Cultural image retrieval is essential for organizing and analyzing large-scale digital heritage resources. However, existing multimodal retrieval methods usually encode a query as a holistic embedding, making it difficult to preserve logical operators and cultural relations during retrieval. Therefore, we introduce Multimodal Boolean Logic Culture Retrieval, a novel task that moves beyond general visual similarity by enforcing strict Boolean constraints over structured cultural elements. To tackle this, we propose *CLAIR*, a training-free framework that reformulates the retrieval objective to seamlessly integrate logical operations directly within the multimodal embedding space. Furthermore, we construct CulTi2.0, a dedicated benchmark designed to evaluate diverse logical query forms based on cultural visual atoms. Extensive experiments show that while current multimodal models struggle with explicit logical queries, *CLAIR* provides consistent and generalizable improvements as a plug-and-play solution across various datasets and model families. Code and dataset are publicly available at https://github.com/AI4Culture/CulTi2.0.