
Induction (Latin, â€˜leading onâ€™), in philosophy, is the method of reasoning by which we move from premises concerning what we have observed to a conclusion concerning what we have not. Every emerald that has been observed has been green. From this we infer that the next emerald to be dug out of the ground will be green. Indeed, from the fact that every emerald that has been observed has been green we infer that every emerald, observed or unobserved, is green. Each of these is an inference from premises concerning what has been observed to a conclusion about something that has not.
Inductive arguments are not deductive. A deductive argument is such that if its premises are true, then its conclusion must be true. It is impossible for all of the premises of a deductive argument to be true and yet its conclusion be false. The following argument is deductive: Gabriel is a man; all men are mortal; therefore Gabriel is mortal. This argument is deductive because it is such that if its premises are true, then its conclusion must be true. Its premises may not be true Gabriel may be an angel, not a man, but it is impossible for all of its premises to be true and yet its conclusion be false.
Inductive arguments are not deductive. No inductive argument is such that if its premises are true, then its conclusion must be true. It is possible for all of the premises of an inductive argument to be true and yet its conclusion be false. The claim that every emerald that has been observed was green is consistent with the claim that the next emerald dug out of the ground will not be green. And the statement that every emerald that has been observed was green is consistent with the claim that not every emerald is green. It may just be coincidence that every emerald that has been observed was green, that we have not yet stumbled upon buried deposits of nongreen emeralds.
It is possible for all of the premises of an inductive argument to be true and yet its conclusion be false. So if one believes the conclusion of an inductive argument on the grounds that its premises are true, it is always possible that one has acquired a false belief. If one infers from one\'s true belief that every emerald that has been observed was green that every emerald is green, it is always possible that one has acquired a false belief. This raises the worry that a method of inference which can lead one from true beliefs to false ones cannot be rational.
Is induction rational? That is, do the premises of an inductive argument ever give one reason to believe its conclusion? Since the truth of the premises of an inductive argument is compatible with the falsity of its conclusion, the premises of an inductive argument cannot decisively establish that its conclusion is true. The truth of the premises of an inductive argument cannot make its conclusion certain. But can the truth of the premises of an inductive argument nevertheless give one good reason to believe its conclusion?
Some philosophers claim that induction is rational because it is a form of inference to the best explanation. Given certain phenomena, it is legitimate to infer to the best explanation of those phenomena. There are various competing attempts to explain the fact that every observed emerald has been green. One is that it is simply coincidence that every emerald that has been observed was green, and that we have not yet stumbled upon the buried deposits of nongreen emeralds. Another is the explanation that, as a matter of natural law, every emerald is green. Every observed emerald is green because the laws of nature make it necessary that every emerald be green.
Of these two attempted explanations, the second seems best. Indeed, the first is not so much an explanation of the fact that every observed emerald was green as the claim that this fact is just an inexplicable coincidence. So we can infer that the best explanation of the fact that every observed emerald was green is that, as a matter of natural law, every emerald is green. But clearly, if the laws of nature make it necessary that every emerald be green, we can also infer that the next emerald to be dug out of the ground will be green, and that every emerald, observed or unobserved, is green. Induction is rational, because it is a form of inference to the best explanation.
In mathematics, induction is one of the major methods used to prove results. (Another is proof by contradiction.) Induction only applies to certain kinds of sets, and is typically used to prove results about the natural numbers, which form the prototypical example of an inductive set (one in which induction can be carried out).
For the natural numbers, induction works as follows. Suppose P(n) is a statement about the natural number n in symbolic logic, and we wish to show that P(k) is true for every possible value of k (so P(0) is true, and P(1), and so on). What we actually show is that P(0) is true, and that if P(n) is true, then so is P(n+1); the principle of induction is then invoked to tell us that P(k) is true for every natural number k. So induction is used to go from a large set of particular statements to a single general statement.
This use of the principle of induction as a means of mathematical proof is based on the idea of induction in the philosophy of science. This idea is that the universe is basically uniform, so that, for example, it is a scientific â€˜factâ€™ that the Sun will rise tomorrow, because it has done so every day in your experience so far. The idea is that the universe in the near future will be like the universe is now, and is the basis for the idea of the scientific method, in which only those experiments which are repeatable are to be studied. Unlike induction in science, howeverâ€”which is not so much a proof that the Sun will rise tomorrow as a suggestion that this is the most likely hypothesisâ€”proof by induction in mathematics is absolute. AJ SMcL
See also deduction; law of nature.Further reading David Hume, Enquiries Concerning Human Understanding; , Bertrand Russell, Problems of Philosophy; , N.W. Gowar, Basic Mathematical Structures. 
