Abstract: The ability of graph neural networks (GNNs) for distinguishing nodes in graphs has been recently characterized in terms of the Weisfeiler-Lehman (WL) test for checking graph isomorphism. This characterization, however, does not settle the issue of which Boolean node classifiers (i.e., functions classifying nodes in graphs as true or false) can be expressed by GNNs. We tackle this problem by focusing on Boolean classifiers expressible as formulas in the logic FOC2, a well-studied fragment of first order logic. FOC2 is tightly related to the WL test, and hence to GNNs. We start by studying a popular class of GNNs, which we call AC-GNNs, in which the features of each node in the graph are updated, in successive layers, only in terms of the features of its neighbors. We show that this class of GNNs is too weak to capture all FOC2 classifiers, and provide a syntactic characterization of the largest subclass of FOC2 classifiers that can be captured by AC-GNNs. This subclass coincides with a logic heavily used by the knowledge representation community. We then look at what needs to be added to AC-GNNs for capturing all FOC2 classifiers. We show that it suffices to add readout functions, which allow to update the features of a node not only in terms of its neighbors, but also in terms of a global attribute vector. We call GNNs of this kind ACR-GNNs. We experimentally validate our findings showing that, on synthetic data conforming to FOC2 formulas, AC-GNNs struggle to fit the training data while ACR-GNNs can generalize even to graphs of sizes not seen during training.