Monday 3 January 2022

Inductive versus probabilistic reasoning

Deductive arguments have a poor track record. Even if the argument is sound, the premises may be wrong, or the argument may not map well to the real world.

But inductive reasoning does not much appeal to deductivists. It sounds wishy-washy and doesn't produce the certainty that deductive reasoning appears to give. This is because so-called inductive reasoning is not how most people reason in the real world.

The classic inductive argument goes: The sun has risen 100 times before, therefore it will rise again tomorrow. Expressed in deductive terms, the implicit inductive premise is "if something has happened a lot in the past, it will probably happen again tomorrow". That's reasonable, but the key word is "probably". So the argument would be better expressed as "The sun has risen 100 times before, therefore it will probably rise again tomorrow." If we make the argument stronger "The sun has always risen before, therefore it will definitely rise again tomorrow", that's false.

The problem with this sort of reasoning is, like Laplace's rule of thumb, it ignores our detailed knowledge about the world. Like Laplace's rule of thumb, it's just a rule of thumb.

If the sun has risen a thousand times before, we do not just estimate the probability of it rising tomorrow as one in a thousand. We can be much more confident, because we understand that "sunrise" is a term for the sun moving above the horizon, as the earth orbits the sun in a solar system. For the sun to fail to rise tomorrow, something extraordinarily improbable would have to happen. For example, the sun would have to explode, in a way we did not predict.

In practice, we can use our knowledge about the sun to predict that it will engulf Mercury & Venus, and render Earth uninhabitable, in about five billion years.

Inductive reasoning should be replaced by probabilistic reasoning.