That header got your attention, huh?? John Hull writes:

Reading an article on “non-Aristotelean” logic, where P(A) is my confidence of A being true, I found (on page 10) the equation P(B=>C)=P(B[AND]C)/P(B). Since I work in municipal government, an obvious interpretation of this is the following:

My confidence that if a person thinks the world is “flat” then they are dangerously stupid is the same as my confidence that a person believes the world is “flat” and is dangerously stupid, divided by my confidence that a person thinks the world is “flat.”

Setting aside the fact that when people’s welfare is in the balance, I tend to become rather passionate and use rather strong language, I simply cannot wrap my head around this idea. For example, my confidence that if it’s a lion, then it eats gazelles equals my confidence that it’s is a lion eating a gazelle, divided by my confidence that it’s is a lion. The left side of that equation is a (near) certainty — lions eat gazelles — but the right-hand side of the equation…how do I even begin to establish my confidence it’s a lion, let alone the rest of it?

Can you make this more understandable? Any help will be appreciated.

My response:

1. I find the if-then connection to Aristotelian logic confusing. I’d prefer to start with probabilities as first principles, and then interpret conditional probabilities Bayeisanly or, equivalently, in a frequentist way as the long-run proportion of cases in a “reference set.” (The choice of reference set is equivalent to the choice of what to condition on in a Bayesian calculation.) We discuss this in chapter 1 of Bayesian Data Analysis.

2. Right now, I’m realizing how nonintuitive many principles of probability are to some people. See this discussion here where one of the commenters want to assign a zero probability to an event (that of a tied congressional election) because it has never happened yet. That sounds commonsensical–but not if p=1/80,000 and n=20,000.