Inductive inference

The term is first found in Socrates ^[3] ( ancient Greek. Ἐπαγωγή ). But Socrates' induction has little to do with modern induction. Socrates by induction means finding a general definition of a concept by comparing particular cases and eliminating false, too narrow definitions.

Aristotle pointed to the peculiarities of inductive inference (Analyt. I, Vol. 2, § 23, Anal. II, Vol. 1, § 23; Vol. 2, § 19, etc.). He defines it as an ascent from the particular to the general. He distinguished complete induction from incomplete, pointed out the role of induction in the formation of first principles, but did not clarify the basis of incomplete induction and its rights. He regarded it as a method of inference, the opposite of the syllogism. Syllogism , according to Aristotle, indicates by means of the average concept that the higher concept belongs to the third, and induction with the third concept indicates that the highest average belongs.

In the Renaissance , the struggle against Aristotle and the syllogistic method began, and at the same time they began to recommend the inductive method as the only fruitful in the natural sciences and opposite to the syllogistic. In Bacon, one usually sees the ancestor of modern induction, although justice also requires mentioning of his predecessors, for example, Leonardo da Vinci et al. Praising induction, Bacon denies the meaning of syllogism (“syllogism consists of sentences, sentences consist of words, words are concept signs; if the concepts that make up the case are indistinct and hastily distracted from things, and built on them can not have any strength "). This negation did not follow from the theory of induction. Bacon's induction (see his Novum Organon) not only does not contradict syllogism, but even requires it. The essence of Bacon’s teaching comes down to the fact that with gradual generalization one must adhere to well-known rules, that is, one needs to make three reviews of all known cases of manifestation of a known property in different subjects: a review of positive cases, a review of negative ones (that is, a review of subjects similar to the first in which However, the property under study is absent) and a review of cases in which the property under investigation manifests itself in various degrees, and from this we can make a generalization (“Nov. Org.” LI, aph. 13). According to the method of Bacon, it is impossible to draw a new conclusion without summing up the object under study under general judgments, that is, without resorting to syllogism. So, Bacon could not establish induction as a special method, the opposite of the deductive .

The next step was taken by J. Art. Millem Every syllogism , according to Mill, contains the petitio principii; any syllogistic conclusion is actually from the particular to the particular, and not from the general to the particular. Considering induction, Mill firstly asks a question about the basis or right for an inductive conclusion and sees this right in the idea of ​​a uniform order of phenomena, and, secondly, reduces all methods of inference in induction to four basic ones: the method of consent (if there are two or more than the case of the phenomenon being studied converges in only one circumstance, then this circumstance is the cause or part of the cause of the phenomenon being studied, the method of difference (if the case in which the phenomenon under study occurs and the case in which it does not occur is completely in all details, except for the studied one, the circumstance that occurs in the first case and is absent in the second, is the reason or part of the cause of the phenomenon being studied), the residual method (if in the phenomenon under investigation some of the circumstances can be explained by certain reasons, then the rest is explained from the remaining preceding facts) and the method of corresponding changes (if, following a change in one phenomenon, a change in another is noticed, then we can infer a causal relationship between them). It is characteristic that, upon closer examination, these methods turn out to be deductive methods; for example the residual method is nothing but determination by exception. Aristotle, Bacon and Mill are the main points in the development of the induction theory; only for the sake of detailed elaboration of some questions one has to pay attention to Claude Bernard (“Introduction to Experimental Medicine”), to Esterlena (“Medicinische Logik”), Herschel, Liebig, Vevel, Apelt and others.

Distinguish double induction:

• full ( lat. inductio completa ) and
• incomplete ( lat. inductio incompleta or per enumerationem simplicem ) ^[4] .

Complete induction

In full induction, we conclude from a complete enumeration of species of a certain genus to the whole genus; Obviously, with such a method of inference we get a quite reliable conclusion, which at the same time expands our knowledge to a certain extent; This method of reasoning can not cause any doubts. Having identified the subject of a logical group with objects of private judgments, we will get the right to transfer the definition to the whole group.

Scheme full induction:

The set A consists of the elements: A ₁ , A ₂ , A ₃ , ..., A _n .

• A ₁ has the sign B
• A ₂ has the sign B
• All elements from A ₃ to A _n also have the sign B

---

Consequently, all elements of the set A have the sign B.

Incomplete induction

The method of summarizing the characteristics of some elements for the whole set in which they belong Incomplete induction is not evidence-based from the point of view of formal logic , it can lead to erroneous conclusions. However, incomplete induction is the main way of obtaining new knowledge. The probative power of incomplete induction is limited, the conclusion is probabilistic in nature, it requires adduction of additional proof.

