In this paper, we address multi-robot path planning in partially unknown environments with Boolean task specifications. In such environments, the workspace is modeled as a map that contains both known and unknown regions. Robots initially have only a priori information regarding the connectivity confidence within the unknown region, while the exact connectivity is revealed only when the robot physically traverses the region. First, a Petri net system that contains existing transitions and hypothetical transitions is designed to represent the partially unknown environments. Then, an improved sequential single-item auction algorithm that provides cost-efficient task allocation based on the expected task cost is proposed using the Petri net system. Additionally, a precomputed motion strategy graph that enables robots to autonomously select paths online using real-time sensor feedback is developed. Finally, a hardware case study involving multi-robot collaboration demonstrates the effectiveness and adaptability of the proposed method.