  We present positive coalgebraic logic in full generality, and show how to
obtain a positive coalgebraic logic from a boolean one. On the model side this
involves canonically computing an endofunctor $T': Pos\to Pos$ from an
endofunctor $T: Set\to Set$, in a procedure previously defined by the second
author et alii called posetification. On the syntax side, it involves
canonically computing a syntax-building functor $L': DL\to DL$ from a
syntax-building functor $L: BA\to BA$, in a dual procedure which we call
positivication. These operations are interesting in their own right and we
explicitly compute posetifications and positivications in the case of several
modal logics. We show how the semantics of a boolean coalgebraic logic can be
canonically lifted to define a semantics for its positive fragment, and that
weak completeness transfers from the boolean case to the positive case.
