There are theories and laws in every branch of science. Every theory in physical science is a generalization.
By observing events and processes theories are formulated for their explanation. Observation of regularities in Nature also prompts to make generalization. Generalizations like water freezes at zero degree temperature, water expands when freezes, water splits into hydrogen and oxygen etc. also require explanation by some theories or laws.
Whether it is a theory or law, it is basically a generalization. Every significant generalization in science possesses explanatory power. All generalizations are obtained by induction.
Looking at all the generalizations and their explanatory power a distinction has been made between primary induction and secondary induction. A primary induction is in the form- ” All the observed A’s are B’s; therefore all A’s are B’s.
That is on the basis of observation of a limited number of cases a generalization is made about all the cases. Here the conclusion is a general real proposition. In case of secondary induction certain theories are formulated but not on the basis of observation of particular facts.
Here there is systematization of the theories by help of a higher theory. That is secondary induction aims at explaining theories by systematizing the existing theories or laws.
Let us analyse them in some detail and bring out their characteristics.
a). Primary induction depends upon observation of facts. At first a number of cases are observed either from nature or under controlled conditions to establish a general proposition. We observe instances of birds laying eggs, whales as mammals, iron wires conducting electricity etc.
On the basis of our experience of facts we make generalization that ” all birds lay egg and are not mammals”, “all whales are mammals” or ” iron is a conductor or electricity” etc. To reach these generalizations we have to depend on actual observation or experiment of facts.
b). The conclusion in primary induction is a general real proposition. A general proposition covers cases of unlimited generality. The subject here denotes a class having innumerable number of individuals.
It represents a whole class. To say “all men are mortal” or “no cow is a biped” is to refer to a class with unlimited individuals. Further a real proposition is one in which the predicate gives some new information about the subject. Hence a real proposition is either true or false as a matter of fact. But in a verbal proposition the predicate simply repeats either a part of the meaning or the whole meaning of the subject.
A bachelor is unmarried, a triangle is a three sided plane figure, two and two equals four etc. are all verbal propositions .A verbal proposition is necessarily true and its opposite is necessarily false and self contradictory. But the opposite of a real proposition is a real proposition.
The opposite of a real proposition may be false but never self- contradictory. Thus a real proposition is completely different from a verbal proposition. The conclusion in primary induction is always a real proposition.
c). Primary induction involves an inductive leap. When we say “All A’s are B’s” it indicates that every A is B always and everywhere. Since the conclusion is a general real proposition of unrestricted totality there is a genuine inference here.
For we observe some cases of A being B and on the basis of this experience when we assert that ‘all A’s are B’s’, it involves a leap. It is an inference since we draw a conclusion on the basis of some observed facts.
When we pass from some cases to all cases, from present experience to a general theory it is said to be the inductive leap. To pass from some to all, from observed to unobserved, from known to unknown is not possible by any deduction, rather the reasoning here is inductive. So it is known as inductive leap.
d). Primary induction is based upon resemblance of facts. Resemblance or similarity is the very basis of an inductive inference. When we observe some instances for a generalization, they belong to a category or class.
The unobserved individuals belong to the same class of observed individuals. This jump from the known to the unknown covering a class of unlimited individuals is on the basis of structural or fundamental similarity between the observed individuals and the unobserved ones.
Since we observe some whales to be mammals we generalize that all whales are mammals for whales constitute a biological class having fundamental resemblance. So on the basis of resemblance a generalization is made.-
This is the very basis of induction according to Mill. He maintains, “Induction then is that operation of the mind, by which we infer that what we know to be true in a particular case or cases will be true in all cases which resemble the former in certain assignable respects.” e).
The conclusion established by induction is probable. In primary induction the conclusion involves a leap. The very presence of an inductive leap renders the conclusion to be probable. Since the conclusion here is a real proposition having factual import, it cannot be logically certain. To say all men are moral or no bird is a mammal is to make an assertion of fact.
Here even though not a single contrary evidence is ever known and even if such generalizations are founded on well-established theories of science, still they are not to be taken as analytically true. The degree of probability may be very high, but it is not logically certain.
There cannot be logical certainty in case of synthetic propositions, which describe empirical facts. Only an analytic or verbal proposition can assume logical certainty. But a real proposition by definition lacks logical certainty for its opposite is also a possibility in the empirical world. Since the conclusion established by primary induction is a real proposition it enjoys novelty but not logical certainty. It has novelty for it is factually- informative or descriptive. Where there lies novelty, there cannot have logical certainty.
