Every swan observed for centuries in Europe was white, so Europeans concluded that all swans are white. The premises were true, the reasoning was exactly what reasoning of that kind is supposed to look like, and the conclusion was false.

That is induction, and the swan case is not a failure of it. It is what induction is: the conclusion says more than the premises do.

What it looks like

Enumerative. These cases had a property, so all of them do. Every invoice from this client was paid late, so the next one will be.

Statistical. A proportion in a sample is taken to hold in a population.

To the best explanation. The data would make sense if some hypothesis were true, so the hypothesis is probably true. This is what a diagnosis is, and what most detective work is, and it carries more weight than enumeration while having exactly the same gap.

Analogical. Two things are alike in several respects, so they are alike in another. The weakest form and the most used.

Why it cannot be made safe

An inductive argument is ampliative — its conclusion contains information not in the premises. That is the point of it: a conclusion that contained only what was already stated would tell you nothing new.

The cost is that no inductive argument, however good, guarantees its conclusion. Adding evidence can strengthen it without closing the gap, because the gap is structural rather than a matter of degree. A thousand white swans and a million make the conclusion better supported and no more certain.

This is the defeasibility of induction: a further observation can overturn a conclusion that was perfectly well supported. Deduction is not like this. If a deductive argument is valid and its premises are true, nothing later can make the conclusion false.

Whether the practice can be justified at all — without assuming the very regularity it is trying to establish — is the problem of induction, and it is a separate question from what induction is and how it works.

Why nobody can stop

Every claim that reaches beyond what has been observed is inductive. Every scientific law, every forecast, every expectation that the next day resembles this one.

There is no non-inductive substitute. You cannot deduce tomorrow from today, because nothing about today logically entails anything about tomorrow. So the alternative to reasoning inductively is not reasoning more rigorously; it is having no beliefs about anything unobserved.

That is why the scientific method is built to manage induction rather than to avoid it: controlled comparison, replication, and the deliberate search for the case that would refute — all of which narrow the gap without closing it. Popper went further and argued that science does not use induction at all, only conjecture and falsification. Whether that description survives contact with how science is actually done is still argued, and the position is a real one rather than a dodge.

What to take from it

The useful discipline is to notice how far past the evidence a conclusion reaches, and to be explicit about it.

Three cases that behaved a certain way is a thin basis for a general claim, and people state such conclusions with the confidence of somebody who has thirty. Base rate neglect and survivorship bias are both failures of induction with specific shapes — one ignores the denominator, the other reasons from a sample selected by the outcome.

And the strongest inductive move available is the one that costs something: look for the case that would refute the generalization rather than for another that fits. A confirming instance was always going to be easy to find.