  We study a real valued propositional logic with unbounded positive and
negative truth values that we call R-valued logic. Such logic slightly extends
continuous propositional logic which, in turn, builds on Lukasiewicz
many-valued logic. After presenting the deduction machinery and the semantics
of R-valued logic, we prove a completeness theorem for finite theories. Then we
define unital and Archimedean theories, in accordance with the theory of Riesz
spaces. In the unital setting, we prove the equivalence of consistency and
satisfiability and an approximated completeness theorem similar to the one that
holds for continuous propositional logic. Eventually, among unital theories, we
characterize Archimedean theories as those for which strong completeness holds.
We also point out that R-valued logic provides alternative calculi for
Lukasiewicz and for propositional continuous logic.
