induction(redirected from Inductional)
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induction, in electricity and magnetism
induction, in logic
See also R. Swinburne, ed., The Justification of Induction (1974); J. Cohen, An Introduction to the Philosophy of Induction and Probability (1989).
a form of generalization, associated with the anticipation of results of observations and experiments on the basis of data from previous experience. Accordingly, one speaks of empirical, or inductive, generalizations, truths of experience, and, finally, empirical laws. One of the justifications of induction in the practice of scientific research is the cognitive necessity of an overall view of groups of homogeneous facts, making possible the explanation and prediction of natural and social phenomena. In induction this overall view is usually expressed by means of new concepts, which seem to decipher the “hidden meaning” of observed results, and is fixed in formulas of causal or statistical laws.
Induction generally begins with analysis and comparison of observational or experimental data. As the set of these data is expanded, the regular recurrence of a specific property or relationship may be identified. When frequent repetition without exceptions is observed in experience, this prompts confidence in the universality of such a repetition and leads naturally to an inductive generalization, the supposition that such will be the case in all similar occurrences. If all of these occurrences are exhausted through experimental studies, then inductive generalization is trivial and represents only a condensed report of the facts. Such induction is called full or complete induction and is frequently viewed as deduction, since it may be represented as a scheme of deductive inference. This, in particular, is done in the idealized form of induction referred to as infinite induction.
The practice of both everyday and scientific thinking is characterized by generalizations based on the investigation not of all occurrences but only of some, since as a rule the number of all occurrences is limitless in practice and theoretical proof for an infinite number of these occurrences is impossible. Such generalizations are called incomplete induction. Incomplete induction does not represent logically founded reasoning. From the point of view of logic, founding reasoning means finding a corresponding logical law, but no logical law corresponds to a transition from the particular to the universal. From the point of view of logic, only those conclusions are valid that do not require any information beyond that which is contained in the premises in order to be obtained. But the conclusions of incomplete induction always tell more than that which is contained in the premises. This, essentially, is the cognitive sense of induction: the abstracting work of thinking advances despite insufficient practical knowledge.
Incomplete induction may result not only from the number of premises (incompleteness with respect to the number of premises) but from the nature of such premises as well (incompleteness with respect to the nature of the premises). For example, the nature of premises—experimental data—may be defined by the experimental procedure of measurement, which, as is known, cannot in principle yield “absolutely exact” results. In this sense any induction associated with the generalization of results of measurement (that is, essentially any empirical law of quantitative correlation between values) is incomplete. Assuming freedom from “shifts in space and time,” a law is an abstract form of expression of universality in nature and thereby of infinity. But in relation to the infinite character of phenomena encompassed by a law, our experience can never be concluded—it is impossible to traverse the infinite. This means that induction leading to the formulation of a law of nature is incomplete with respect to both premises and the verifiability of conclusions drawn. Generally speaking, this makes such induction problematic.
These problematic aspects are viewed by philosophical criticism as the weak point of incomplete induction. Therefore, incomplete induction is generally understood to be a source of assumed propositions—hypotheses—which are then verified by other means. Nevertheless, a positive answer to the question of whether or not it is necessary to attempt to increase the number of instances confirming incomplete induction, if no increase in this number is capable of overcoming epistemological skepticism associated with the incomplete nature of our experience, is prompted by the fact that given fully rational assumptions there exist several confirming instances for which, from the point of view of minimization of expected loss, incomplete induction is a “fully suitable” form of generalization. Of course, this answer, in a certain sense, is a pragmatic one and cannot serve as an answer to other questions about principles of induction, such as, for example, the epistemological or ontological problems that form the “induction problem,” posed as an object of philosophical discussions even in ancient times.
