  The present paper deals with complemented lattices where, however, a unary
operation of complementation is not explicitly assumed. This means that an
element can have several complements. The mapping $^+$ assigning to each
element $a$ the set $a^+$ of all its complements is investigated as an operator
on the given lattice. We can extend the definition of $a^+$ in a natural way
from elements to arbitrary subsets. In particular we study the set $a^+$ for
complemented modular lattices, and we characterize when the set $a^{++}$ is a
singleton. By means of the operator $^+$ we introduce two other operators $\to$
and $\odot$ which can be considered as implication and conjunction in a certain
propositional calculus, respectively. These two logical connectives are
``unsharp'' which means that they assign to each pair of elements a non-empty
subset. However, also these two derived operators share a lot of properties
with the corresponding logical connectives in intuitionistic logic or in the
logic of quantum mechanics. In particular, they form an adjoint pair. Finally,
we define so-called deductive systems and we show their relationship to the
mentioned operators as well as to lattice filters.
