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 premise 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.
Dylan is 99 and in a coma.
Therefore, Dylan will not finish the marathon tomorrow.
Again, it is not logically impossible for Dylan to recover from his coma and join the marathon and finish, 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:
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 small number, say, 5, 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.
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 them all to Tinky-winky. Obviously this new argument will be a lot weaker than the old one.
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 guarantee 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.