Modern probability theory is typically derived from the Kolmogorov axioms, using measure theory with concepts like events and sample space. In one way, it's intuitive to understand how this works as Laplace wrote:
The probability of an event is the ratio of the number of cases favorable to it, to the number of all cases possible, when [the cases are] equally possible. ... Probability is thus simply a fraction whose numerator is the number of favorable cases and whose denominator is the number of all the cases possible.
However, the intuition of this view of probability breaks down when we want to do more complex reasoning. After learning probability from the lens of coins, dice and urns full of red and white balls, I still didn't feel that I had have a strong grasp about how to apply it to other situations -- especially ones where it was difficult or too abstract to apply the idea of "a fraction whose numerator is the number of favorable cases and whose denominator is the number of all the cases possible". And then I read Probability Theory: The Logic of Science by E. T. Jayne.
Jayne takes a drastically different approach to probability, not with events and sample spaces, but rather as an extension of Boolean logic. Taking this view made a great deal of sense to me since I spent a lot of time studying and reasoning in Boolean logic. The following post is my attempt to explain Jayne's view of probability theory, where he derives it from "common sense" extensions to Boolean logic. (Spoiler alert: he ends up with pretty much the same mathematical system as Kolmogorov's probability theory.) I'll stay away from any heavy derivations and stick with the intuition, which is exactly where I think this view of probability theory is most useful.