Inductive Reasoning
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Inductive Reasoning

Inductive reasoning (as opposed to deductive reasoning or abductive reasoning) is a method of reasoning in which the premises are viewed as supplying some evidence for the truth of the conclusion. While the conclusion of a deductive argument is certain, the truth of the conclusion of an inductive argument may be probable, based upon the evidence given.[1]

Many dictionaries define inductive reasoning as the derivation of general principles from specific observations, though some sources disagree with this usage.[2]

The philosophical definition of inductive reasoning is more nuanced than simple progression from particular/individual instances to broader generalizations. Rather, 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. In this manner, there is the possibility of moving from general statements to individual instances (for example, statistical syllogisms, discussed below).

## Description

Inductive reasoning is inherently uncertain. It only deals in degrees to which, given the premises, the conclusion is credible according to some theory of evidence. Examples include a many-valued logic, Dempster-Shafer theory, or probability theory with rules for inference such as Bayes' rule. Unlike deductive reasoning, it does not rely on universals holding over a closed domain of discourse to draw conclusions, so it can be applicable even in cases of epistemic uncertainty (technical issues with this may arise however; for example, the second axiom of probability is a closed-world assumption).[3]

An example of an inductive argument:

All biological life forms that we know of depend on liquid water to exist.
Therefore, if we discover a new biological life form it will probably depend on liquid water to exist.

This argument could have been made every time a new biological life form was found, and would have been correct every time; however, it is still possible that in the future a biological life form not requiring liquid water could be discovered.

As a result, the argument may be stated less formally as:

All biological life forms that we know of depend on liquid water to exist.
All biological life probably depends on liquid water to exist.

## Inductive vs. deductive reasoning

Argument terminology

Unlike deductive arguments, inductive reasoning allows for the possibility that the conclusion is false, even if all of the premises are true.[4] Instead of being valid or invalid, inductive arguments are either strong or weak, which describes how probable it is that the conclusion is true.[5] Another crucial difference is that deductive certainty is impossible in non-axiomatic systems, such as reality, leaving inductive reasoning as the primary route to (probabilistic) knowledge of such systems.[6]

Given that "if A is true then that would cause B, C, and D to be true", an example of deduction would be "A is true therefore we can deduce that B, C, and D are true". An example of induction would be "B, C, and D are observed to be true therefore A might be true". A is a reasonable explanation for B, C, and D being true.

For example:

A large enough asteroid impact would create a very large crater and cause a severe impact winter that could drive the non-avian dinosaurs to extinction.
We observe that there is a very large crater in the Gulf of Mexico dating to very near the time of the extinction of the non-avian dinosaurs
Therefore it is possible that this impact could explain why the non-avian dinosaurs became extinct.

Note however that this is not necessarily the case. Other events also coincide with the extinction of the non-avian dinosaurs. For example, the Deccan Traps in India.

A classical example of an incorrect inductive argument was presented by John Vickers:

All of the swans we have seen are white.
Therefore, we know that all swans are white.

The correct conclusion would be, "We expect that all swans are white".

The definition of inductive reasoning described in this article excludes mathematical induction, which is a form of deductive reasoning that is used to strictly prove properties of recursively defined sets.[7] The deductive nature of mathematical induction is based on the non-finite number of cases involved when using mathematical induction, in contrast with the finite number of cases involved in an enumerative induction procedure with a finite number of cases like proof by exhaustion. Both mathematical induction and proof by exhaustion are examples of complete induction. Complete induction is a type of masked deductive reasoning.

## History

### Ancient philosophy

For a move from particular to universal, Aristotle in the 300s BCE used the Greek word epagogé, which Cicero translated into the Latin word inductio.[8] In the 300s CE, Sextus Empiricus maintained that all knowledge derives from sensory experience--concluded in his Outlines of Pyrrhonism that acceptance of universal statements as true cannot be justified by induction.[8]

### Early modern philosophy

In 1620, early modern philosopher Francis Bacon repudiated mere experience and enumerative induction, and sought to couple those with neutral and minute and many varied observations before[further explanation needed] to uncover the natural world's structure and causal relations beyond the present scope of experience via his method of inductivism, which nonetheless required enumerative induction as a component.