Inductive logic, which is indebted to Socrates for the very concept “inductive reasoning,” grew out of an attempt to solve the problem of induction. Socrates, however, did not view induction as a generalization of data of experience but as a method of definition, a “path” to the true (philosophical) meaning of concepts through analysis of individual examples from “everyday” use. Aristotle was the first to view induction in association with generalization of observations and signifying, essentially, a method of inferring, by means of which “the universal is demonstrated on a basis of the fact that the particular is known” (Posterior Analytics, 71al-71al3; Russian translation, Moscow, 1952). This Aristotelian view was adopted by philosophers of the Epicurean school, who defended induction in a dispute with the Stoics as the only authoritative method of proving the laws of nature. It was then that the induction problem emerged for the, first time. Specifically, in substantiating induction, the Epicureans proposed what appeared to them to be an empirical criterion but what was in fact a completely logical one: the absence of facts interfering with inductive generalization, that is, contradictory examples.
This criterion, revived by F. Bacon, became the basis of the form of inductive logic whose first historical variant was the inductive methods of Bacon and Mill. The importance of the contradictory example results from the fact that observations (facts) favoring inductive generalization may only in varying degrees confirm induction but can never serve as proof, while a single contradictory example, from a purely logical point of view, necessarily refutes the results of induction. If observational data allow us to propose several inductive generalizations or hypotheses based on such data, then the refutational force of a contradictory example may be used in a completely positive manner for corroboration of one (or several) of them. The only requirement for this is alternative hypotheses, that is, ones linked in such a way that refutation of one of them serves to confirm all others. It is natural then to attempt to create an experimental situation that eliminates all hypotheses save one. The process of eliminating hypotheses by means of a refutational experiment was referred to by J. S. Mill as eliminative or scientific induction. If from a number of possible hypotheses all are eliminated save one, elimination will be complete. If there remain several unrefuted hypotheses, that is, ones for which contradictory examples were not able to be constructed, elimination will be partial. Assume, for example, that a group of events αβγ is followed by a group of events ABC. Observational data make it possible to propose a number of alternative hypotheses: either “α is a result of A” or “α is a result of B,” or “α is a result of C.” Which of these hypotheses is correct? It is evident that an experiment which establishes that only βγ are results of BC will also refute the last two hypotheses, and elimination will be complete.
Both Bacon and Mill attempted to reveal the apodictic (necessary) bases of induction within a framework of the methodology of empiricism. It appeared that the refutational experiment would serve as precisely such a basis. However, in encroaching upon the sphere of empirical facts, the theory of the refutational experiment proves to be “too logical”; it does not take into account, first, that in such a case results obtained through logic depend on the nature of “extralogical” assumptions and cannot exceed the accuracy of the latter, and, second, that observations and experiments always provide only a “relative demonstration.” As an example of this it is sufficient to compare the experiments of A. J. Fresnel and J. B. L. Foucault, which refute the corpuscular model of light in favor of a wave model, with the photoelectric effect and the Millikan experiment on dislodging electrons from small specks of dust, which refute the wave model in favor of the corpuscular. Moreover, further analysis of Mill’s methods showed that they all, essentially, represent a unification of methods of deductive inference with incomplete induction. If the former give demonstrative force to these methods, then the latter eliminates it, so that, in this sense, the degree of persuasiveness of scientific induction cannot exceed the degree of persuasiveness of incomplete induction.
Recognition of this fact has led the majority of “empirically minded” researchers to look for probabilistic bases of induction. Attempts were undertaken to relate theories of induction to theories of probability and inductive logic to probability logic. Prominent among systematic attempts of such a kind are theories in which only the plausibility of inductive transfer from observational data to inductive generalizations is evaluated by probability measure, while no probability is attributed to the inductive generalization itself: the inductive generalization may be either true or false but only one of the two. It may be said that such an approach preserves the principles of classical logic to the detriment of certain principles of empiricism. Indeed, if our relationship to propositions is based on the principle of bivalence, then the problematical nature of the results of induction must have only a subjective meaning, reflecting the transient fact of our knowledge or ignorance of the genuine state of affairs independent of our experience. If, on the contrary, one is to rely exclusively on the data of experience in relation to the premises of induction, inductive generalizations, and consequences of such generalizations, then in any “probability approach” to induction the laws of nature must be seen only as more or less probable hypotheses, and facts confirming them must also be viewed as random ones. This then makes any proposition about the world “problematic in principle,” and removes it from the sphere of classical logic. Reliance on the “approximately valid” nature of inductive generalizations does not alter the state of affairs, since, from the theoretical point of view, the smallest inaccuracy is, in principle, an absolute inaccuracy.
