We present a new logic-based framework for modeling and automatically verifying trust in Multi-Agent Systems (MASs). We start by refining TCTL, a temporal logic of trust that extends the Computation Tree Logic (CTL) to enable reasoning about trust with preconditions. A new vector-based version of interpreted systems is defined to capture the trust relationship between the interacting parties. We introduce a set of reasoning postulates along with formal proofs to support our logic. Moreover, we present new symbolic model checking algorithms to formally and automatically verify the system under consideration against some desirable properties expressed using the proposed logic. We fully implemented our proposed algorithms as a model checker tool called MCMAS-T on top of the MCMAS model checker for MASs along with its new input language VISPL (Vector-extended ISPL). We evaluated the tool and reported experimental results using a real-life scenario in the healthcare field.