The supposedly radical empiricist David Hume's 1740 stance found enumerative induction to have no rational, let alone logical, basis but to be a custom of the mind and an everyday requirement to live, although observations could be coupled with the principle uniformity of nature--another logically invalid conclusion, thus the problem of induction--to seemingly justify enumerative induction and reason toward unobservables, including causality counterfactually, simply that[further explanation needed] modifying such an aspect prevents or produces such outcome.

Awakened from "dogmatic slumber" by a German translation of Hume's work, Kant sought to explain the possibility of metaphysics. In 1781, Kant's Critique of Pure Reason introduced the distinction rationalism, a path toward knowledge distinct from empiricism. Kant sorted statements into two types. The analytic are true by virtue of their terms' arrangement and meanings--thus are tautologies, merely logical truths, true by necessity--whereas the synthetic arrange meanings to refer to states of facts, contingencies. Finding it impossible to know objects as they truly are in themselves, however, Kant found the philosopher's task not peering behind the veil of appearance to view the noumena, but simply handling phenomena.

Reasoning that the mind must contain its own categories organizing sense data, making experience of space and time possible, Kant concluded uniformity of nature a priori.[9] A class of synthetic statements was not contingent but true by necessity, then, the synthetic a priori. Kant thus saved both metaphysics and Newton's law of universal gravitation, but incidentally discarded scientific realism and developed transcendental idealism. Kant's transcendental idealism prompted the trend German idealism. G F W Hegel's absolute idealism flourished across continental Europe and fueled nationalism.

### Late modern philosophy

Developed by Saint-Simon, and promulgated in the 1830s by his former student Comte was positivism, the first late modern philosophy of science. In the French Revolution's aftermath, fearing society's ruin again, Comte opposed metaphysics. Human knowledge had evolved from religion to metaphysics to science, said Comte, which had flowed from mathematics to astronomy to physics to chemistry to biology to sociology--in that order--describing increasingly intricate domains, all of society's knowledge having become scientific, as questions of theology and of metaphysics were unanswerable. Comte found enumerative induction reliable by its grounding on experience available, and asserted science's use as improving human society, not metaphysical truth.

According to Comte, scientific method frames predictions, confirms them, and states laws--positive statements--irrefutable by theology or by metaphysics. Regarding experience to justify enumerative induction by having shown uniformity of nature,[9]Mill welcomed Comte's positivism, but thought laws susceptible to recall or revision, and withheld from Comte's Religion of Humanity. Comte was confident to lay laws as irrefutable foundation of other knowledge, and the churches, honoring eminent scientists, sought to focus public mindset on altruism--a term Comte coined--to apply science for humankind's social welfare via Comte's spearheaded science, sociology.

During the 1830s and 1840s, while Comte and Mill were the leading philosophers of science, William Whewell found enumerative induction not nearly so simple, but, amid the dominance of inductivism, described "superinduction".[10] Whewell proposed recognition of "the peculiar import of the term Induction", as "there is some Conception superinduced upon the facts", that is, "the Invention of a new Conception in every inductive inference". Rarely spotted by Whewell's predecessors, such mental inventions rapidly evade notice.[10] Whewell explained,

"Although we bind together facts by superinducing upon them a new Conception, this Conception, once introduced and applied, is looked upon as inseparably connected with the facts, and necessarily implied in them. Having once had the phenomena bound together in their minds in virtue of the Conception, men can no longer easily restore them back to detached and incoherent condition in which they were before they were thus combined".[10]

These "superinduced" explanations may well be flawed, but their accuracy is suggested when they exhibit what Whewell termed consilience--that is, simultaneously predicting the inductive generalizations in multiple areas--a feat that, according to Whewell, can establish their truth. Perhaps to accommodate prevailing view of science as inductivist method, Whewell devoted several chapters to "methods of induction" and sometimes said "logic of induction"--and yet stressed it lacks rules and cannot be trained.[10]