The conclusion about the probabilistic character of laws of nature is to a certain degree indebted to the view that knowledge of “the universal” is essentially inductive and possible only on a basis of empirical observations and that empirical observations in themselves are not sufficient to demonstrate necessity. However, it is known that many inductive generalizations are founded not only on observations but on purely speculative principles, such as the principle of inertia or the general theory of relativity, which enter into formulations of theories and are accepted as axioms of our scientific picture of the world. By means of such principles, both inductive generalizations and confirmations of their results—observed phenomena—are derived by purely logical means. In other words, human reason does not trust the “factual basis” of inductive generalizations in an a priori manner. It strives to provide a logical foundation for a majority of such generalizations, subordinating them to purely theoretical postulates. These postulates are more indebted to the heuristic, or creative, work of thinking, so that for inductive generalizations, no matter how broad, they not only are based on experimental data but also manifest (frequently in an unrecognized fashion) an amazing confidence in the capacity of thought to divine the “course of nature.” The objective significance of this purely psychological confidence is also seen in the probability model of induction. A conclusion justifying the search for examples confirming an incomplete induction is based on the premise that confirmation is possible only if the inductive generalization, independent of this confirmation, has some a priori plausibility.
The advisability of trusting inductive generalizations, in addition to reasons considered in inductive logic, has yet another, purely epistemological basis, prompted by the difference between the epistemological precision of an empirical law—its practical applicability in a corresponding (infinite, but always limited) objective sphere—and the precision of measurement of its inductive basis. When the law of universal gravitation was discovered, empirical bases (observations and experiments) allowed I. Newton to verify this law with an exactness of only approximately 4 percent. Nevertheless, verification more than two centuries later showed the law to be correct with an exactness of up to 0.0001 percent. Generally speaking, as soon as we speak of a law of nature, the epistemological precision of a generalization (law of nature) for increasing precision of measurement for an inductive premise in a sufficiently broad interval is continuous during such an interval. Therefore, it would be unwise to make every step of application of the law dependent upon measurement procedures, although the precision of measurement of generalization cannot, of course, exceed the precision of its empirical basis.
In certain cases of “inductive discovery” the basis of induction is adequate in terms of the significance generally attributed to its results. For example, the experience of Newton’s contemporaries was fully sufficient to confirm his second law, as well as to make convincing its universal validity. In order to observe that the mass of a body is a function of its velocity, experiments with velocities almost equal to the velocity of light were required; this was the experience of a different historical age. This means that if it is true that experience is the source and touchstone of all our knowledge, then it is true only with the stipulation that experience is seen in its historic perspective as the historical praxis of man and not only as experience “on a given day.” Insofar as “experience on a given day” remains the only empirical source of generalizations, induction requires, at least psychologically, the support of principles not dependent on this basis.
One of these principles is that of the knowability of the world, which defines all purposeful activity of scientific thinking. The fundamental content of this principle is eloquently expressed in the idea of Galileo that human reason perceives certain truths as perfectly and with the same degree of absolute reliability as nature itself. At first glance it would appear that numerous changes in scientific views and reformulations of old laws find little in common with this idea. Nevertheless, a circumstance fundamental to the viability of “old” theories is that the epistemological precision and completeness of scientific abstractions are unambiguously defined by experience within extremely broad limits, so that each scientific abstraction is associated with a corresponding interval, within which an increase in precision of experimental data changes nothing in the theoretical evaluation of the generalization and in its practical use. Discovery of the “fallibility” of abstraction—inductive generalization—is, essentially, only a manifestation of the limits of this interval, of the limits of the applicability of abstraction. And although these limits are not known beforehand, this does not change the fact that within these limits, that is, within the interval of epistemological precision of abstraction, it has the same absolute reliability that nature itself possesses.
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M. M. NOVOSEIOV
(1) In physiology, induction is the dynamic interaction of the nerve processes of excitation and inhibition, expressed in the fact that inhibition in a group of nerve cells induces excitation (positive induction) and, conversely, an initially produced process of excitation induces inhibition (negative induction).
