With the wide-adoption of edge computing, the functionalities of Internet of Things (IoT) devices can be encapsulated as edge services, to facilitate domain applications through edge service compositions. Considering the capacity-fluctuating and resource-varying of IoT devices, edge service monitoring is essential to guarantee the healthy of their compositions at runtime. Current techniques focus mostly on the monitoring of atomic edge services, which, however, are inadequate for that of inter-and composite services. Besides, constraints to be monitored are usually pre-specified, although certain parameters may have to be adapted online according to execution context. To address these challenges, this paper proposes a novel logic-based adaptive monitoring mechanism, to achieve the interpretation of temporal constraints and time-dependent QoS constraints upon intra-, inter-, and composite services. Leveraging our proposed Compositional Signal Temporal Logic (CSTL) with extended compositional modalities and online parameter settings, constraints can be converted to CSTL formulae, and QoS variations and temporal violations are interpreted qualitatively and quantitatively at runtime. Extensive experiments are conducted upon publicly-available datasets, and evaluation results demonstrate that our CSTL performs better than baseline techniques in terms of expressiveness, applicability, and robustness.