[A08] Inductive Reasoning

§1. What is induction?

Consider the following argument :

Dipsy bought one ticket in a fair lottery with 10000 tickets.
So, Dipsy is not going to win the lottery.

This argument is of course not valid, since Dipsy might be so lucky that he wins the lottery. But this is quite unlikely to happen if the lottery is indeed a fair one. In other words, the conclusion can be false even when the premise is true. However, even though the argument is not valid, if you believe that the premise is true, you probably will accept the conclusion as well on that basis. In other words, the conclusion is highly likely to be true given that the premise is true.

Here is another example :

Dylan is a man.
He is 99 years old and is in a coma.
Therefore, Dylan will not win the Boston marathon tomorrow.

Again, it is not logically impossible for Dylan to recover from his coma and join the Boston marathon and win, but this is unlikely to happen. Again, given that the premise is true, the conclusion is likely to be true also.

Although the two arguments above are not valid, we would normally still regard them as good arguments. What is special about them is that they are inductively strong arguments : the conclusion is highly likely to be true given that the premises are true. With an inductively strong argument, although the premises do not logically entail the conclusion, they provide strong inductive support for it.

There are at least three main differences between an inductively strong argument and a valid argument :

1. As already noted, in a valid argument, the conclusion follows logically from the premises, but this is not the case in an inductively strong argument. It is logically possible for the premises to be true while the conclusion is false.
2. Deductive validity is not a matter of degree. An argument is either deductively valid, or it is not. But inductive support is a matter of degree, depending on the probability of the conclusion being true given the premises.

   For example, consider this slightly modified argument :

   Dipsy bought X tickets in a fair lottery with 10000 tickets.
   So Dipsy is going to win the lottery.

   If we replace X by a very small number, say, 5, then the argument is obviously not at all convincing, since it is most unlikely that Dipsy can win by buying so few tickets. If X is 6000, then the conclusion is more likely to be true then false, but the argument is still rather weak. But if X is 9999, then the argument is extremely strong, since it is now most likely that Dipsy will win. So you can see that inductive strength is not an all-or-nothing matter.

3. A related point is that inductive strength is defeasible, whereas validity is not. To say that validity is not defeasible is to say that if you have a valid argument, adding new premises will not make it invalid. If it is indeed true that three people have died, then it follows that at least two people died, and this will remain the consequence whatever new information you acquire.

   However, new information can be added to an inductively strong argument to make it weak. Consider the second lottery argument again, and suppose we add the new premise that Dipsy bought 9999 lottery tickets, but gave all but one of them to Tinky-winky. Obviously this new argument will no longer be a strong argument.

Inductive reasoning is very important in ordinary life and science. We believe lots of things on the basis of limited evidence. The evidence might not logically gaurantee that the belief is correct, but the belief can still be reasonable. For example, we see dark clouds in the sky and think it is likely to rain so we bring an umbrella. We see mould on our bread and think we will be sick if we eat it.

Further reading : Chapter 2 "Probability and Inductive Logic" in Brain Skyrms (2000) Choice and Chance : An Introduction to Inductive Logic Wadsworth.