Abstract: Isomorphism invariance is a popular and well motivated criterion of logicality. Its extensional adequacy, however, has been contested on different grounds. Some authors have argued that the criterion overgenerates by counting as logical notions which, intuitively, seem to belong to the realm of mathematics. Other authors have contested that the criterion undergenerates by failing to count as logical notions which ought to be so regarded. I argue that isomorphism invariance also faces a more radical objection, from indeterminacy. In the context of positive free logic (PFL), there appear to be two admissible precisifications of isomorphism invariance to test quantification for logicality. These warrant contradictory claims about the logical status of quantifiers. I conclude that the logical status of PFL-quantifiers, given isomorphism invariance, is unknowable and thus indeterminate. I will consider two objections aiming to establish one precisification of isomorphism invariance as the intended one, but I found both of them unconvincing.