Abstract: The problem of reasoning about statements whose truth is uncertain has been open since the time of Boole. In 20th century logical foundations of probability theory were studied by Carnap, Reichenbach, Keisler many other authors. Since the mid 80's the problem was studied by people from Artificial Intelligence and Computer Science, starting with Nilsson, who were looking for tools to deal with uncertain knowledge. Many systems have been designed but some of the problems persist. One of these is the problem of providing a complete axiomatization. In the approach to be presented (which is due to M.Raskovic, Z.Ognjanovic and Z.Markovic) this problem is solved by introducing an infinitary rule of inference. Although this may seem to be a serious disadvantage for possible applications, it is proved that the system is decidable and that the decision problem is tractable. Another problem is that of entailment. Namely, classical (material) implication behaves sometimes paradoxically in the context of probability so it cannot serve to model the notion of "follows from" when applied to statements with different probability of truth. In our approach the problem is dealt with in two ways. One is to use the conditional probabilities, which is the standard approach, but here it has a complete axiomatization. Another way is to start with intuitionistic propositional calculus, which is very nonstandard but provides a much better behaved implication. Again the system is decidable, with tractable decision problem.