How can induction be valid? Even if all of a large set of instances are true, the general conclusion is not guaranteed to be true. The usual example: One billion black crows leads to the conclusion that all crows are black, and then someone finds a white crow (the do exist because of the gene for albinism).
Several scientific theories based on induction turned out to be false. For example the inductive conclusion that heat is a fluid (caloric) was falsified by Wm. Thompson (Lord Kelvin).
Inductive arguments are not guaranteed to turn out true conclusions even if all the premises are true. Whereas valid deductive arguments are guaranteed to produce true conclusions if the premises are true.
The best an induction can produces is a reasonable probability or possibility that the conclusion is true. Which is not an argument against using induction. It is the only way we can go from a finite set of observations to a general statement or a law.
Induction is the process of inferring conclusions based on observation. Deduction is the process of deriving new conclusions from existing knowledge. Induction is like earning money; deduction is like spending it. You cannot have deduction without induction.
Induction is simply an observation of a causal connection. The law of causality has two aspects: (1) An event is determined by the circumstances in which it occurs, including the nature of the entities involved, and (2) the same cause has the same effect. Humeís assault on induction was really an attack on causality.
Humeís mistake was to see sensation, rather than perception, as the base of all knowledge. On the level of sensations, we only have momentary awareness of disparate qualities. He saw entities as constructed inventions of the mind, and events as autonomous occurrences disconnected from the entities that act. This is the basis for suggesting that establishing inductive connections involves finding more and more instances of that connection. This is how induction gets confused with probability theory.
In fact, the base of human knowledge is perception, which involves the direct awareness of entities, or objects as unified collections of properties. Actions depend on the nature of the entities that act. An entity with the same nature, under the same circumstances, will act in the same way. From this perspective, once a cause has been isolated, we can generalize on the base of one instance. Repetition plays no role in the reasoning process beyond replication as a test of accuracy.
We know that water freezes when the temperature gets sufficiently cool that the molecular movement slows down and the molecules stick to each other to form crystals. But how do we know that cooling temperatures caused this result and not some other factor, such as light or sound? We use Millís Methods: Agreement, Difference, Concomitant Variations, et. al. The Method of Agreement says: If two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which alone all the instances agree, is the cause (or effect) of the given phenomenon.
Once you have isolated temperature as the cause of the slowing molecules under the microscope, the job is done.
The error in the Black Swans example consists in the conclusion that color is an essential attribute somehow connected to the nature of Swans. That is an inductive hypothesis based on enumeration, not an inductive inference based on an analysis of the nature of Swans.