Degrees of Disbelief

Meehl's Philosophical Psychology, Lecture 9, part 1.

This post digs into Lecture 9 of Paul Meehl’s course “Philosophical Psychology.” Technically speaking, this lecture starts at minute 82 of Lecture 8. The video for Lecture 9 is here. Here’s the full table of contents of my blogging through the class. 

After weeks of extracting hot takes from Lecture 8, I now turn to the very uncontroversial topic of Lecture 9: Probability. Woo boy. Please don’t tell The Bayesians.

It’s a bit odd that Meehl waited until the 2nd to last lecture of the quarter to dive into probability, as every lecture has touched on this foundational concept. But it’s also brilliant how far he got playing fast and loose with probability concepts. That’s what makes probability so compelling and so dangerous. Like causality, we all have a casual understanding of probability and uncertainty that undergirds our everyday lives. Chance, likelihood, and probability are all useful but slippery words. But then trying to make it rigorous becomes a mathematical and philosophical mess. We’ve spent hundreds of years in a quixotic quest to make everyday concepts about opinion into rigorous mathematics.

Having spent a career using it and endless hours blogging about it, I find myself less comfortable with probability than ever. The mathematics is daunting, and the connections between the mathematics and reality feel so tenuous. But at least I have found clarity in understanding that there is a problem! Meehl succinctly presents the concept and problem of probability in a single lecture, and let me try to do him justice in a few blog posts.

Everyone learns about probability by thinking about games of chance. Our initial notion is that probability quantifies a degree of confidence about something happening in the future. The probability that I will roll snake eyes in craps is 1 in 1/36. The probability I will roll seven is 1/6. These numbers quantify how much I’d be willing to be on an outcome in a game where every round is more or less the same.

But we use the word probability to describe a lot of other things. We can ask “What is the probability Donald Trump will win the 2024 election?” People now get paid a lot of money to put numbers on this. Nate Silver puts the number at 71%. Where does that number come from exactly? I unfortunately don’t get paid enough to tell you.

We also use probability in courts of law. In criminal trials, we ask jurors to decide if defendants are guilty “beyond a reasonable doubt.” In civil trials, we only ask if they believe it is “more likely than not” that the defendant violated the law. These are also statements about degrees of belief. About probabilities. Given the evidence, quantify how much you believe some statement of fact.

Throughout Meehl’s course, a running theme is that theories, though always wrong, have some accordance with truth. We could ask “Given everything we know, what is the probability this theory is true?” Is that a probability? Can this colloquial question be quantified too?

Can we actually answer all of these different questions with concrete numbers? Meehl broaches this problem using a dichotomy he attributes to Carnap. For Carnap, there are two types of probability, conveniently named like errors in statistics, Probability 1 and Probability 2.  

Before defining the two probabilities, we have to go back to the first lecture to remember the distinction between object language and metalanguage. Object language speaks about entities you can observe or measure:  blood, protons, libido. Metalanguage speaks about statements, properties about statements, and logic: truth, confirmation, validity.

Probability 1, often called logical probability, maps metalinguistic statements into numbers. It measures our certainty about some given statement. Probability 1 refers to the relationship between a hypothesis and its evidence. It quantifies how much credence we put into a particular hypothesis given everything we’ve seen so far. It is a relationship between propositions and beliefs. 

Probability 2, the one we all learn in grade school, measures the relative frequency of some property in a set of objects. Probability 2 just amounts to counting. The probability a coin flip is heads is just the fraction of the time the coin comes up as heads. 

But wait, you might ask, if I haven’t flipped this coin I just got from the mint, what’s its probability of coming up with heads? Is this a question of object language or metalanguage? Herein, we find ourselves in a pickle.

The confusing part is that the laws of fractions can be made to work for a calculus of belief. Even with one-off statements with no frequencies or probabilities, I can imagine a way to put numbers on degrees of certainty.

Here, let me confuse you some more. Let’s say I have a big set of stuff, like a bunch of cards laid face down on the table, and I want to understand some properties about the proportions of subsets. Here are a few things that are true

• Any subset has proportion greater than 0

• If I take the entire set, the proportion is equal to 1.

• If I take two nonoverlapping subsets, the proportion of their union is equal to the sums of their proportion

Using the cards example, in a standard deck of 52 cards, the proportion of diamonds is 1/4, the proportion of aces is 1/13, the proportion of face cards is 3/13. With a little more mathematical abstraction, these are Kolmogov’s axioms of probability. From these, we get every property we want of probability.

Surprisingly, we can do this for logic too! Let’s find us a function that maps metalinguistic propositions into numbers. I could say

• Any syntactically valid statement has probability bigger than zero

• Any statement that is certain has probability 1.

• If two statements describe mutually exclusive outcomes, then the probability of either one or both of the outcomes is equal to the sum of the individual probabilities.

Lo and behold I have Kolmogorov’s axioms again, and now I can do all sorts of probability calculus on degrees of belief or verisimilitude. No frequencies in sight.

Which one is more epistemically fundamental for understanding the unknown? Fractions or degrees of belief? You get to choose a side in the war.

Carnap thought he could end the war. He spent the last decades of his life attempting to find formal rules so that you could, in inductive logic, grind out precise probabilities by looking at the propositions in a particular formalized language. He didn’t succeed. Meehl suggests that if Carnap, one of the smartest people he’d ever met, couldn’t do it, this was evidence that it was impossible. Tomorrow, let me at least describe the weird roadblocks we run into when we try to give primacy to one camp versus the other.