We introduce Riesz Logic, whose models are abelian lattice ordered groups, which generalise Riesz spaces (vector lattices), and show soundness and completeness. Our motivation is to provide a logic for distributional semantics of natural language, where words are typically represented as elements of a vector space whose dimensions correspond to contexts in which words may occur... This basis provides a lattice ordering on the space, and this ordering may be interpreted as "distributional entailment". Several axioms of Riesz Logic are familiar from Basic Fuzzy Logic, and we show how the models of these two logics may be related; Riesz Logic may thus be considered a new fuzzy logic. In addition to applications in natural language processing, there is potential for applying the theory to neuro-fuzzy systems. (read more)