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[Tutorial A07] Inductive Reasoning

§ A07.1 What is induction?

Consider the following argument :

Dipsy bought one ticket in a fair lottery with ten million 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. If you believe that the premise is true, you probably will accept the conclusion as well. 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 and is in a coma.
Therefore, Dylan will not run in the marathon tomorrow.

Again, it is not logically impossible for Dylan to recover from his coma and join the marathon, but if the premises are true this is unlikely to happen.

Although the two arguments above are not valid, we would 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 ten thousand tickets.
    So Dipsy is going to win the lottery.

    If we replace X by a very small number, say, 10, then the argument is obviously very weak, since it is very unlikely that Dipsy can win by buying so few tickets. However, if we increase X to say 2000, then the inductive strength of the argument will of course increase. If X is 9999, then the argument is even stronger, since it is extremely likely now 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 lottery argument again, and suppose we add the new premise that Po has bought 9000 lottery tickets, and have given all of them to Dipsy. Obviously this new argument will is a lot stronger than the old one.

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

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Argument analysis

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Even when all the experts agree, they may well be mistaken.

Bertrand Arthur William Russell

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