11.2 Varieties of Induction Incomplete Induction “Inverted Laws of Logic”

In inductive inference, the relation of premises and conclusions is not based on a logical law, and the conclusion follows from the received assumptions not with logical necessity, but only with a certain probability. Induction can give a false conclusion from true premises; its conclusion may contain information missing in the parcels. The concept of induction (inductive inference) is not quite clear. Induction is defined, in essence, as “non-deduction,” and is an even less clear concept than deduction. It is possible, nevertheless, to indicate a relatively solid "core" of inductive methods of reasoning. It includes, in particular, incomplete induction, the so-called "inverted laws of logic", confirmation of the consequences, targeted justification and confirmation of the general situation with the help of an example. A typical example of inductive reasoning is also an analogy.

Incomplete induction

Inductive reasoning, the result of which is the general conclusion about an entire class of objects based on the knowledge of only certain objects of this class, is called incomplete, or popular, induction.

For example, from the fact that inert gases helium, neon and argon have a valence equal to zero, we can conclude that all inert gases have this valence. This is an incomplete induction, since knowledge of the three inert gases applies to all such gases, including those not specifically considered krypton and xenon.

Sometimes the enumeration is quite extensive and, nevertheless, the generalization based on it turns out to be erroneous.

“Aluminum is a solid body; iron, copper, zinc, silver, platinum, gold, nickel, barium, potassium, lead - also solids; therefore, all metals are solids. ” But this conclusion is false, since mercury - the only metal of all - is liquid.

Many interesting examples of such erroneous, hasty generalizations that have been encountered in the history of science are given in his works by V. I. Vernadsky.

Until the 17th century, until the concept “force”, “some forms of objects, and by analogy some forms of the paths described by objects, were finally included in science, they were considered essentially capable of producing infinite motion. In fact, let us imagine the shape of an ideally regular ball, put this ball on a plane; theoretically he cannot keep still and will be in motion all the time. This was considered a consequence of the ideally round shape of the ball. For the closer the shape of the figure to the ball, the more accurate the expression will be that such a material ball of any size will stay on the ideal mirror plane on one atom, i.e. will be more able to move, less stable. Ideally round shape, believed then, in its essence is capable of supporting the movement once reported. This way was explained by the extremely fast rotation of the celestial spheres, epicycles. These movements were once communicated to them by the deity and then continued for centuries as a property of an ideal ball shape. ” “How far these scientific views are from modern, and meanwhile, in essence, these are strictly inductive constructions based on scientific observation. And even now, among research scientists, we see attempts to revive, essentially, similar views. ”

A hasty generalization, i.e. generalization without sufficient reason is a common mistake in inductive reasoning.

Inductive generalizations require some care and caution. Much here depends on the number of cases studied. The more extensive the base of induction, the more likely the inductive conclusion. Also important is the diversity and heterogeneity of these cases.

But the most essential is the analysis of the nature of the connections of objects and their attributes, the proof of the non-randomness of the observed regularity, its rootedness in the essence of the objects under study. The identification of the causes that give rise to this regularity makes it possible to supplement pure induction with fragments of deductive reasoning and thereby strengthen and strengthen it.

General statements, and in particular the scientific laws obtained by the inductive method, are not yet full truths. They have to go a long and difficult way, while from probabilistic assumptions they will turn into the constituent elements of scientific knowledge.

Induction finds application not only in the sphere of descriptive statements, but also in the field of assessments, norms, advice, and similar expressions.

The empirical justification of assessments has a different meaning than in the case of descriptive statements. Ratings may not be supported by references to what is given in direct experience. At the same time, there are ways to substantiate assessments that are in some respects similar to those used to substantiate descriptions and which can therefore be called quasi-empirical. These include various inductive reasoning, among the premises of which there are estimates and the conclusion of which is also an estimate or a similar statement. Among these methods are incomplete induction, analogy, reference to a sample, targeted justification (confirmation), etc.

Values ​​are not given to man in experience. They do not speak about what is in the world, but about what should be in it, they cannot be seen, heard, etc. Knowledge of values ​​cannot be empirical, the procedures for obtaining it can only look like the procedures for obtaining empirical knowledge.

The simplest and at the same time unreliable method of inductive justification of estimates is incomplete (popular) induction. Her general scheme:

S1 should be R.

S2 should be R.

Sn should be R.

All S1, S2, ..., Sn are R.

All S must be R.

Here, the first packages are estimates, the last package is a descriptive statement; conclusion - assessment. For example:

Suvorov should be persistent and courageous. Napoleon must be persistent and courageous. Eisenhower should be persistent and courageous. Suvorov, Napoleon, Eisenhower were the generals.

Every commander must be persistent and courageous.

Along with incomplete induction, it is customary to single out complete induction as a special kind of inductive reasoning . In her statements about each of the objects included in the considered set, it is stated that he has a certain property. In conclusion, it is said that all objects of this set possess this property.

For example, the teacher, reading the list of students of a class, makes sure that everyone named by him is present. On this basis, the teacher concludes that all students are present.

In full induction, conclusion with necessity, and not with some probability, follows from the premises. This induction is thus a kind of deductive reasoning.

The so-called mathematical induction, widely used in mathematics , also applies to deduction.

F. Bacon, who initiated the systematic study of induction, was very skeptical of popular induction, based on a simple listing of supporting examples. He wrote: “Induction, which is accomplished by simple enumeration, is a child’s thing, it gives shaky conclusions and is endangered by conflicting particulars, deciding for the most part on the basis of a smaller number of facts, and only those that are available ".

Bacon contrasted this “children's thing” with the specific inductive principles of establishing causal relationships described by him. He even believed that the inductive path of discovery of knowledge offered by him, which is a very simple, almost mechanical procedure, "... almost equalizes talents and leaves little to their superiority ...". Continuing his thought, one can say that he hoped almost to create a special “inductive machine”. Entering into this kind of computing machine all the proposals related to the observations, we would get at the output an exact system of laws explaining these observations.

Bacon’s program was, of course, pure utopia. No "inductive machine" that processes facts into new laws and theories is impossible. Induction, leading from particular statements to general ones, will give only probable, but not reliable, knowledge.

All this once again confirms the simple idea: the cognition of the real world is always creative. Standard rules, principles and techniques, no matter how perfect they are, do not guarantee the authenticity of new knowledge. The most strict adherence to them does not protect against errors and delusions.

Any discovery requires talent and creativity. And even the use of various techniques, to some extent facilitating the path to discovery, is a creative process.

"Inverted laws of logic"

Under the "inverted laws" refers to formulas derived from having the form of implication (conditional utterance) of the laws of logic by changing the places of the base and effect. It has been suggested that all "inverted laws of logic" can be attributed to inductive reasoning schemes. For example, if the expression:

"If A and B, then A"

there is a law of logic, the expression:

“If A, then A and B”

There is an inductive inference scheme. Similarly for:

"If A, then A or B" and schemes:

"If A or B, then A".

Similarly for the laws of modal logic. Since expressions:

“If A is possible , then A” and “If A is necessary , then A” are laws of logic, then the expressions:

“If A is possible , then A” and “If A is necessary, then A is necessary” are inductive reasoning schemes. The laws of logic are infinitely many. This means that inductive reasoning schemes are infinite.

The assumption that “inverted laws of logic” are inductive reasoning schemes, however, comes up with serious objections: some “inverted laws” remain laws of deductive logic; a series of “inverted laws”, when interpreted as induction schemes, sounds quite paradoxical. “Inverted laws of logic” does not, of course, exhaust all possible induction schemes.