Łukasiewicz logic and Riesz spaces

Abstract

We initiate a deep study of Riesz MV-algebras which are MV-algebras endowed with a scalar multiplication with scalars from $[0,1]$. Extending Mundici’s equivalence between MV-algebras and $\ell$-groups, we prove that Riesz MV-algebras are categorically equivalent to unit intervals in Riesz spaces with strong unit. Moreover, the subclass of norm-complete Riesz MV-algebras is equivalent to the class of commutative unital C$^*$-algebras. The propositional calculus ${\mathbb R}{\mathcal L}$ that has Riesz MV-algebras as models is a conservative extension of Łukasiewicz $\infty$-valued propositional calculus and is complete with respect to evaluations in the standard model $[0,1]$. We prove a normal form theorem for this logic, extending McNaughton theorem for Ł ukasiewicz logic. We define the notions of quasi-linear combination and quasi-linear span for formulas in ${\mathbb R}{\mathcal L},$ and relate them with the analogue of de Finetti’s coherence criterion for ${\mathbb R}{\mathcal L}$.