The notion of grounding is usually conceived as an objective and explanatory relation. It connects two relata if one-the ground-determines or explains the other-the consequence. In the contemporary literature on grounding, much effort has been devoted to logically characterize the formal aspects of grounding, but a major hard problem remains: defining suitable grounding principles for universal and existential formulae. Indeed, several grounding principles for quantified formulae have been proposed, but all of them are exposed to reflexivity and symmetry paradoxes in some very natural contexts of application. We introduce in this paper a first-order formal system that captures the notion of grounding and avoids, in a novel and non-trivial way, both reflexivity and symmetry paradoxes. The presented system formally develops Bolzano's ideas on grounding by employing Hilbert's ε-terms and an adapted version of Fine's theory of arbitrary objects.