In complete induction it is assumed that all the individuals...

In complete induction it is assumed that all the individuals under consideration are investigated and that their common property is observed in all of them. It is clear that in practice this is impossible, for even if all the present instances of a whatness could be investigated, there would be no way to investigate all past and future instances. At the very least, the possibility would remain that in the past or in the future there could be instances of that essence.

Incomplete induction occurs when many of the instances of a whatness are observed and a property common to them is attributed to all individuals of that essence. But this intellectual inference will not lead to certainty, for there will always be the possibility, no matter how weak, that some of the individuals which have not been investigated lack this property. Therefore, in practice, certain and indubitable conclusions cannot be obtained by induction.

Inference from universals to a particular, that is, first a predicate is proved for a universal subject and on the basis of this the judgment about the particulars of that subject becomes clear. In logic this sort of intellectual inference is called qiyās (deduction), and it yields certainty under the conditions that its premises are certain and the deduction also has a valid form.

Logicians have allocated an important section of classical logic to the explanation of the material and formal conditions of certain deduction, proof. There is a famous problem which has been raised regarding deduction. If a judgment is known to hold generally, the application of that judgment to all instances of the subject will also be known. But then there would be no need for the formulation of a deductive argument.

The scholars of logic have answered that a judgment for a major premise may be known in summary form, but in the conclusion it becomes known in detail. Meditation on the problems of mathematics and the ways of solving them shows how useful deduction is, for the method of mathematics is that of deduction, and if this method were not useful, none of the problems of mathematics could be solved on the basis of mathematical principles.

A point which must be mentioned here is that in analogy and induction there is a hidden form of deduction, but nevertheless, in the cases of analogy and incomplete induction this deduction does not constitute a proof and is of no use for obtaining certainty.