In philosophy, we doubt everything, even the things which seem obvious and intuitive. So why do we accept deductive and inductive reasoning without doubting their validity?
1Since Hume, at least, inductive reasoning is problematic.– Mauro ALLEGRANZAMay 6, 2017 at 18:29
And also deductive reasoning is discusses, at least for some aspect; see Intuitionistic Logic and Paraconsistent Logic.– Mauro ALLEGRANZAMay 6, 2017 at 18:32
1But the correct approach, IMO, is not "doubting" but "discussing"...– Mauro ALLEGRANZAMay 6, 2017 at 18:32
1we do not doubt everything. we do question everything. big difference. we question because we want to understand, not because we doubt.– user20153May 7, 2017 at 19:27
Deductive reasoning is easy to argue for. Deductive reasoning is meaningless until you have a true statement about a group. Until you can confidently state "all swans are white," you cannot use deduction to make a statement about any one swan's color. I would argue that this can be used to argue for deduction in reverse. One should not accept the phrase "all swans are white" until one is willing to accept the deductive logic that comes from it.
Inductive reasoning has been tricky, as mentioned in comments. Many do not actually consider it "obvious" and demand an argument for why we can engage in inductive reasoning.
To me, the interesting one is abduction, which can be grossly paraphrased as "assuming the best hypothesis is true." It's what you use to go from "LIGO detectors have produced some really interesting squiggles which conform to that of gravity wave detection within a statistical error bound" to "LIGO has detected gravity waves." What's interesting to me is that that kind of reasoning is taken for granted with such a degree in today's scientific world that many have trouble even realizing that it was a logical reasoning step in the first place.
The limiting case or the simplest form of induction is the example. Even by only one example we may already begin to deduce. Hence induction depends on the example, and deduction depends also on the example, which it uses by circularly reproducing it into a conclusion. – Hence nothing must be accepted or be taken for granted, it is quite easy, if you just let me explain it without a highbrow style.
If we add to the above example in comparison a second one, the analogy is born, and hence analogy is analogous to induction, since we just take not one example but another one in association.
By reversing the deduction we arrive at abduction or criminal cause.
So, there is no necessity to doubt the validity of induction and deduction because they are so well embedded into example, analogy and cause.
(P.S.: The circular, deductive reproduction of the example into a conclusion is already an analogy, because it is its utilization to a further event that occurs after the first example. – But I have omitted this in the answer, as a little riddle.)
The distinction between deductive and inductive reasoning is not as simple as you seem to think. Sure: Aristotle's syllogism is a paradigm case of deductive reasoning, and the inference of "All ravens are black" from the observation of many black ravens is that of inductive reasoning. But what about probabilistic reasoning?
Inference methods that are involved with probability is categorized under Inductive Logic in SEP. In light of its involvement with uncertainly of the conclusion it looks like an inductive reasoning. But, the conclusion of the probabilistic reasoning is necessarily true. Given a fair coin, the probability of observing two heads in two tosses is one fourth. In this light, probabilistic reasoning looks deductive. So to determine whether probabilistic reasoning is deductive or inductive, we need the defining characteristics of deductive and inductive reasoning.
Consider Aristotle's syllogism: if A then B; if B then C; thus if A then C. In this deductive reasoning, the truths of the premises are transmitted to the truth of the conclusion. When done right (i.e, in a valid argument), if all the premises are true, then the the conclusion must be true. For this reason, deductive reasoning is called the truth preserving inference.
What does the raven inference reveal? It moves from particulars (sample) to universals (population). The characteristic of inductive reasoning is generalization. Generalization is always involved with uncertainly, or making a mistake. That is, even if the particulars (premises) are true, a statement regarding the population (conclusion) might not be necessarily true. Inductive reasoning is not truth preserving.
Traditionally, there is a division of labor in the study of two reasoning methods. Logicians study deductive reasoning: e.g., Aristotle's syllogism, Frege's first-order logic. In light of truth preserving-ness, a probability theory can be said to be deductive. That is, given the probability values of premises, the probability value of the conclusion must be true. Thus, Carnap's probabilistic logic can be also included in the deductive reasoning.
Statisticians study inductive reasoning. Since an inference from sample to population (from sample value to parameter value) is always involved with fallibility, statisticians need to quantify uncertainty, inherent in their inference. How to quantify the uncertainty led to different schools of statistics: the Neyman-Pearson school, Fisher-ians, and Baysians.