Permalink : http://hdl.handle.net/2115/75452

Birkhoff and von Neumann introduced Quantum Logic, in which the commonly agreed definition of the implicational connective has not yet achieved. Kotas proposed six formulations to define six implicational connectives. Ozawa introduced symmetrical relations among these implicational connectives. NFL is a Gentzen-style natural deduction for non-commutative substructural logic, which excludes three structural inference rules, i.e. contraction, weakening and exchange. We wll construct proof figures of NFL, augmented with other inference rules, to establish relations among the implicational connectives, so that the relevancies of inference rules, including structural rules such as exchange rule, are clarified.