Information retrieval plays a crucial role in resource localization. Current dense retrievers retrieve the relevant documents within a corpus via embedding similarities, which compute similarities between dense vectors mainly depending on word co-occurrence between queries and documents, but overlook the real query intents. Thus, they often retrieve numerous irrelevant documents. Particularly in the scenarios of complex queries such as negative-constraint queries, their retrieval performance could be catastrophic. To address the issue, we propose a neuro-symbolic information retrieval method, namely NS-IR, that leverages first-order logic (FOL) to optimize the embeddings of naive natural language by considering the logical consistency between queries and documents. Specifically, we introduce two novel techniques, logic alignment and connective constraint, to re-rank candidate documents, thereby enhancing retrieval relevance. Furthermore, we construct a new dataset NegConstraint including negative-constraint queries to evaluate our NS-IR’s performance on such complex IR scenarios. Our extensive experiments demonstrate that NS-IR not only achieves superior zero-shot retrieval performance on web search and low-resource retrieval tasks, but also performs better on negative-constraint queries. Our scource code and dataset are available at https://github.com/xgl-git/NS-IR-main.