Towards a Standard Completeness for a Probabilistic Logic on Infinite-Valued Events

Abstract

MV-algebras with internal states, or SMV-algebras for short, are the equivalent algebraic semantics of the logic SFP( L, L) which allows to represent and reason about the probability of infinite-valued events. In this paper we will make the first steps towards establishing completeness for SFP( L, L) with respect to the class of standard SMV-algebras, a problem which has been left open since the first paper on SMV-algebras was published. More precisely we will prove that, if we restrict our attention to a particular, yet expressive, subclass of formulas, then theorems of SFP( L, L) are the same as tautologies of a class of SMV-algebras that can be reasonably called "standard".