# Probability as Extended Logic

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.