  Probabilistic team semantics is a framework for logical analysis of
probabilistic dependencies. Our focus is on the axiomatizability, complexity,
and expressivity of probabilistic inclusion logic and its extensions. We
identify a natural fragment of existential second-order logic with additive
real arithmetic that captures exactly the expressivity of probabilistic
inclusion logic. We furthermore relate these formalisms to linear programming,
and doing so obtain PTIME data complexity for the logics. Moreover, on finite
structures, we show that the full existential second-order logic with additive
real arithmetic can only express NP properties. Lastly, we present a sound and
complete axiomatization for probabilistic inclusion logic at the atomic level.
