Runtime monitors analyze system execution traces for policy compliance. Monitors for propositional specification languages, such as metric temporal logic (MTL), produce Boolean verdicts denoting whether the policy is satisfied or violated at a given point in the trace. Given a sufficiently complex policy, it can be difficult for the monitor’s user to understand how the monitor arrived at its verdict. We develop an MTL monitor that outputs verdicts capturing why the policy was satisfied or violated. Our verdicts are proof trees in a sound and complete proof system that we design. We demonstrate that such verdicts can serve as explanations for end users by augmenting our monitor with a graphical interface for the interactive exploration of proof trees. As a second application, our verdicts serve as certificates in a formally verified checker we develop using the Isabelle proof assistant.