Both positive and negative induction may have two forms: (a) simultaneous induction: excitation in one area induces and intensifies simultaneous inhibition in surrounding areas, and a focus of inhibition induces the process of excitation; and (b) successive induction: alternation of relationships proceeds with time—excitation at the point of its development is replaced by inhibition after cessation of the action of the stimulus, and vice versa. The degree of expression and the duration of induction depend on such conditions as the strength of the excitation or inhibition and the distance of the focus of initial activity from the point to be induced. The phenomenon of induction is characteristic of all parts of the nervous system. It limits the spread (irradiation) of nerve processes and promotes their concentration. Negative induction may be seen, for example, when a strong stimulation of the auditory center (an abrupt ring) produces inhibition in other nerve centers, such as the alimentary center, which is expressed in cessation of saliva secretion.
I. V. ORIOV
(2) In embryology, induction is the influence of some parts of a developing embryo (inductors) on other parts of it (the reacting system), which is effected upon their contact with one another and which determines the direction of development of the reacting system, similar to the direction of differentiation of the inductor (homotypic induction) or different from it (heterotypic induction).
Induction was discovered in 1901 by the German embryologist H. Spemann while he was studying the formation of the eye lens from ectoderm in amphibian embryos. When the rudiment of the eye was removed, the lens did not develop. The rudiment of the eye, transplanted to the side of the embryo, induced formation of the lens from ectoderm, which should normally have become differentiated into epidermis. Later Spemann discovered the inductive influence of the chordamesoderm on the formation from the ectoderm of the gastrula of the rudiment of the central nervous system—the neural plate; he called this phenomenon primitive embryonic induction and the inductor—the chordamesoderm—the organizer. Subsequent research with removal of parts of the developing organism and their cultivation, separately or in combination, and transplantation to a part alien to the embryo showed that the phenomenon of induction is common in all chordates and in many invertebrates. Induction can be effected only on condition that the cells of the reacting system are “competent” to receive a given influence, that is, capable of receiving a stimulus and responding to it with the formation of the appropriate structures.
A chain of inductive influences is realized in the process of development: the cells of a reacting system, having received a stimulus for differentiation, often in turn become inductors of other reacting systems. Inductive influences are necessary also for further differentiation of the reacting system in a given direction. It has been established in the process of induction that in many cases not only does the inductor influence the differentiation of the reacting system, but the reacting system exerts an effect on the inductor, which is necessary for its own differentiation and for the realization of its inductive influence, that is, induction is the interaction of groups of cells of a developing embryo with each other. For a number of organogeneses it has been shown that in the process of induction, substances (inductive agents) are transferred from the cells of the inductor to the cells of the reacting system, which participate in activating the synthesis of specific messenger RNA’s necessary for the synthesis of corresponding structural proteins in the cell nuclei of the reacting system.
The term “induction” is also used to designate a wider range of phenomena in the individual development of animal and plant organisms: for example, induction of the differentiation of secondary sex characteristics by sex hormones, induction of molting in insect larvae by the hormone ecdysone, and induction of differentiation and growth of plants by plant hormones, light, temperature, and other factors.
G. M. IGNAT’EVA
IF for all t in S, t < s => P(t) THEN P(s)
I.e. if P holds for anything less than s then it holds for s. In this case we say P is proved by induction.
The most common instance of proof by induction is induction over the natural numbers where we prove that some property holds for n=0 and that if it holds for n, it holds for n+1.
(In fact it is sufficient for "<" to be a well-founded partial order on S, not necessarily a well-ordering of S.)
inductionThe process of generating an electric current in a circuit from the magnetic influence of an adjacent circuit as in a transformer or capacitor.
Electrical induction is also the principle behind the write head on magnetic disks and earlier read heads. To create (write) the bit, current is sent through a coil that creates a magnetic field which is discharged at the gap of the head onto the disk surface as it spins by. To read the bit, the magnetic field of the bit "induces" an electrical charge in the head as it passes by the gap. See inductor.