In some advanced sciences there is attempt to build some higher theories to systematize the generalizations. No observation of facts is made for this. When a higher theory is imagined for enhancing the explanatory power of some theories it is a case of secondary induction. For here there is no observation of facts unlike that of primary induction to think of a hypothesis.
The hypothesis imagined in secondary induction will provide possible explanatory models to some existing theories or laws of science. Since the hypothesis here has nothing to do with the cases of factual instance, it is called non-instantial hypothesis.
It is done in developed sciences. Thus when there is induction of a non-instantial hypothesis to provide explanations to the theories of science, it is called secondary induction.
It is different from primary induction for unlike the latter it is not making an induction on the basis of observation of facts. But the induction here aims at systematizing the theories of science by some higher theory. Let us see what are the important characteristics of secondary induction.
It establishes a transcendental or non-instantial hypothesis. Here a higher theory is imagined to systematize the existing theories. This higher theory is supposed to provide explanatory framework to the existing theories.
The supposed theory is a hypothesis having no instantial exemplification. Therefore it is sometimes characterized as a transcendental or non-instantial hypothesis.
The laws of universal gravitation, the theory of relativity, the black-whole cosmological theory to account for the origin of the universe are such theories.
These hypotheses have no direct instances to be observed, but they are non-instantial Ones to systematize some existing theories. Thus when the supposed hypothesis is considered to provide a systematic explanation to the existing theories or laws it is considered to be a case of secondary induction.
Secondary induction applies a hypothetico-deductive method. In secondary induction a non-instantial hypothesis is formulated which is supposed to possess great explanatory power. Forming a non-instantial hypothesis is formulated which is supposed to possess great explanatory power. Forming a non-instantial hypothesis requires a great insight.
Only a genius can think of such a hypothesis for it requires a sharp mind and powerful ability to imagine. When such a hypothesis is made it is not subject to ordinary confirmation for it has no instance or exemplification. But such a hypothesis is verified in some indirect way by the deductive method.
By applying the method of deduction to it, theories are brought out and tallied with the existing laws. If the well established laws are deduced from the hypothesis, then only the theory can be accepted.
Thus a non-instantial hypothesis established by secondary induction must have high explanatory power in order to be accepted: Its explanatory power gets confirmed more and more if the consequences brought out from it agree with the existing laws.
The existing laws are well founded by it. They coherently make a system with the help of the non-instantial hypothesis. That establishes its veracity.
c. The conclusion obtained by secondary induction is tested only indirectly. When a hypothesis is formed to account for an explanation of a phenomenon or a class of phenomena, the hypothesis is normally put to verification before it is accepted.
Verification may be direct or indirect in case of instantial hypothesis. Direct verification consists in observing or experimenting the fact to see whether the hypothesis is a legitimate one or not.
Where direct verification is not possible because of the complexity of the situation, lack of necessary equipment for the purpose or absence of instantial exemplifications indirect verification is resorted to.
Thus in case of non-instantial hypothesis not only there is absence of instances, there is great complexity for its verification by any direct method. So indirect method of verification is the only available way to test its veracity.
Indirect verification includes deduction or accumulation of evidences. But accumulation of evidences is also not that easy in non-instantial hypotheses for there is no instance exemplifying it. So the method of deduction is applied here. From the non-instantial hypothesis laws of lower or the lowest variety are drawn by help of deduction.
They are tested by tallying them with facts. This test is done under varying circumstances to provide a possible basis for the acceptance or rejection of a non-instantial hypothesis.
d. The conclusion” is probable in nature. All inductions, whether primary or secondary, are probable. Since induction makes generalization which ultimately relates to facts it cannot possess logical certainty. A generalized theory or law admits innumerable instances which can never be exhaustively verified. So all inductions are probable though the degree of probability differs.
When a single contrary evidence is found or some discrepancy is observed with regard to any generalized theory or law it arouses suspicion and leads to imagine some new hypothesis.
Any theory or law, therefore, is subject to change or modification as there is no limit to scientific discovery or innovation. Thus any hypothesis, whether instantial or non-instantial, is probable in nature and that is an accompanying characteristic of any empirical generalization.
Thus the distinction between primary induction and secondary induction throws insight to describe the nature of induction. The descriptions of induction as given by Aristotle, scholastic logicians, Bacon, Mill etc. are inadequate.
That is because their descriptions do not cover all aspects of the concept as it is understood in recent time. At present it is maintained that inductive logic is basically concerned with the procedure and methodology of all sciences.
It also discusses other allied and auxiliary issues associated with the procedure of science. An empirical science aims at discovering exceptionless regularities- called theories or laws- and the systematization of the laws by a coherent system or order. That is done by the help of the procedure of induction. So let us turn to the issue of the procedure of induction.