The Brouwer Zadeh logics

We now study some stronger examples of unsharp quantum logic, that have been called Brouwer Zadeh logics (also fuzzy intuitionistic logics). These logics represent natural abstractions from the class of all BZ-lattices (defined in Chapter 4). As expected, a characteristic property of Brouwer Zadeh logics is a splitting of the connective “not” into two forms of negation: a fuzzy-like negation, that gives rise to a paraconsistent behavior and an intuitionistic-like negation. The fuzzy “not” represents a weak negation, that inverts the two extreme truth-values (truth and falsity), satisfies the double negation principle but generally violates the noncontradiction principle. The second “not” is a stronger negation, a kind of necessitation of the fuzzy “not”.