A propositional logic whose formulas are read as assertions that an element of an abelian lattice-ordered group is positive: implication is interpreted as subtraction, disjunction as the lattice join, and the constant 0 as the group identity. Students learn how a preferred basis on a word-vector space induces a componentwise lattice order that can be read as entailment between word meanings, why that makes vector lattices (Riesz spaces) a semantics for a logic rather than only a geometry, and how soundness and completeness are established for such a system by mutual translation with the Logic of Equilibrium. The system also shows where an ordered-group logic diverges from t-norm fuzzy logic: vector addition plays the role of strong conjunction, and there are no constants for true or false, only 0 for complete uncertainty.
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 f…