present syntactic characterisations for the union closed fragments ofexistential second-order logic and of logics with team semantics. Since unionclosure is a semantical and undecidable property, the normal form we introduceenables the handling and provides a better understanding of this fragment. Wealso introduce inclusion-exclusion games that turn out to be precisely thecorresponding model-checking games. These games are not only interesting intheir own right, but they also are a key factor towards building a bridgebetween the semantic and syntactic fragments. On the level of logics with teamsemantics we additionally present restrictions of inclusion-exclusion logic tocapture the union closed fragment. Moreover, we define a team based atom thatwhen adding it to first-order logic also precisely captures the union closedfragment of existential second-order logic which answers an open question byGalliani and Hella.

On the Union Closed Fragment of Existential Second-Order Logic and Logics with Team Semantics