Developing neural architectures that are capable of logical reasoning has become increasingly important for a wide range of applications (e.g., natural language processing). Towards this grand objective, we propose a symbolic reasoning architecture that chains many join operators together to model output logical expressions. In particular, we demonstrate that such an ensemble of join-chains can express a broad subset of ''tree-structured'' first-order logical expressions, named FOET, which is particularly useful for modeling natural languages. To endow it with differentiable learning capability, we closely examine various neural operators for approximating the symbolic join-chains. Interestingly, we find that the widely used multi-head self-attention module in transformer can be understood as a special neural operator that implements the union bound of the join operator in probabilistic predicate space. Our analysis not only provides a new perspective on the mechanism of the pretrained models such as BERT for natural language understanding but also suggests several important future improvement directions.