Meehl’s second lecture is almost entirely about Karl Popper and his program of refutation. Popper is the scientist’s favorite philosopher, as he conceives the scientist as a heroic truth-seeker, carving out understanding with the sword of falsification. I’ve been guilty of falling for Popper’s flirting! If you don’t think too deeply about it, Popper’s view of science is very romantic. Great thinkers put theories up for scrutiny and do ingenious experiments rendering them false, rapidly revealing an essential theoretical core. But, as we’ll see, not only does no science work this way, but if you probe a scientist for more than a few minutes, they’ll agree they aren’t in the falsification business. Let me return to these social issues after first describing the logical idea behind falsification.

As is always the case with logic, we have to start with some stodgy formal notation. I’m not going to use Meehl’s notation as I’d like to avoid the Emoji & Symbols Viewer when possible. But I think what I’ve chosen should be fine. If P and Q are statements, then I’ll write ~P to denote the negation of P, and P-->Q will denote material implication. Material implication is a logical rule that we colloquially read as “if P, then Q.” You could say that Q is necessary for P, or P is sufficient for Q. If you really like logic, the implication is equivalent to “~P or Q.” Or, more instructively, “~(P and ~Q).” Bah, I don’t like logic! But fortunately we won’t need much more than this to discuss Popper.

The final piece we need is the hypothetical syllogism.

The way to read a chart like this is “A is true. B is true. Therefore C is true.” A is some rule, B is a truth statement, and C is the implication.

There are four combinations from the “if P, then Q” relationship.

In the second line of each of these syllogisms, the truth of one of the propositions is asserted. Only two of these correspond to valid logical deduction. The top left corner is called *modus ponens* or affirming the antecedent. If P implies Q and P is true, then Q must be true. This is all well and good.

The lower left corner is called denying the antecedent. It doesn’t get a fancy Latin name as it’s not valid. Certainly, just because P implies Q doesn’t mean that Q can’t happen when P doesn’t happen. When the Patriots win a lot of games, it makes me happy. I’m happy. *Checks the Patriots’ record in 2023*…

Now, the really interesting cell in this table is the upper right. It is not valid. Just because P implies Q does not mean that Q implies P. “If I listen to Taylor Swift, I get a headache. I have a headache. Therefore I listened to Taylor Swift.” Or whatever. This implication is called the “converse fallacy” or “affirming the consequent.” We learn that it’s invalid in high school geometry at the latest.

But when you think about it, science and engineering is entirely built upon affirming the consequent. Typical reasoning in science goes something like this: “If my theory is true, then I’ll observe this outcome of my experiment. I observe exactly this outcome. Therefore, my theory is true.” We do this all the time. “If Newton’s Laws are true, bowling balls and feathers drop at the same rate in a vacuum. I see that bowling balls and feathers drop at the same rate in a vacuum. Therefore, I conclude Newton’s Laws are awesome.”

Huh, this can’t be the way things work, can it? Science can’t be built upon irrationality! Popper certainly didn’t think so. But let me get back to Popper in a second.

Even if it’s the first logical fallacy we learn, our entire society is built upon affirming the consequent. We all agree to believe the future will resemble the past, at least somewhat. This belief will always just be your opinion, man. Postmodern Machine Learning Dude Ben learned to stop worrying and embrace Hume’s Problem of Induction. It’s unavoidable. The sun will come up tomorrow. We build technology around prediction, assuming the future is like the past. Our society affirms the consequent. We’re delightfully arational. The Dude abides.

If you don’t want to embrace inductive anarchy like me, Meehl offers a probabilistic fix, one I incessantly write about on this blog:

“All empirical inference is only probable. That's why it differs from inference in mathematics set theory, pure logic. That's why no matter how much evidence you have about facts, the theory can never be said to be proved in the strong sense of Euclid. It's only proved in the sense of rendered more likely, rendered more credible, supported, whatever you want to say.“

For this reason, probability will necessarily play a central role in Meehl’s course.

OK, but let’s get back to Popper. Popper hated inductive reasoning. He knew it was logically invalid. And he thought that we were just confused by Hume’s ramblings and could do science with purely logical deduction. Popper’s scientific logic is based on the fourth hypothetical syllogism: “If P, then Q. Not Q. Therefore Not P.” This is denying the consequent, also known as *modus tollens*. It gets a fancy Latin name as it’s logically valid. It forms the logical basis of our proofs by contradiction. And Popper tried to make it the basis of scientific inference.

Popper figured that he could solve the problem of induction, by denying induction exists. Bold! Or, at least, you could do science without induction. “You don’t support theories with facts.” For Popper. what is essential about science is its falsifiability. A scientist honorably tells their colleagues what sorts of observations undermine their theory. And then the other scientists do these experiments. The irrefutable theories are left standing.

I get why this is appealing, of course. We like to teach the scientific method as about generating alternative hypotheses and finding clever experiments to show these are false. After all, didn’t Galileo actually do that Leaning Tower of Pisa experiment to prove Aristotle wrong? Though Popper and the Logical Positivists were allergic to history, they were clearly inspired by certain historical anecdotes. But tomorrow, I’ll dive into some alternative anecdotes showing how science has never been about falsification. How scientists cling to theories despite significant evidence against them. And how Popper and others tried to patch this up.

You’re missing something. One key aspect of Popper’s theory on falsification is the argument it acts as a criterion for science. Most people fixate on this as being a way to distinguish between science and non-science, however there is another way to view it. To return to your example of bowling balls in space, when said bowling ball falls at the same speed as feathers, you are not proving Newton’s Second Law, you are failing to disprove it. However, what that means is that insofar as you are in the business of dropping bowling balls, your failure to disprove the law means that it does not matter if Newton’s Second Law is accurate or not, only that the difference between it and the truth is such that it does not impact on life. Insofar as a theory in a given environment is unfalsifiable, it means you do not need to care about whether it is true or not. This observation I’d argue is far more valuable that the argument of repeated falsification.

Looking forward to more in this series. But in the reconstructed version I first learned, the logical positivists believed in affirming the consequent ( the possibility of generating confirmatory hypotheses was their criterion for a meaningful statement). Popper then improved on them by replacing verification with falsification, and by making it a criterion for a scientific statement rather than for a meaningful one (as mentioned by CW III below).

Then Lakatos fixed up a lot of the problems with Popper, avoiding the difficulty of the critical test etc. And Bayesian reasoning helped a bit more.

As I said, a reconstructed version, but seems reasonably close to the mark.