Induction is one of the three classical modes of inference. Induction goes from the particular to the general; it generalizes to all cases findings and information gathered from a limited number of cases.

I draw a marble from the bag; then a second marble, … then still another marble…
The bag is still not empty; nonetheless, I conclude that this is a bag of marbles.

To conclude with certainty, all the remaining items would have to be examined, but it would take a long time. A trade-off must be found between 1) the margins of uncertainty I can tolerate and 2) the amount of time that would be needed to check the entire bag. I decide to save time I check some items and conclude, “this is a bag of marbles”.
Induction rests on similarity between the individuals, possibly based just on one feature, deemed relevant by the arguer.

An induction based on just one case is an example.

1. Forms of induction

Complete induction — Induction is said to be complete and its conclusion positive (valid, certain), if one proceeds by an exhaustive inspection of each individual. Such a process is possible only if one has access to all the members of the set.

Induction from a representative subset to the set — A proposition found true in a carefully selected sample can be tentatively extended to the whole:

40 per cent of a representative sample of voters polled declared their intention to vote for candidate Joni. So Joni will get 40 per cent of the vote on Election Day.

Depending on whether or not the sample is truly representative, whether or not people have given fanciful answers, the conclusion varies from almost certain to vaguely probable.

Induction from an essential characteristic — The generalization from an accidental property of a specimen to all other specimen is hazardous, but when based on an essential property, the conclusion is positive, S. Example:

This is a normal Syldavian passport.
This passport mentions the religious affiliation of the holder.
So all Syldavian passports mention religious affiliation.

2. Refutation of induction

A conclusion obtained by induction is refuted by showing that it proceeds from a hasty generalization, based on the examination of an insufficient number of cases. To that end, one exhibits members of the collection that do not possess the property.

3. Induction in mathematics: recursive reasoning

In mathematics, recursive reasoning is a form of induction which leads to positive conclusions (Vax 1982, [Mathematical induction or recursive reasoning]). It is operative in domains such as arithmetic, where a relation of succession can be defined. First, it must be shown that the property holds for 1; then, that if it holds for an individual “i”, it also holds for its successor “i + 1”. The conclusion is that all the members of the set possess the tested property.

4. Induction as a positive method in literary history

An inductive argument consists of establishing a general law or tendency and applying this to a large number of examples. This process is typical of the positivist science of literature and ideas.

§ 2 Diffusion of Irreligion in the Nobility and the Clergy
Diffusion of irreligion is considerable in the high nobility. General testimonies abound, ‘Atheism’, says Lamothe-Langon, ‘was universally spread in what was called high society; to believe in God was becoming ridiculous, and we were careful to guard ourselves’. The Memoirs of Ségur, those of Vaublanc, those of the Marquise de la Tour du Pin confirm what Lamothe-Langon writes. At Madame d’Hénin’s, the Princess de Poix, the Duchesse de Biron, the Princess de Bouillon, and in the ranks of officers, people are, if not atheist, at least deist. Most members of the salons were “philosophers”, and adopted the spirit of the philosophers, and the great philosophers are their most beautiful ornament. This may be seen not only in the salon of the philosophers themselves, at d’Holbach’s, Mme Helvetius’s, Mme Necker’s, Fanny de Beauharnais’s (where we see Mably, Mercier, Cloots, Boissy d’Anglas), but also among the great nobility. At the Duchesse d’Enville’s, one meets Turgot, Adam Smith, Arthur Young, Diderot, Condorcet; at the Count de Castellane’s, D’Alembert, Condorcet, and Raynal. In the salons of the Duchesse de Choiseul, the Maréchale de Luxembourg, the Duchesse de Grammont, Madame de Montesson, the Comtesse de Tessé, the Comtesse de Ségur (her mother), Ségur meets or hears Rousseau, Helvétius, Duclos, Voltaire, Diderot, Marmontel, Raynal, Mably. The Hôtel de la Rochefoucauld is the meeting place of the more or less skeptical and liberal great lords, Choiseul, Rohan, Maurepas, Beauvau, Castries, Chauvelin, Chabot, who meet with Turgot, d’Alembert, Barthélémy, Condorcet, Caraccioli, Guibert. There are many others who might be mentioned here: the salons of the Duchesse d’Aiguillon, who was ‘very infatuated with modern philosophy, that is to say, with materialism and atheism’, Madame de Beauvau, the Duke of Levis, Madame de Vernage, the Comte de Choiseul-Gouffier, the Vicomte de Noailles, the Duke de Nivernais, the Prince de Conti, etc.
Daniel Mornet, [The Intellectual Origins of the French Revolution], 1933[1]

The claim to be justified asserts that, “the diffusion of irreligion is considerable in the high nobility”. It is supported by an explicit testimony, accompanied by three others, which are merely evoked. This is followed by an affirmation of the same order, “most members of the salons are “philosophers” and philosophers are their most beautiful ornament”, supported by twenty-eight names of philosophers. The reasoning is irresistible, but the reading can be boring.

The strength of the asserted principle depends on the number of cases considered. Their small number gives some reasons for skepticism:

It hasn’t been sufficiently appreciated how insignificant is the number of these historical examples upon which are asserted the “laws” claiming to be valid for all the past and future evolution of the humanity. [Vico] claims that history is a succession of alternations between a period of progress and a period of regression; he gives two examples. [Saint-Simon] that it is a succession of oscillations between an organic epoch and a critical epoch; he gives two examples. A third, [Marx], that it is a series of economic regimes, each of which eliminates its predecessor by violence; he gives one single example!
Julien Benda, The Treachery of the Clerks, [1927].[2] Our emphasis.

It should be noted that Benda’s own claim that, “the number of these historical examples upon which is asserted a “law” claiming to be valid for all the past and future evolution of humanity is insignificant”, is itself backed by three examples.

[1] Daniel Mornet (1933). Les origines intellectuelles de la Révolution Française, 1715-1787. Paris: Armand Colin, p. 270-271.
[2] Quoted after Julien Benda, La Trahison des clercs. Paris: Grasset, 1975, p. 224-225.