  In this paper, a short survey about the concepts underlying general logics is
given. In particular, a novel rigorous definition of a fuzzy negation as an
operation acting on a lattice to render it into a fuzzy logic is presented.
According to this definition, a fuzzy negation satisfies the weak double
negation condition, requiring double negation to be expansive, the antitony
condition, being equivalent to the disjunctive De Morgan law and thus
warranting compatibility of negation with the lattice operations, and the
Boolean boundary condition stating that the universal bounds of the lattice are
the negation of each other. From this perspective, the most general logics are
fuzzy logics, containing as special cases paraconsistent (quantum) logics,
quantum logics, intuitionistic logics, and Boolean logics, each of which given
by its own algebraic restrictions. New examples of a non-contradictory logic
violating the conjunctive De Morgan law, and of a typical non-orthomodular
fuzzy logic along with its explicit lattice representation are given.
