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.　©　2023　IEEE.