Inductive reasoning

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Inductive reasoning is reasoning in which the premises seek to supply strong evidence for (not absolute proof of) the truth of the conclusion. This is in opposition to deductive reasoning or abductive reasoning. While the conclusion of a deductive argument is certain, provided the premises are certain, the truth of the conclusion of an inductive argument is probable, based upon the evidence given, and assumes the the uniformity, lawfulness, or repeatability of the course of nature. The premises of an inductive logical argument indicate some degree of support (inductive probability) for the conclusion but do not entail it; that is, they suggest truth but do not ensure it. Mathematical induction is not considered a form of inductive reasoning, but may include processes which serve to generalize, e.g., reach conclusions about infinite sequences, from a finite number of particular instances, so a few of the quotes which follow may include discussions of induction in mathematics.

A-D, E, F, G-H, J, K, L-M, N-Z - See also



  • The mistake frequently committed... has its origin in the vague and erroneous definition usually given of the inductive process... the method of induction... is indeed the only process of reasoning whereby it is possible to pass, at will, and in a given direction, beyond the boundaries of our knowledge.
  • [T]he human reason discovers new relations between things not by deduction, but by that unpredictable blend of speculation and insight... induction, which—like other forms of imagination—cannot be formalized.
  • [S]ymbols have a reach and a roundness that goes beyond their literal and practical meaning. They are the rich concepts under which the mind gathers many particulars into one name, and many instances into one general induction.
  • Inductive reasoning is reasoning from particular facts to a general law, or proposition, called the conclusion. Inductive reasoning is synthetic; that is, it builds up the law (the proposition) by giving particular instances in which that law is true. It is by inductive reasoning that we have established most of our laws in the natural sciences. We have proved the proposition, Wild geese fly south in winter, by inductive reasoning, for we have noted particular instances, and from the particular facts we have reasoned to the proposition.
  • Induction... proposes to have to do with things; and, as a mode or principle of argumentation, it may perhaps be correctly defined as a process of reasoning from particulars to a general: a method which requires a scrupulous, accurate, and comprehensive examination of all the cases which come within the range of the subject of inquiry, and from these instances infers the great axiomatic truth, or the universal and invariable law, in which they are found to meet, and which they will be always found to obey. ...This is unquestionably the nature of the principle of induction as proposed by Lord Bacon. Its useful and successful application, however, to the various departments of knowledge,—and there is scarcely any department to which, under suitable modifications, it may not be advantageously applied,—requires much care, attention, and assiduous patience.


  • There is no inductive method which could lead to the fundamental concepts of physics. Failure to understand this fact constituted the basic philosophical error of so many investigators of the nineteenth century. It was probably the reason why the molecular theory, and Maxwell's theory were able to establish themselves only at a relatively late date. Logical thinking is necessarily deductive; it is based upon hypothetical concepts and axioms. How can we hope to choose the latter in such a manner as to justify us in expecting success as a consequence?
  • It will seem a little paradoxical to ascribe a great importance to observations even in that part of the mathematical sciences which is usually called Pure Mathematics, since the current opinion is that observations are restricted to physical objects that make impression on the senses. As we must refer the numbers to the pure intellect alone, we can hardly understand how observations and quasi-experiments can be of use in investigating the nature of numbers. Yet, in fact, as I shall show here with very good reasons, the properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations. There are many properties of the numbers with which we are well acquainted, but which we are not yet able to prove; only observations have led us to their knowledge. Hence we see that in the theory of numbers, which is still very imperfect, we can place our highest hopes in observations; they will lead us continually to new properties which we shall endeavor to prove afterwards. The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful.
    • Leonhard Euler, Opera Omnia, ser. 1, vol. 2, p. 459 Spcimen de usu observationum in mathesi pura, as quoted by George Pólya, Induction and Analogy in Mathematics Vol. 1, Mathematics and Plausible Reasoning (1954)


