A Double Deduction System for Quantum Logic Based On Natural Deduction

The author presents a deduction system for Quantum Logic. This system is a combination of a natural deduction system and rules based on the relation of compatibility. This relation is the logical correspondant of the commutativity of observables in Quantum Mechanics or perpendicularity in Hilbert spaces.Contrary to the system proposed by Gibbins and Cutland, the natural deduction part of the system is pure: no algebraic artefact is added. The rules of the system are the rules of Classical Natural Deduction in which is added a control of contexts using the compatibility relation.The author uses his system to prove the following theorem: if propositions of a quantum logical propositional calculus system are mutually compatible, they form a classical subsystem.