From Basic Logic to Quantum Logics with Cut-elimination

The results presented in this paper were born in the framework of basic logic, a new logic aiming at the uniication of several logical systems. The rst result is a sequent formulation for orthologic which allows to use methods of proof theory in quantum logic. Such formulation admits a very simple procedure of cut-elimination and hence, because of the subformula property, also a method of proof search and an eeective decision procedure. By using the framework of basic logic, we also obtain a cut-free formulation for orthologic with implication, for linear orthologic and, more in general, for a wide range of new quantum-like logics. These logics meet some requirements expressed by physicists and computer scientists. In particular, we propose a good candidate for a linear quantum logic with implication.