  • My intention is not to replace one set of general rules by another such set: my intention is, rather, to convince the reader that all methodologies, even the most obvious ones, have their limits. The best way to show this is to demonstrate the limits and even the irrationality of some rules which she, or he, is likely to regard as basic. In the case of induction (including induction by falsification) this means demonstrating how well the counterinductive procedure can be supported by argument.
  • Counter-induction is... both a fact—science could not exist without it—and a legitimate and much needed move in the game of science.
  • Induction, I maintain, may or may not employ hypothesis, but what is essential to it is the inference from the particular to the general, from the known to the unknown, and the nature of this inference it is impossible to represent adequately by reference to the forms of deduction.
  • I maintain, as against Mr. Jevons, that many of our inductive inferences have all the certainty of which human knowledge is capable. ...Still, it must be confessed that all our inferences from the present to the future are, in one sense, hypothetical, the hypothesis being that the circumstances on which the laws themselves depend will continue to be the same as now, that is, in the present case, that the constitution of nature, in its most general features, will remain unchanged; or, to put it in still another form, that the same causes will continue to produce the same effects. What would happen if this expectation were ever frustrated, it is absolutely impossible for us to say, so completely is it assumed in all our plans and reasonings.
  • There is... no special uncertainty attaching to the truths arrived at by induction. They are, indeed, like all other truths, relative to the present constitution of nature and the present constitution of the human mind, but this is a limitation to which all our knowledge alike is subject, and which it is vain for us to attempt to transcend.


  • Inductive reasoning corresponds to probabilistic, uncertain, approximate reasoning, and as such, it corresponds to everyday reasoning. ...induction is related to, and it could be argued, is central to, a number of other cognitive activities, including categorization, similarity judgement, probability judgement, and decision making. ...Because so much of people's reasoning is actually inductive reasoning, and because there is such a rich data set associated with induction, and because induction is related to other central cognitive activities, it is possible to find out a lot about not only reasoning but cognition more generally by studying induction.
    • Evan Heit, Inductive Reasoning: Experimental, Developmental, and Computational Approaches (2007)
  • Of the three great skeptics I interviewed, Popper was the first to make his mark. His philosophy stemmed from his effort to distinguish pseudoscience, such as Marxism or astrology or Freudian psychology, from genuine science, such as Einstein's theory of relativity. The latter, Popper decided, was testable; it made predictions about the world that could be empirically checked. The logical positivists had said as much. But Popper denied the positivist assertion that scientists can prove a theory through induction, or repeated empirical tests or observations. One never knows if one's observations have been sufficient; the next observation might contradict all that preceded it. Observations can never prove a theory but can only disprove, or falsify it. Popper often bragged that he had "killed" logical positivism with this argument.
    • John Horgan, The End of Science (1996) citing Karl Popper's "Who Killed Logical Positivism" in Unended Quest: An Intellectual Autobiography (1976)


  • In a certain sense all knowledge is inductive. We can only learn the laws and relations of things in nature by observing those things. But the knowledge gained from the senses is knowledge only of particular facts, and we require some process of reasoning by which we may collect out of the facts the laws obeyed by them. Experience gives us the materials of knowledge: induction digests those materials, and yields us general knowledge. When we possess such knowledge, in the form of general propositions and natural laws, we can usefully apply the reverse process of deduction to ascertain the exact information required at any moment. In its ultimate foundation, then, all knowledge is inductive—in the sense that it is derived by a certain inductive reasoning from the facts of experience.
  • I shall endeavor to show that induction is really the inverse process of deduction.
    • William Stanley Jevons, The Principles of Science (1874) p. 14
  • Neither in deductive nor inductive reasoning can we add a tittle to our implicit knowledge, which is like that contained in an unread book or a sealed letter. ...Reasoning explicates or brings to conscious possession what was before unconscious. It does not create, nor does it destroy, but it transmutes and throws the same matter into a new form.
    • William Stanley Jevons, The Principles of Science (1874) p.136
  • By induction we gain no certain knowledge; but by observation, and the inverse use of deductive reasoning, we estimate the probability that an event which has occurred was preceded by conditions of specified character, or that such conditions will be followed by the event. ...I have no objection to use the words cause and causation, provided they are never allowed to lead us to imagine that our knowledge of nature can attain to certainty. ...We can never recur too often to the truth that our knowledge of the laws and future events of the external world are only probable.


