In this paper, we consider Turing machines based on unsharp quantum logic. For a latticeordered quantum multiplevalued (MV) algebra E, we introduce Evalued nondeterministic Turing machines (ENTMs) and Evalued deterministic Turing machines (EDTMs). We discuss different Evalued recursively enumerable languages from widthfirst and depthfirst recognition. We find that widthfirst recognition is equal to or less than depthfirst recognition in general. The equivalence requires an underlying E value lattice to degenerate into an MV algebra. We also study variants of ENTMs. ENTMs with a classical initial state and ENTMs with a classical final state have the same power as ENTMs with quantum initial and final states. In particular, the latter can be simulated by ENTMs with classical transitions under a certain condition. Using these findings, we prove that ENTMs are not equivalent to EDTMs and that ENTMs are more powerful than EDTMs. This is a notable difference from the classical Turing machines.
