 In this article the probabilistic inductive logic we will examine is a Bayesian inductive logic in this broader sense. This logic will not presuppose the subjectivist Bayesian theory of belief and decision, and will avoid the objectionable features of the syntactic version of Bayesian logicism. We will see that there are good reasons to distinguish inductive probabilities from degree-of-belief probabilities and from purely syntactic logical probabilities. So, the probabilistic logic articulated in this article will be presented in a way that depends on neither of these conceptions of what the probability functions are. However, this version of the logic will be general enough that it may be fitted to a Bayesian subjectivist or Bayesian syntactic-logicist program, if one desires to do that.