Qubit semantics and quantum trees

In the qubit semantics the meaning of any sentence alpha is represented by a quregister: a unit vector of the n-fold tensor product circle times C-n(2), where n depends on the number of occurrences of atomic sentences in alpha (see Cattaneo et al.). The logic characterized by this semantics, called quantum computational logic (QCL), is unsharp, because the noncontradiction principle is violated. We show that QCL does not admit any logical truth. In this framework, any sentence alpha gives rise to a quantum tree, consisting of a sequence of unitary operators. The quantum tree of alpha can be regarded as a quantum circuit that transforms the quregister associated to the occurrences of atomic subformulas of alpha into the quregister associated to alpha.