Positive logics

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.