Hume criticised the reliability of inductive reasoning on the grounds that in order to be sure that an inductively deduced universal claim is true one would have to empirically verify all its logical implications, which is impossible. For example, the rising of Sun every single day cannot make one certain of the universal claim that the Sun will rise every day, because to do so requires verifying that the Sun does indeed rise every single in the future. Thus, one cannot be certain of the claims that are inductively deduced. This is in contrast to deductive reasoning, which is beyond any reasonable doubt. For example, the claim the Sun will rise tomorrow deductively follows from the claim that the Sun will rise everyday. If the universal claim that the Sun will rise everyday is known to be true, any claim that can be deductively derived from it is also true beyond any reasonable doubt. The truth of deductive reasoning is grounded in the truth of logic, while the truth of inductive reasoning is either ungrounded or can only be grounded in an empirical verification that is humanly impossible to achieve.

But now consider another example of inductive reasoning. A straight line is know to pass from two points A and B. From the fact that the line passes from these two specific points, one may inductively infer that the line will pass from several other points besides A and B. On the face of it, this situation is not very different from that of the rising Sun, where one inductively infers the rising of Sun on several other days from a known rising of the Sun on a few specific days, just as one infers the passing of a line from several other points from a known passing of the line from two specific points. However, in case of a straight line, the knowledge of any two distinct points from which the line passes is sufficient to know all the other points from which the line will pass. That is, the seemingly inductive claim that the line will pass from several other points is actually deductive in nature!

However, the fact that this generalisation is deductive in nature, i.e. one can derive the equation of a straight line from two points, and know all possible points from which the line will pass, requires one to know all the mathematical/logical rules embedded in the concept of a straight line. In general, it would not be surprising for someone to be unfamiliar with all these rules and all their implications, and thus be unable to deduce the general claim about the points from which the line will pass. For such a person, the generalisation will appear inductive in nature, even though in reality it is not.

A similar argument holds for essentially all inductive claims. To know whether a claim is truly inductive in nature, one would have to know all the possible logical rules embedded in all the concepts contained in the claim, and all the logical implications that can be deductively deduced from these rules and concepts. For example, in case of the universal claim that the Sun will rise everyday, one has to know all possible implications of the concept of Sun, which might further require knowing the concepts of planets and stars, as well as the concept of time, among other possible things. If one takes a relativistic-philosophical view of time, where there is no objective distinction between past and future, the claim that the Sun will always rise every day may indeed be deducible from a few observed instances of a rising of the Sun, as the information about the future can in principle be available at all times, under this relativistic view of time. Thus, even a single observation of the rising of the Sun may be sufficient to deduce that the Sun will rise every single day. For, to put it more poetically, every single day is nothing but a single day, and in every single day there is every single day!

Therefore, whether or not a universal claim follows inductively or deductively from a set of few particular (non-universal) claims cannot be determined with certainty, as one cannot know what are the right definitions of all the different concepts entailed in those claims, and what are all the logical (deductive) implications of whatever the right definitions are. As a result, one cannot be sure that a claim is truly inductive in nature (in contrast to being deductive in nature), just as one cannot be sure that an inductive claim (meaning a claim that is inductively inferred) is actually justified. Verifying the validity of an inductive claim requires one to empirically verify all the possible implications of the claim (as Hume argued), and verifying that the claim is inductive in nature requires one to analyse all the possible implications of all the possible concepts entailed in the claim (as this post is arguing) — two tasks, the former of which is more empirical and latter of which is more analytical in nature, but both of which are nevertheless equally humanly impossible to accomplish.

Conversely, the claims that may be deemed deductive in nature, may not be so in reality, if some of the implications of some of the concepts are not correctly understood. For example, the claim about the other points from which a line will pass may indeed be inductive in nature if one uses a non-Euclidean notion of space, i.e. a notion of non-flat or curved space, in which case even ‘straight’ lines will be inevitably curved, and two points will no longer be sufficient to determine what other points the line will pass through. But at the time when one is deciding whether this claim is deductive in nature or not, one may not even be aware that there are other, non-Euclidean notions of space, let alone know which notion of space is really relevant. Thus, the claim about all the points from which a straight line passes can be said to deductively follow only under assumptions, which may neither be explicitly known nor necessarily justified.

Hence, to reiterate the main point, knowing whether a claim can can be deductively derived from a set of other claims or not, hinges on one’s view of all the implications of various concepts used to formulate those claims, and one’s view of all these concepts can neither be made explicit every time one faces a question about whether or not a claim deductively follows from a set of other claims, nor any given view can be fully justified (as justifying any such view would almost necessarily fall into those broader philosophical questions that cannot be settled through merely deductive reasoning).

In summary, this whole discussion blurs the line between deductive and inductive reasoning, as the demarcation of these two realms of reasoning depends on answering questions that cannot be objectively settled. Two people may reasonable differ about whether a claim is deductive or inductive in nature, without disavowing any of the objectively valid logical or empirical facts. This blurring raises a question about the validity of any view that relies on a sharp distinction between deductive and inductive reasoning, such as Popper’s deductivism, which we will specifically address at another time.