Inductive reasoning

We continue our initial foray into proper ways of reasoning by looking today at induction. In the previous post, we talked about deductive reasoning, where we use logic to derive conclusions from premises. That form of reasoning ensures that if the premises are true, then the conclusion necessarily follows from the premises (“necessity” is important here). Induction does not have the same degree of necessity, or support for its conclusions. For deduction, we say that the premises give total support to the conclusion. For inductive reasoning, we weaken the strength of that support, which becomes only probable. But what does an argument based on inductive reasoning look like?

A famous example goes: “The sun has always risen in the East. Therefore the sun will rise tomorrow in the East”. It is actually not certain that the sun will rise tomorrow in the East. The conclusion is warranted to some degree by the empirical evidence amassed so far, but is it true that something that happened many times in the past will happen tomorrow? Our argument actually hides an implicit premise: that the past always predicts the future. We should question that premise. We can reasonably expect that the sun will rise in the East tomorrow, maybe even with high probability (although Hume might disagree), but we do not know that with complete certainty, because there is no guarantee that the past always predicts the future. The premises could be true and the conclusion false.

Let’s look at another famous example, which involves a generalization from particulars, one type of inductive argument. Suppose you see a lot of white swans while walking around in various parks. You say: “Swans are white”. That would be a mistake. There is no guarantee that tomorrow, or elsewhere (maybe on a trip to Australia where there are black swans), you will not see a swan of another color. We can’t escape the fact that we do not find necessary support for the conclusion, we have only accumulated evidence, and that evidence could be defeated tomorrow by a new observation. We cannot be certain of the conclusion in the same way as in deductive reasoning.

A case could be made that inductive arguments usually have implicit premises that we could state to make the argument into a deductive one. We could say: “All the swans I’ve seen are white. If all the things of a certain category I’ve seen share some property, then all the things of that category have that property. Therefore all swans are white”. Now we have a deductive form for the argument, and we have made explicit a hidden premise that underpins how the inductive argument works. That argument is actually logically valid, there is no way its premises would be true and its conclusion false. However, the premise we have made explicit is not true. In inductive arguments that we put in “deductive form”, with all their premises explicit, there will always be at least one premise that is either false, or whose truth value we cannot ever ascertain. This usually happens because the premise attempts to say something about the future, which we cannot know (as in the first example), or about a class of objects in its totality, when that totality is not accessible (second example).

Note that depending on how many swans you have seen, you can be more or less confident about the conclusion. If you have seen only one, you would typically not be very confident in the conclusion that “all swans are white”, but if you have seen 1,000, you might be more confident, and if you’ve traveled to all the continents you should be even more confident … that not all swans are white! New observations have led you to revise your knowledge about swans, making it more precise, but contradicting your earlier theory that “all swans are white”. The confidence we have in the conclusion of an inductive argument has a probabilistic, and provisional flavor. We do not know what new observations we will make tomorrow.

Deductive and inductive reasoning are both used to acquire knowledge. In mathematics, we derive theorems from axioms via deductive proofs, relying on logic, in a first approximation. Deductive reasoning relies on logic to give us certainty about the theorems we derive, but that mechanism is very tautological. For example, to solve 2x = 6, we use logic at each step to infer that x = 3, but the fact that x = 3 is already present in the equation 2x = 6, it is not a new fact. In a sense, the theorems contain nothing that is not already contained in the axioms. That is the price to pay for certainty. We say that logical deduction is “analytical”, it analyzes the premises/axioms to deduce their logical consequences.

In the natural sciences however, we observe nature, and typically use inductive reasoning to infer general laws (of course, inference from particulars to universals is not the only activity in the natural sciences). For example, based on the observation of many swans, we try to say something about all swans. We say that induction is “synthetic” (or “ampliative”): it attempts to increase our knowledge by deriving new facts that are not necessarily contained in the observations/premises. We are more or less confident of the conclusions in the natural sciences, but never certain as we can be of a theorem given some axioms in mathematics: a contradictory observation can always be made tomorrow. We evaluate deductive arguments on their validity - it is not possible for the premises to be true and the conclusion false, but inductive arguments allow their premises to be true and their conclusion false at the same time. We evaluate inductive arguments only on their strengths, on how convincingly they persuade us of the conclusion, rather than their validity. We evaluate inductive arguments only on their strengths, i.e. how convincingly they persuade us of the conclusion, rather than their validity.

This is only the beginning of our exploration of deductive and inductive reasoning, we will have a lot more to say as we look at Hume, Kant, and many other philosophers who have weighed on these questions in epistemology.

Addendum - going back to the idea of inductive arguments in “deductive form”, and the claim that they will always have a false premise or a premise that cannot be verified, I think the deductive arguments in contrast have either premises whose truth value can in principle be verified, or they eject the problem of establishing the truth value of the premises altogether (in mathematics), by simply declaring the axioms to be “true” by definition, without referring back to the world. What makes the inductive argument less certain than a deductive one is these premises (often suppressed) that are unsound. But we overlook that and choose to believe in the conclusion of the inductive argument anyway because, provisionally at least, it is quite a productive way to acquire knowledge.

Addendum 2 - there is a mapping between inductive and deductive reasoning and types of algorithms in computer science: deductive reasoning would correspond to usual algorithms, which in the end, just derive consequences from initial conditions, and inductive reasoning, would correspond to machine learning, where a model is inferred from repeated observations. 

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