  • In reasoning the complex whole is consciously analyzed, and what one has found true of objects possessing certain characteristics is said to be true of all objects possessing those characteristics, and that truth is affirmed of any object found to possess such characteristics. ...Note that there are all gradations, from a simple inferred judgment to the most exact reasoning, the difference being largely an increased consciousness of the general truth and intentional analysis to find the exact element to which it applies. ...Primarily analysis means separating into parts and synthesis putting together. ...Since in induction the particular things and conditions must be analyzed in order to determine what ones are the basis of the universal affirmation, that kind of reasoning has been called analytic. In deductive reasoning two things are put together, and what is known to be true of one is affirmed of the other; hence that kind of reasoning is often called synthetic. In reality, however, the words analytic and synthetic should not be applied to reasoning at all. Analysis is necessary in induction, but its function is ended when a thing is separated into its parts; and the inference that what is true of the thing possessing these characteristics will be true of all things possessing those characteristics, is an induction, and, properly speaking, analysis has nothing to do with the reasoning phase of the process. Analysis plays almost as essential a part in deductive reasoning as in inductive, for the object must be analyzed to determine whether it possesses the characteristics of the class; hence calling inductive reasoning analytic reasoning tends only to produce confusion, with no corresponding advantage.
  • Inductive reasoning in mathematics is more closely related to deduction and the conclusions more certain than in the natural sciences, for the concepts in mathematics are not gained directly from observation (e.g., square, right angle, circle), but are made by putting together in the definition certain simple characteristics that are already known, while in the natural sciences the essential characteristics of any class of things are determined by observation and experiment, and may be changed at any time by examination of other specimens.


  • Induction, analogy, hypotheses founded upon facts and rectified continually by new observations, a happy tact given by nature and strengthened by numerous comparisons of its indications with experience, such are the principal means for arriving at truth.
    If one considers a series of objects of the same nature one perceives among them and in their changes ratios which manifest themselves more and more in proportion as the series is prolonged, and which, extending and generalizing continually, lead finally to the principle from which they were derived. But these ratios are enveloped by so many strange circumstances that it requires great sagacity to disentangle them and to recur to this principle: it is in this that the true genius of sciences consists. Analysis and natural philosophy owe their most important discoveries to this fruitful means, which is called induction. Newton was indebted to it for his theorem of the binomial and the principle of universal gravity. It is difficult to appreciate the probability of the results of induction, which is based upon this that the simplest ratios are the most common; this is verified in the formulae of analysis and is found again in natural phenomena, in crystallization, and in chemical combinations. This simplicity of ratios will not appear astonishing if we consider that all the effects of nature are only mathematical results of a small number of immutable laws.
    Yet induction, in leading to the discovery of the general principles of the sciences, does not suffice to establish them absolutely. It is always necessary to confirm them by demonstrations or by decisive experiences; for the history of the sciences shows us that induction has sometimes led to inexact results.
  • The mental operation by which one achieves new concepts and which one denotes generally by the inadequate name of induction is not a simple but rather a very complicated process. Above all, it is not a logical process although such processes can be inserted as intermediary and auxiliary links. The principle effort that leads to the discovery of new knowledge is due to abstraction and imagination.
    • Ernst Mach Erkenntnis und Irrtum: Skizzen zur Psychologie der Forschung (1905) 3rd edition, p. 318ff, translated as Knowledge and Error: Sketches Toward a Psychology of (Scientific) Research; as quoted by Phillip Frank, Philosophy of Science: The Link Between Science and Philosophy (1957)
  • Although... there is not yet extant a body of Inductive Logic, scientifically constructed; the materials for its construction exist, widely scattered, but abundant: and the selection and arrangement of those materials is a task with which intellects of the highest order, possessed of the necessary acquirements, have at length consented to occupy themselves. Within a few years three writers, profoundly versed in every branch of physical science, and not unaccustomed to carry their speculations into still higher regions of knowledge, have made attempts, of unequal but all of very great merit, towards the creation of a Philosophy of Induction: Sir John Herschel, in his [A Preliminary] Discourse on the Study of Natural Philosophy; Mr. Whewell, in his History and Philosophy of the Inductive Sciences; and, greatest of all, M. Auguste Comte, in his Cours de Philosophic Positive, a work which only requires to be better known, to place its author in the very highest class of European thinkers. That the present writer does not consider any of these philosophers, or even all of them together, to have entirely accomplished this important work, is implied in his attempting to contribute something further towards its achievement...
  • A complete logic of the sciences would be also a complete logic of practical business and common life. Since there is no case of legitimate inference from experience, in which the conclusion may not legitimately be a general proposition; an analysis of the process by which general truths are arrived at, is virtually an analysis of all induction whatever. Whether we are inquiring into a scientific principle or into an individual fact, and whether we proceed by experiment or by ratiocination, every step in the train of inferences is essentially inductive, and the legitimacy of the induction depends in both cases upon the same conditions.
    • John Stuart Mill, A System of Logic, Ratiocinative and Inductive (1858) p. 172
  • Induction may be defined the operation of discovering and proving general propositions.
    • John Stuart Mill, A System of Logic, Ratiocinative and Inductive (1858) (1858) p. 172
  • Induction... is a process of inference; it proceeds from the known to the unknown; and any operation involving no inference, any process in which what seems the conclusion is no wider than the premises from which it is drawn, does not fall within the meaning of the term. ...A general proposition is one in which the predicate is affirmed or denied of an unlimited number of individuals; namely, all, whether few or many, existing or capable of existing, which possess the properties connoted by the subject of the proposition.
    • John Stuart Mill, A System of Logic, Ratiocinative and Inductive (1858) p. 175
  • There are... in mathematics, some examples of so called induction, in which the conclusion does bear the appearance of a generalization grounded upon some of the particular cases included in it. A mathematician, when he has calculated a sufficient number of the terms of an algebraical or arithmetical series to have ascertained what is called the law of the series, does not hesitate to fill up any number of the succeeding terms without repeating the calculations. But I apprehend he only does so when it is apparent from à priori considerations (which might be exhibited in the form of demonstration) that the mode of formation of the subsequent terms, each from that which preceded it, must be similar to the formation of the terms which have been already calculated. And when the attempt has been hazarded without the sanction of such general considerations, there are instances upon record in which it has led to false results. ...Even, therefore, such cases as these, are but examples of what I have called induction by parity of reasoning, that is, not really induction, because not involving any inference of a general proposition from particular instances. ...I am happy to be able to refer, in confirmation of this view of what is called induction in mathematics, to the highest English authority on the philosophy of algebra, Mr. Peacock. See pp. 107-8 of his profound Treatise on Algebra.
    • John Stuart Mill, A System of Logic, Ratiocinative and Inductive (1858) p. 176
  • The first and second editions of the "System of Logic" contained a passage which purported to controvert the views of Archbishop Whately respecting Inductive Syllogism. In the third edition the most controversial portions of this passage are omitted, and additions are made, which materially modify the result. But it is still implied, that "Archbishop Whately's must be held to be the correct account" of no more than "the immediate major-premise in every inductive argument;" and it is still maintained, that "if we throw the whole course of any inductive argument into a series of syllogisms, we shall arrive, by more or fewer steps, at an ultimate syllogism, which will have for its major-premiss the principle, or axiom, of the uniformity of the course of nature;" which principle or axiom is regarded by Mr. Mill as known to us only by "induction."
  • The distinction between deductive and inductive reasoning is thus: In deductive reasoning the process is from the whole to the parts; in inductive reasoning, on the other hand, the process is from the parts to the whole.


