Designing expressive Graph neural networks (GNNs) is an important topic in graph machine learning fields. Traditionally, the Weisfeiler-Lehman (WL) test has been the primary measure for evaluating GNN expressiveness. However, high-order WL tests can be obscure, making it challenging to discern the specific graph patterns captured by them. Given the connection between WL tests and first-order logic, some have explored the logical expressiveness of Message Passing Neural Networks. This paper aims to establish a comprehensive and systematic relationship between GNNs and logic. We propose a framework for identifying the equivalent logical formulas for arbitrary GNN architectures, which not only explains existing models, but also provides inspiration for future research. As case studies, we analyze multiple classes of prominent GNNs within this framework, unifying different subareas of the field. Additionally, we conduct a detailed examination of homomorphism expressivity from a logical perspective and present a general method for determining the homomorphism expressivity of arbitrary GNN models, as well as addressing several open problems.