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 \emph{negative-constraint queries}, their retrieval performance could be catastrophic. To address the issue, we propose a neuro-symbolic information retrieval method, namely \textbf{NS-IR}, that leverages first-order logic (FOL) to optimize the embeddings of naive natural language by considering the \emph{logical consistency} between queries and documents. Specifically, we introduce two novel techniques, \emph{logic alignment} and \emph{connective constraint}, to rerank candidate documents, thereby enhancing retrieval relevance. Furthermore, we construct a new dataset \textbf{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.