The consistent histories, also known as decoherent histories, approach to quantum interpretation is broadly compatible with standard quantum mechanics as found in textbooks. However, the concept of measurement by which probabilities are introduced in standard quantum theory no longer plays a fundamental role. Instead, all quantum time dependence is probabilistic (stochastic), with probabilities given by the Born rule or its extensions. By requiring that the description of a quantum system be carried out using a well-defined probabilistic sample space (called a “framework”) this approach resolves all the well-known quantum paradoxes of quantum foundations. In particular, quantum mechanics is local and consistent with special relativity. Classical mechanics emerges as a useful approximation to the more fundamental quantum mechanics under suitable conditions. The price to be paid for this is a set of rules for reasoning resembling, but also significantly different from, those which comprise quantum logic. An important philosophical implication is the lack of a single universally-true state of affairs at each instant of time. However, there is a correspondence limit in which the new quantum logic becomes standard logic in the macroscopic world of everyday experience.