Originator of pragmatism, C S Peirce who, as did Gottlob Frege independently, in the 1870s performed vast investigations that clarified the basis of deductive inference as mathematical proof, recognized induction but continuously insisted on a third type of inference that Peirce variously termed abduction or retroduction or hypothesis or presumption.[11] Later philosophers gave Peirce's abduction, etc, the synonym inference to the best explanation (IBE).[12]

### Contemporary philosophy

#### Bertrand Russell

Having highlighted Hume's problem of induction, John Maynard Keynes posed logical probability as its answer--but then figured not quite.[13]Bertrand Russell found Keynes's Treatise on Probability the best examination of induction, and if read with Jean Nicod's Le Probleme logique de l'induction as well as R B Braithwaite's review of it in the October 1925 issue of Mind, to provide "most of what is known about induction", although the "subject is technical and difficult, involving a good deal of mathematics".[14] Two decades later, Russell proposed enumerative induction as an "independent logical principle".[15][16] Russell found,

"Hume's skepticism rests entirely upon his rejection of the principle of induction. The principle of induction, as applied to causation, says that, if A has been found very often accompanied or followed by B, then it is probable that on the next occasion on which A is observed, it will be accompanied or followed by B. If the principle is to be adequate, a sufficient number of instances must make the probability not far short of certainty. If this principle, or any other from which it can be deduced, is true, then the casual inferences which Hume rejects are valid, not indeed as giving certainty, but as giving a sufficient probability for practical purposes. If this principle is not true, every attempt to arrive at general scientific laws from particular observations is fallacious, and Hume's skepticism is inescapable for an empiricist. The principle itself cannot, of course, without circularity, be inferred from observed uniformities, since it is required to justify any such inference. It must therefore be, or be deduced from, an independent principle not based on experience. To this extent, Hume has proved that pure empiricism is not a sufficient basis for science. But if this one principle is admitted, everything else can proceed in accordance with the theory that all our knowledge is based on experience. It must be granted that this is a serious departure from pure empiricism, and that those who are not empiricists may ask why, if one departure is allowed, others are forbidden. These, however, are not questions directly raised by Hume's arguments. What these arguments prove--and I do not think the proof can be controverted--is that the induction is an independent logical principle, incapable of being inferred either from experience or from other logical principles, and that without this principle, science is impossible".[16]

#### Gilbert Harman

In a 1965 paper, Gilbert Harman explained that enumerative induction is not an autonomous phenomenon, but is simply a masked consequence of inference to the best explanation (IBE).[12] IBE is otherwise synonym to C S Peirce's abduction.[12] Many philosophers of science espousing scientific realism have maintained that IBE way that scientists develop approximately true scientific theories about nature.[17]

## Criticism

Inductive reasoning has been criticized by thinkers as far back as Sextus Empiricus.[18] The classic philosophical treatment of the problem of induction was given by the Scottish philosopher David Hume.[19]

Although the use of inductive reasoning demonstrates considerable success, its application has been questionable. Recognizing this, Hume highlighted the fact that our mind draws uncertain conclusions from relatively limited experiences. In deduction, the truth value of the conclusion is based on the truth of the premise. In induction, however, the dependence on the premise is always uncertain. As an example, let's assume "all ravens are black." The fact that there are numerous black ravens supports the assumption. However, the assumption becomes inconsistent with the fact that there are white ravens. Therefore, the general rule of "all ravens are black" is inconsistent with the existence of the white raven. Hume further argued that it is impossible to justify inductive reasoning: specifically, that it cannot be justified deductively, so our only option is to justify it inductively. Since this is circular he concluded that our use of induction is unjustifiable with the help of Hume's Fork.[20]

