  We use modal logic as a framework for coalgebraic trace semantics, and show
the flexibility of the approach with concrete examples such as the language
semantics of weighted, alternating and tree automata, and the trace semantics
of generative probabilistic systems. We provide a sufficient condition under
which a logical semantics coincides with the trace semantics obtained via a
given determinization construction. Finally, we consider a condition that
guarantees the existence of a canonical determinization procedure that is
correct with respect to a given logical semantics. That procedure is closely
related to Brzozowski's minimization algorithm.
