  Lindstr\"om's Theorem characterizes first order logic as the maximal logic
satisfying the Compactness Theorem and the Downward L\"owenheim-Skolem Theorem.
If we do not assume that logics are closed under negation, there is an obvious
extension of first order logic with the two model theoretic properties
mentioned, namely existential second order logic. We show that existential
second order logic has a whole family of proper extensions satisfying the
Compactness Theorem and the Downward L\"owenheim-Skolem Theorem. Furthermore,
we show that in the context of negationless logics, positive logics, as we call
them, there is no strongest extension of first order logic with the Compactness
Theorem and the Downward L\"owenheim-Skolem Theorem.
