We study a hierarchy of logics where each formula of each logic in the hierarchy consists of a formula of a certain fragment of transitive closure logic prefixed with an existentially quantified tuple of unary relation symbols. By playing an Ehrenfeucht-Fraïssé game specifically developed for our logics, we prove that there are problems definable in the second level of our hierarchy that are not definable in the first; and that if we are to prove that the hierarchy is proper in its entirety (or even that the third level does not collapse to the second) then we shall require substantially different constructions than those used previously to show that the hierarchy is indeed proper in the absence of the existentially quantified second-order symbols.

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