An inductive inference is prbable as it is concerned with the material truth of the conclusion. The truth of the conclusion depends upon the veracity of the premises which act as the basis for the induction.
If the conclusion in an induction is a general proposition, it is verified by some sample instances as observation of all instances is not possible. These particular instances should represent the whole class to which they belong.
That means the particular instances should be the fair samples of the entire generalized class. Unless the particular evidences are the representative samples, no inductive generalization can have higher degree of probability.
So the specific problem of induction is to decide to what extent the evidential samples are fair. Hence our attempt to understand the role of fair samples in induction will be of much help to comprehend the probability of inductive generalization.
We have already noticed that the theories or laws established in science are basically generalizations.
Further the theories or generalizations in every branch of science mutually support one another and make a coherent system. But in the early state of development of sciences the generalizations were considered as somewhat isolated.
Even today there are some areas of science-particularly the social sciences- in which generalizations are somewhat isolated from each other and do not constitute a coherent system.
Where the theories or laws are mutually supporting they are highly probable forms of generalizations. But where they are relatively isolated, there the idea of fair sample is very important because on the basis of these instantial representations, generalizations are made.
We have already discussed that an inductive generalization is made on the basis of observation of particular instances. The instances which are observed or verified constitute only a limited or small part of the conclusion which is a proposition of unlimited totality.
The more instances covered or verified add to the probability of the conclusion. The problem of inductive leap has been dealt with in the first chapter. Here we shall embark upon on a specific issue of the role of "fair sample" in induction.
It is remarkable that at timfes a large number of instances is inadequate to establish a generalization whereas in some other cases a few instances are sufficient to firmly establish a generalisition. Suppose I have seen some bald-headed people and all of them have crossed fifty.
From this I cannot make a generalization that all bold headed people must have crossed the age of fifty. But observation of a single whale as mammal provides the ground to firmly establish the conclusion that all whales are mammal.
Mill has pointed out this problem in a very befitting way in a passage. " why is a single instance, in some cases, sufficient for a complex induction, while in others myriads of concurring instances, without a single exception known or presumed, go such a very little way towards establishing a universal proposition? Whoever can answer this question knows more of the philosophy of logic than the wisest of the scientists, and has solved the problem of induction".
This problem presented by Mill is a problem in philosophy of logic but to find an answer to it is not that difficult or impossible. When a universal proposition is inferred on the basis of our experience of facts, it requires gradual confirmation.
Confirmation is made by objectively verifying fair samples. A fair sample must be a representative of the class having all the defining properties with which the class is generally associated. For the observed instance is the representative of all possible instances.
Thus any one instance is as good as another. If we find through an experiment the atomic number and weight of silver, then we can comfortably generalize the atomic number and weight of silver. And any object that answers to the definition of silver is supposed to possess the same atomic number and weight like the examined instances.
In such case the probability will be too high. Similarly if it is found that frogs collected from different sources are cold-blooded, we become pretty sure that all frogs are cold-blooded because the examined instances are fair samples of all possible instances.
In other words the unobserved instances have the same homogeneity with the observed instances. As all frogs make a homogeneous class, so it is not"necessary to observe each frog to establish the conclusion. For the verified instances are the fair samples.
They represent the whole class. In this example the observed instance is as good as the unobserved or unknown instances. If different instances do not differ in their defining or representative nature, they matter as one instance. Because the members of the class form one homogeneous kind, so to examine a few cases as fair samples will be the basis for generalization.
In some advanced sciences theories or laws established by inductive generalization mutually support one another and form a coherently organized system of propositions. In some areas a generalization is not only based on verifying instances of fair samples, but also support some far-reaching results of the system.
That means in science each generalization is not an isolated theory of its kind, rather part of a comprehensive system where theories mutually support each other. For an isolated theory cannot stand firm by itself. Rather as part of a unifying system it supports each generalization to stand on a firm basis and as such helps the system as well.
Thus the probability of an advanced theory is not influenced by further verifying instances but by the supporting theories of the system. But if there is a contrary hypothesis which can systematize the facts or the prevailing theories in a superior way that poses doubt to the veracity of the prevailing theory in question. So the probability of a theory is influenced by another hypothesis if it has more systematizing and explanatory power.
If direct verification of such a theory is not possible, it is verified by random sampling of its consequences. Any non-instantial theory may not be verified directly, but by verifying the sampling of its results its veracity can be accepted provided it has explanatory power.
Thus the idea of fair sample is significant in inductive general isaions.