Induction (logical)

See also: Induction

With the difference of the Deduction which imposes proposals for a departure not presumedly true, the induction proposes to seek general laws starting from the observation of particular facts, on a probabilistic basis.

The starting idea of induction was that the repetition of a phenomenon increases the probability of it of seeing it reproducing. It is there properly the way in which reacts the brain at for example the Chien of Pavlov. The accumulation of concordant facts and the absence of counterexamples then make it possible to increase the level of plausibility of the law until the moment when one chooses by simplification to regard it as near a certainty: thus Second principle of thermodynamics. To in no case, however, one will not reach the certainty, very against example being likely to give this " immediately; loi" in question.

Thereafter, of the theorems as that of Cox will give to this empirical inductive step a mathematical base closes, and will make it possible to calculate the probabilities concerned without any arbitrary except for a starting position.

Example

For example: If the law of the universal gravitation determines that, and how, an apple which is detached from its tree will fall on the ground, the observation of the movement of this same apple makes it possible to establish the general law, but with a probability or a very weak certainty. So then, one observes that all apples and all body fall in the same way, if one then observes that the bodies in space respect the same law, the probability of the law will increase until becoming near a certainty. In the case of the universal gravitation, however, one observed that the Mercury orbit presented an effect of precession which was not not explained by the law (*). The law of the universal gravitation however remained used until Einstein proposes the theory of the General relativity which, it, explains the phenomenon. Despite everything, the universal gravitation remains used because it remains valid in the current cases, and it is simpler to use and understand that the theory of relativity.
(*) not explained by the law is an euphemism usually employed to say that the example contradicts the law, but which one does not have desire for the moment to reject the law because one does not have the best to replace it. To a lesser extent, not explained can as mean as one continues to use the law, because in the very large majority of the cases it is checked and that it is much simpler to use or to understand that another law more exacte.

Old vision of induction

In a general way, induction, contrary to the deduction, is a logically inaccurate reasoning, which is supported by its repeated checking, but which can be contradicted by a counterexample. It is however universally used for two reasons:
  • Other than the Logical and of the Mathématiques which explicitly consist in posing arbitrary axioms on basis of which they reasons by the deduction, all other sciences try to describe reality and cannot make it that exclusively on the basis of checking by the observation, which forces them to call upon induction and any possibility prohibits to them of using the pure deduction.
  • All the alive systems function on the basis of induction. the training by the Brain being based on its confrontation with reality, is primarily inductive, and, by extension, in Artificial intelligence, the system of training to Réseau of neurons are different from the systems Algorithmique S in what they are inductive, whereas the algorithmic systems are deductive. The natural selection, it even, while eliminating weakest by confrontation from the species with the difficulties from the existence, is also a basically inductive phenomenon.
Note: il is rather curious to observe that the principle of deduction is infinitely simpler than the principle of induction, however, the life adapts according to the principle of induction and, paradoxically, the brain which is conceived for induction is not a logical machine: it does not integrate spontaneously and must acquire the deduction which is however more simple.

It should be noticed that if induction is an intrinsically probabilistic reasoning, it is however impossible to evaluate the subjacent probability. Indeed, that Ci is a conditional probability and will remain always subjected to the choices of the conditions of its evaluation, knowing that there can be conditions of which one did not think and who would change the facts of the case completely.

exemple: If I meet only gray cats, it will be easy for me in to induce that all the cats are gray with a strong level of certainty. But if I carry out that the fact that the cats are gray could be specific to the area or I live, and that it could exist another area or all the cats are russet-red and still another with green cats (to adopt a real assumption AND an approach absurdity), my evaluation of this level of certainty will be completely put by it in cause.
Moreover, the level of certainty of my law will depend on the coefficient with which I accept that it is not completely general and admits exceptions.
Je can consider, for example, that general relativity, is a particular case which applies only in real situations, but that in general does not blame the theory of the universal gravitation, or on the contrary, I can decide that the universal gravitation must be precise and exact, in which case, it is fausse.

Some traditional examples

Most famous of inductions is probably the example that Aristote gives some:
the ass, the mule, the horse live a long time;
gold, they are there all the animals without Fiel ;
thus, all the animals without gall live a long time.

It is seen well that induction rests on an assumption: that “they are there all the animals without gall”. The inductive Syllogisme is known as hypothetical (not-scientist):

Socrate is bald person ;
Socrate is a man ;
thus the men are bald people .
has a false conclusion, because Socrate cannot represent the man, on the matter.

An example celebrates induction of Claude Bernard, illustrating the scientific method:

a normally nourished rabbit has a basic urine ;
the same rabbit with jeun has an acid urine ;
thus all the herbivores have a basic urine ;
whereas all the badly nourished animals and the carnivores have an acid urine.

One sees the use of induction there: starting from observations (which are always particular proposals), induction produces hypothetical general proposals which are then testable. It is the analysis of Claude Bernard, like that of Karl Popper.

Hume considered ( Enquête into the human understanding , VII, 2) that the origin of induction (the idea of connection) is the practice. It thus gives a psychological force to the induction, which does not have a logical force.

Karl Popper ( Conjectures and refutations , p. 78) watch that “Hume ever recognized all the range of its own logical analysis”, and proposes an inversion: “instead of explaining our propensity to suppose the existence of regularity like an effect of the repetition, I imagined to explain what is repetition in our eyes like the result of our tendency to suppose and to seek regularity”.

A long time purely empirical, the process of induction was formalized by the Théorème of Cox-Jaynes which confirms the rationality of the method for the update of knowledge, quantifies it, and unifies the universe of the Boolean Logique with that of the Probabilité S (seen either as a passage in extreme cases of frequencies, but like numerical translation of a state of knowledge in this Paradigme).

See too

Related articles

External bonds

  • Frederic Fabre Refutation of probabilistic induction and corroboration

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