INDUCTION
There are three types of classical inference: analogy, deduction, and induction. Induction goes from the particular to the general. It generalizes the knowledge and information gathered from a limited number of cases to all cases.
I take a marble out of the bag; then a second marble, … then still another marble…
Even though the bag is not empty, I conclude that this is a bag of marybles.
Examining all the remaining objects one by one would be necessary to be sure, but that would take a long time.
A trade-off must be made between 1) the margin of uncertainty I can tolerate and 2) the time it would take to check the entire bag. I choose to save time, I check a few more items and conclude, “this is a bag of marbles.”
The induction is based on the similarity of the individuals, possibly based on just one feature, that I consider relevant.
An induction based on a single case is an example.
1. Forms of Induction
Complete induction — Induction is said to be complete and its conclusion is positive (valid, certain), if one proceeds by an exhaustive inspection of each individual. Such a procedure is possible only if one has access to all the members of the set. See definition by enumeration, §2.2.2
Induction from a representative subset to the set — A proposition found to be true in a carefully selected sample can be tentatively extended to the whole:
Forty percent of a representative sample of voters say they will vote for candidate Joni.
Therefore, Joni will receive forty percent of the vote on election day.
Depending on whether the sample is truly representative, and whether the respondents gave sincere answers, as well as assuming they don’t change their minds, the conclusion varies from almost certain to vaguely probable.
Induction from an essential property — Generalizing gfrom an accidental property of one sample to all other samples is risky. If the generalization is based on an essential property, the conclusion is positive, see example:
This is an ordinary Syldavian passport.
This passport mentions the holder’s religious affiliation.
Therefore, all Syldavian passports mention religious affiliation.
2. Refuting induction
A conclusion reached by induction is refuted by showing that it is based on a hasty generalization, i.e., on the examination of an insufficient number of cases. This is done by showing members of the collection who do not possess the that does not possess the desired trait. However, these cases can be considered exceptions, see proof by fact.
3. Induction in Mathematics: Recursive Reasoning
In mathematics, recursive reasoning is a form of induction that leads to positive conclusions (Vax 1982, [Mathematical induction or recursive reasoning]). It is effective in domains such as arithmetic, where a relation of succession can be defined between numbers. First, show that the investigated property holds for 1. Then, demonstrate that if it holds for an individual “i”, it also holds for its successor “i + 1”. The conclusion is that all the members have the tested property.
4. Induction as a Positive Method in Literary History
An inductive argument involves postulating a general law or tendency and testing it on a large number of cases. This process is typical of the positivist science of literature and ideas.
§ 2 The Diffusion of Irreligion among the Nobility and the Clergy
The diffusion of irreligion among the high nobility is considerable. General testimonies abound. Lamothe-Langon says, « Atheism was universal in what was called high society; to believe in God became ridiculous, and we were careful to protect ourselves. » The Memoirs of Ségur, those of Vaublanc, those of the Marquise de la Tour du Pin confirm what Lamothe-Langon writes. At Madame de Hénin, the Princess de Poix, the Duchess of Biron, the Princess of Bouillon, and in the officers were, if not atheists, at least deists. Most of the salons members were “philosophers”, who adopted the spirit of the philosophers, and the great philosophers were their most beautiful ornaments. This can be seen not only in the salons of the philosophers themselves, at d’Holbach’s, Madame Helvetius’s, Madame Necker’s, Fanny de Beauharnais’s (where we find 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. At 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 met or listened to Rousseau, Helvétius, Duclos, Voltaire, Diderot, Marmontel, Raynal, Mably. The Hôtel de la Rochefoucauld was the meeting place for the more or less skeptical and liberal noblemen, Choiseul, Rohan, Maurepas, Beauvau, Castries, Chauvelin, Chabot, who met with Turgot, d’Alembert, Barthélémy, Condorcet, Caraccioli, Guibert. Many others that could be mentioned here: the salons of the Duchesse of Aiguillon, who was « very much in love with modern philosophy, that is to say, with materialism and atheism »; Madame de Beauvau, the Duke of Levis, Madame de Vernage, the Count of 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, that are merely evoked. This assertion is followed by a similar statement, “most of the members of the salons are “philosophers” and philosophers are their most beautiful ornaments”, supported by twenty-eight names of philosophers. The lists of names are open; note that the last quoted name is never preceded by a final « and ».
The argument is compelling, but ad nauseam; reading the lists can be tedious.
The strength of the asserted principle depends on the number of cases considered. A small number gives reason for skepticism:
It hasn’t been sufficiently appreciated how insignificant is the number of these historical examples is on which the “laws” are claimed to be valid for all the past and future evolution of mankind. [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] says that history is a succession of oscillations between an organic epoch and a critical epoch; he gives two examples. A third, [Marx], that it is a succession of economic regimes, each of which violently eliminates its predecessor; he gives one single example!
Julien Benda, The Treachery of the Clerks, [1927].[2] Our emphasis.
It should be noted that Benda’s own assertion that, “the number of these historical examples on the basis which a “law” is claimed to be valid for all the past and future evolution of humanity is insignificant”, is supported by three examples.
[1] Daniel Mornet (1933). Les origines intellectuelles de la Révolution Française, 1715-1787. Paris: Armand Colin, pp. 270-271.
[2] Quoted after Julien Benda, La Trahison des clercs. Paris: Grasset, 1975, pp. 224-225.