Cloud API recommender system has emerged as a promising solution to address the overload problem caused by the overwhelming growth of cloud APIs, aiming to improve software development efficiency. However, the incremental development nature of the service-oriented software necessitates an adaptive complementary cloud API recommender system, while existing systems purely focus on generating single-function, high-quality, and personalized recommendations based on retrieval content, quality of service, or user preferences, and overlook the practical scenarios that recommend high-order complementary cloud APIs for developers. This paper addresses this gap by providing a logical reasoning network for high-order complementary cloud API recommendation (LRN4HCAR). We first construct three relation graphs to characterize cloud API co-invocation, function co-occurrence, and substitute relations, capturing the strong complementarity, weak complementarity, and non-substitute relations among cloud APIs. Next, we develop a high-order complementary logical reasoning network with three sub-networks: strong complementary logical reasoning, weak complementary logical reasoning, and non-substitute logical reasoning. This network enables the recommendation of cloud APIs with high-order complementary relationships while excluding substitutes. Utilizing LRN4HCAR, we evaluate classic and SOTA baselines across various complementary recommendation scenarios on two real-world datasets. Experimental results demonstrate that LRN4HCAR outperforms others in all experimental settings for complementary cloud API recommendations. We also verify the ability of LRN4HCAR to handle substitute noise and improve the visibility of long-tail cloud APIs in the recommendation results. The implementation code has made publicly available athttps://github.com/hey-mem/LRN4HCAR.