Qubit semantics and quantum trees

Abstract

In the qubit semantics the \emph{meaning} of any sentence $\alpha$ is represented by a \emph{quregister}: a unit vector of the $n$--fold tensor product $\otimes^n \C^2$, where $n$ depends on the number of occurrences of atomic sentences in $\alpha$. The logic characterized by this semantics, called {\it quantum computational logic} (QCL), is {\it unsharp}, because the non-contradiction principle is violated. We show that QCL does not admit any logical truth. In this framework, any sentence $\alpha$ gives rise to a \emph{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 atomic subformulas of $\alpha$ into the quregster associated to $\alpha$.