Nonmonotonic Logic and Stubborn Thinking

I was struck recently by some similarities between the psychology of stubborn thinking and the history of science and logic. It’s not just individuals that have trouble changing their minds; entire scientific, logical, and mathematical movements suffer from the same problem.


When people think about logic (which I imagine is not very often, but bear with me on this), they probably think about getting from a premise to a conclusion in a straight line of rule-based reasoning — like Sherlock Holmes finding the criminal perpetrator with infallible precision, carving his way through a thicket of facts with the blade of deduction.

Here’s a sample logical proof that would do Holmes proud.

Birds fly.
Tweetie is a bird.
Therefore Tweetie flies.

We have here a general principle, a fact, and a deduction from those to a logical conclusion.

The problem is that the general principle here is just that: general. It is generally the case that birds fly. In fact, some birds do not fly at all. (In fact, there’s not ever a general principle that universally applies: even the laws of physics are arguably fallible. Cf. Nancy Cartwright’s wonderful How the Laws of Physics Lie.) Tweetie could be an ostrich or an emu, or Tweetie could have lost his wings in a window-fan accident, or Tweetie could be dead.

You could shore up your general principle in order to try to make it more universal: Birds that aren’t ostriches, emus, wingless, or dead, fly. But this sort of backpedaling is really an exercise in futility. As the past several decades of research in artificial intelligence through the 90s showed us, the more you expand your general principle to cover explicit cases, the less of a general rule it becomes, and the more you realize you have to keep covering more and more explicit cases, permutations upon permutations that will never end. (E.g., even in the case of death, Tweetie might be able to fly. He could be dead, but also in an airplane at 20,000 feet. Would you amend your general principle to cover this case? It would be a strange sort of “scientific” law that stated “Birds fly, except dead birds that aren’t in airplanes.”)

A brilliant solution to this sort of problem was found via the creation of nonmonotonic logic, a logical system that is what they call defeasible — that is, it allows for making a conclusion that can be undone by information that eventually emerges to the contrary. So the idea is that a nonmonotonic system allows you to conclude that Tweetie flies via the logic above, but also allows you to change that conclusion if you then find out that Tweetie is, in fact, e.g., dead.

This may not seem like a big deal, since this is how a rational human is supposed to react on a regular basis anyway. If we find out that Tweetie is dead, we are supposed to no longer hold to the conclusion, as logical as it may be, that he flies. But for logicians it was huge. The old systems of logic pinned us helplessly to non-defeasible conclusions that may be wrong, just because the logic itself seemed so right. But now logicians have a formal way of shaking free of the bonds of non-defeasibility.


The history of science is rife with examples of this principle-clinging tenacity from which it took logic millennia to escape. A famous case is found in astronomy, where the concept persisted for more than a dozen centuries that the earth was at the center of the universe. As astronomy progressed, it became clear that to describe the motion of the planets and the sun in the sky, a simple model of circular orbits centered around the Earth would not suffice. Eventually, a parade of epicycles was introduced — circles upon circles upon circles of planetary motion spinning this way and that, all in order to explain what we observed in the earth’s sky, while still clinging to the precious assumption that the Earth is centrally located. The simpler explanation, that the Earth was in fact not the center of all heavenly motion, would have quickly done away with the detritus of clinging to a failed theory, but it’s not so easy to change science’s mind.

In fact, one strong line of thought, courtesy of Thomas Kuhn has it that the only way for scientists to break free from such deeply entrenched conceptions is nothing short of a concept-busting revolution. And such revolutions can take years to gather enough momentum in order to be effective in mind-changing. (Examples of such revolutions include the jarring transition from Newtonian to Einsteinian physics, and the leap in chemistry from phlogiston theory to Lavoisier’s theory of oxidation.)

Down to Earth

If even scientists are at the mercy of unchanging minds, and logicians have to posit complicated formal systems to account for the ability to logically change one’s mind, we should be prepared in our daily lives to come up against an immovable wall of opinions. Despite what the facts tell us.

Indeed, it isn’t very hard to find people that have a hard time changing their minds. Being an ideologue is the best way of sticking to an idea despite evidence to the contrary, and ideologues are a dime a dozen these days. What happens in the mind of an ideologue when she is saving her precious conclusion from the facts? Let’s revisit Tweetie. (You can substitute principles and facts about trickle-down economics or global warming for principles and facts about birds, if you like.)

Ideologue: By my reasoning above, I conclude that Tweetie flies.

Scientist: That is some nice reasoning, but as it turns out, Tweetie is dead.

Ideologue: Hmmm. I see. Well, by “flies” I really mean “flew when alive”.

Scientist: Ah, I see. But, actually, Tweetie was an emu.

Ideologue: Of course, of course, but I mean by “flies” really “flew when alive if not an emu”.

Scientist: But so then you’ll admit that Tweetie didn’t actually fly.

Ideologue: Ah, but he could have, if he had had the appropriate physical structure when he was alive.

Scientist: But your conclusion was that Tweetie flies. And he didn’t.

Ideologue: Tweetie was on a plane once.

Scientist: But isn’t that more a case of Tweetie being flown, not Tweetie flying?

Ideologue: You’re just bogging me down in semantics. In any case, Tweetie flies in heaven now. Case closed.