On standard completeness and finite model property for a probabilistic logic on Łukasiewicz events

The probabilistic logic FP(¿, ¿) was axiomatized with the aim of presenting a formal setting for reasoning about the probability of infinite-valued Łukasiewicz events. Besides several attempts, proving that axiomatic system to be complete with respect to a class of standard models, remained an open problem since the first paper on FP(¿, ¿) was published in 2007. In this article we give a solution to it. In particular we introduce two semantics for that probabilistic system: a first one based on Łukasiewicz states and a second one based on regular Borel measures and we prove that FP(¿,¿) is complete with respect to both these classes of models. Further, we will show that the finite model property holds for FP(¿,¿)