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The words ‘valid’ and ‘sound’, if used strictly as most logicians do (but, regrettably, not all) only have application to arguments which purport to be deductive.

What then is an inductive argument?

An inductive argument is one in which it is at least logically possible for every member of the premise set to be true, and the conclusion false.

Something is logically possible if it is not self-contradictory (a good practical test is whether what it states is imaginable). The expression ‘logical possibility’ casts a much wider net than ‘scientific possibility’ or ‘practical possibility’. Something is only scientifically possible if it does not contradict the laws of science. But to be logically possible it is sufficient that it does not contradict the laws of logic,

Thus the following are examples of logical possibilities:

Pigs can fly.
There is a pink elephant in the next room.
There is a perpetual motion machine.

(Here is a more contentious example:
The same surface can be uniformly red and green all over at the same time.
What do you think?)

TASK 2: Review Arguments 2 and 3 above and satisfy yourself that, in each case, it is at least logically possible for every member of the premise set to be true, and the conclusion false.

It is sometimes possible to ‘convert’ an inductive argument into a deductive one by adding an additional premise.  Suppose we were to add the following additional premise to Argument 2:

Apart from George, Helen, Ian, Jane, and Kevin there are no other students in the front row

So that it now read:

(Argument 2’)

George is a student in the front row and is under 30.
Helen is a student in the front row and is under 30.
Ian is a student in the front row and is under 30.
Jane is a student in the front row and is under 30.
Kevin is a student in the front row and is under 30.
Apart from George, Helen, Ian, Jane, and Kevin there are no other students in the front row
So:
All of the students in the front row are under 30.

We now have a valid deductive argument. But the conclusion no longer involves us in taking a risk.

A cautionary note on terminology: Some writers call arguments such as Argument 2’ cases of enumerative induction. This is infuriating because it is actually a variety of deduction. Sadly, it is not the only example of misleading terminology. In mathematics, it is not uncommon, as a means of proving (say) that all the positive whole numbers have a certain property (call it P), to argue by the following pattern.

Step 1: Prove that the number 0 has the property P
Step 2: Prove that if any arbitrarily selected number (call it n) has the property P, then n + 1 must have P also.
Conclusion: Every positive whole number has the property P.

This type of argument is used when you have to prove a conclusion about an infinite set. But, terminological warning, this process is called mathematical induction. Once again it is actually a case of deduction.

We will not be particularly concerned with either enumerative induction or mathematical induction. This is simply mentioned as a terminological warning if you come across these phrases in the literature. Why this confusing terminology? I haven’t the slightest idea.

3. The Hypothetico-Deductive Method

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