  • It is the authority of the less general case, which most commonly prevails, inasmuch as it generally precedes the interpretation of the more general case in the order of investigation, and is more immediately and more essentially connected with the first principles of the science. Assuming therefore the correctness of the interpretation of the less general case, it is by an inductive process of reasoning only, that we pass from it to the interpretation of the more general case, and the existence of one does not determine, in the mathematical sense of the term, the existence of the other: but it is the necessary connection which exists between the interpretation of the more general result and those which are subordinate to it, which makes it so important to examine and ascertain the latter...
  • As an interpreter of nature... Leibnitz stands in no comparison with Newton. His general views in physics were vague and unsatisfactory; he had no great value for inductive reasoning; it was not the way of arriving at truth which he was accustomed to take; and hence, to the greatest physical discovery of that age, and that which was established by the most ample induction, the existence of gravity as a fact in which all bodies agree, he was always incredulous, because no proof of it, a priori could be given.
  • The examples of plausible reasoning collected in this book... may throw light upon a much agitated philosophical problem: the problem of induction. The crucial question is: Are there rules for induction? ...the question should be... treated... in closer touch with the practice of scientists. ...older writers, such as Euler and Laplace, clearly perceived... that the role of inductive evidence in mathematical investigation is similar to its role in physical research. the door opens to investigating induction inductively.
    • George Pólya, Induction and Analogy in Mathematics (1954) Vol. 2 of Mathematics and Plausible Reasoning
  • Induction is the process of discovering general laws by the observation and combination of particular instances. It is used in all sciences, even in mathematics. ...Induction tries to find regularity and coherence behind the observations. Its most conspicuous instruments are generalization, specialization, analogy. Tentative generalization starts from an effort to understand the observed facts; it is based on analogy, and tested by further facts.
    ...many mathematical results are found by induction first and proved later. Mathematics presented with rigor is a systematic deductive science but mathematics in the making is an experimental inductive science. ...In the physical sciences, there is no higher authority than observation and induction but in mathematics there is such an authority: rigorous proof.

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