Incomplete induction scheme:

The set A consists of the elements: A ₁ , A ₂ , A ₃ , ... A _k , ... A _n .

• A ₁ has the sign B
• A ₂ has the sign B
• All elements from A ₃ to A _k also have a sign B

---

Therefore, it is likely that A _{k + 1} and the remaining elements of the set A have the sign B.

An example of an erroneous result:

• In Argentina, Venezuela and Ecuador they speak Spanish.
• Argentina, Venezuela and Ecuador - Latin American countries.

---

Consequently, it is likely that Spanish is spoken in every Latin American country.

Incomplete I. by construction resembles the third figure of the syllogism, differing from it, however, by the fact that I. seeks to general conclusions, while the third figure allows only private ones.

Inference on incomplete I. (per enumerationem simplicem, ubi non reperitur instantia contradictoria) seems to be based on habit and gives the right only to a probable conclusion in the whole part of the statement that goes beyond the number of cases already investigated. Mill in explaining the logical right to conclude with incomplete I. pointed out the idea of ​​a uniform order in nature, by virtue of which our belief in an inductive conclusion should increase, but the idea of ​​a uniform order of things is itself the result of incomplete induction and, therefore, I. cannot serve as a basis . In fact, the basis of the incomplete I. is the same as the full, as well as the third figure of the syllogism, that is, the identity of private judgments about the subject with the whole group of subjects. "In incomplete I. we conclude, on the basis of a real identity, not just some items with some members of the group, but such items whose appearance in front of our consciousness depends on the logical characteristics of the group and which are before us with the authority of the representatives of the group"

The task of logic is to indicate the boundaries beyond which the inductive conclusion ceases to be legitimate, as well as the auxiliary techniques used by the researcher in the formation of empirical generalizations and laws. There is no doubt that experience (in the sense of experiment) and observation serve as powerful tools in the investigation of facts, delivering material by which the researcher can make a hypothetical assumption that should explain the facts.

Any comparison and analogy, pointing to common features in phenomena, serves as the same instrument, and the commonality of phenomena suggests that we are dealing with common causes; thus, the coexistence of phenomena, to which the analogy points, in itself does not yet contain an explanation of the phenomenon, but gives an indication of where to look for explanations. The main relation of phenomena, which I. has in mind, is the relation of causality , which, like the most inductive conclusion, rests on an identity, for the sum of conditions, called a cause, if it is given in its entirety, is nothing more than a cause caused by a consequence . The validity of inductive confinement is beyond doubt; however, logic must strictly establish the conditions under which an inductive conclusion can be considered correct; the absence of negative instances still does not prove the correctness of the conclusion. It is necessary for an inductive conclusion to be based on as many cases as possible, for these cases to be as diverse as possible, to serve as typical representatives of the whole group of phenomena that the conclusion concerns, etc.

For all that, inductive conclusions easily lead to errors, of which the most ordinary result from a multiplicity of causes and from the confusion of temporal order with causal one. In an inductive study, we always deal with the consequences to which we must find the causes; their finding is called the explanation of the phenomenon, but the known consequence may be caused by a number of different reasons; the talent of an inductive explorer lies in the fact that he gradually chooses from the set of logical possibilities only that which is actually possible. For human limited cognition, of course, various causes can produce the same phenomenon; but full adequate knowledge of this phenomenon can detect signs indicating its origin from only one possible reason. The temporal alternation of phenomena is always an indication of a possible causal relationship, but not every alternation of phenomena, even if correctly repeated, should certainly be understood as a causal relationship. Very often we conclude post hoc - ergo propter hoc ^[5] , in this way all the superstitions arose, but here is the correct indication for inductive inference.

• The method of mathematical induction is a deductive method (so named because of the use of the induction axiom ).

• Concept formation
• Deductive reasoning
• Axiom of choice
• The Paradox of the Ravens
• Induction problem
• Statistical syllogism

• Vladislavlev MI. English inductive logic // Journal of the Ministry of National Education. 1879. P.152. November. P.110-154.
• Svetlov V. A. The Finnish School of Induction // Philosophy Questions. 1977. No. 12
• Inductive logic and the formation of scientific knowledge [Coll. articles / USSR Academy of Sciences, Institute of Philosophy]. M., 1987.
• Mikhalenko Yu. P. Antique teachings on induction and their modern interpretations // Foreign Philosophical Antiquity. Critical analysis. M., 1990. P.58-75.