However, Hume then stated that even if induction were proved unreliable, we would still have to rely on it. So instead of a position of severe skepticism, Hume advocated a practical skepticism based on common sense, where the inevitability of induction is accepted.[21]Bertrand Russell illustrated his skepticism in a story about a turkey, fed every morning without fail, who following the laws of induction concludes this will continue, but then his throat is cut on Thanksgiving Day.[22]

Karl Popper.[23] had declared in 1963, "Induction, i.e. inference based on many observations, is a myth. It is neither a psychological fact, nor a fact of ordinary life, nor one of scientific procedure".[24] Popper's 1972 book Objective Knowledge--whose first chapter is devoted to the problem of induction--opens, "I think I have solved a major philosophical problem: the problem of induction".[24] Within Popper's schema, enumerative induction is "a kind of optical illusion" cast by the steps of conjecture and refutation during the problem shift.[24] An imaginative leap, the tentative solution is improvised, lacking inductive rules to guide it.[24] The resulting, unrestricted generalization is deductive, an entailed consequence of all, included explanatory considerations.[24] Controversy continued, however, with Popper's putative solution not generally accepted.[25]

By now, inductive inference has been shown to exist, but is found rarely, as in programs of machine learning in Artificial Intelligence (AI).[26] Popper's stance on induction is strictly falsified--enumerative induction exists--but is overwhelmingly absent from science.[26] Although much talked of nowadays by philosophers, abduction or IBE lacks rules of inference and the discussants provide nothing resembling such, as the process proceeds by humans' imaginations and perhaps creativity.[26]

### Biases

Inductive reasoning is also known as hypothesis construction because any conclusions made are based on current knowledge and predictions.[] As with deductive arguments, biases can distort the proper application of inductive argument, thereby preventing the reasoner from forming the most logical conclusion based on the clues. Examples of these biases include the availability heuristic, confirmation bias, and the predictable-world bias

The availability heuristic causes the reasoner to depend primarily upon information that is readily available to them. People have a tendency to rely on information that is easily accessible in the world around them. For example, in surveys, when people are asked to estimate the percentage of people who died from various causes, most respondents would choose the causes that have been most prevalent in the media such as terrorism, and murders, and airplane accidents rather than causes such as disease and traffic accidents, which have been technically "less accessible" to the individual since they are not emphasized as heavily in the world around them.

The confirmation bias is based on the natural tendency to confirm rather than to deny a current hypothesis. Research has demonstrated that people are inclined to seek solutions to problems that are more consistent with known hypotheses rather than attempt to refute those hypotheses. Often, in experiments, subjects will ask questions that seek answers that fit established hypotheses, thus confirming these hypotheses. For example, if it is hypothesized that Sally is a sociable individual, subjects will naturally seek to confirm the premise by asking questions that would produce answers confirming that Sally is in fact a sociable individual.

The predictable-world bias revolves around the inclination to perceive order where it has not been proved to exist, either at all or at a particular level of abstraction. Gambling, for example, is one of the most popular examples of predictable-world bias. Gamblers often begin to think that they see simple and obvious patterns in the outcomes and, therefore, believe that they are able to predict outcomes based upon what they have witnessed. In reality, however, the outcomes of these games are difficult to predict and highly complex in nature. However, in general, people tend to seek some type of simplistic order to explain or justify their beliefs and experiences, and it is often difficult for them to realise that their perceptions of order may be entirely different from the truth.[27]

## Types

### Generalization

A generalization (more accurately, an inductive generalization) proceeds from a premise about a sample to a conclusion about the population.

The proportion Q of the sample has attribute A.
Therefore:
The proportion Q of the population has attribute A.
Example

There are 20 balls--either black or white--in an urn. To estimate their respective numbers, you draw a sample of four balls and find that three are black and one is white. A good inductive generalization would be that there are 15 black and five white balls in the urn.

How much the premises support the conclusion depends upon (a) the number in the sample group, (b) the number in the population, and (c) the degree to which the sample represents the population (which may be achieved by taking a random sample). The hasty generalization and the biased sample are generalization fallacies.

### Statistical syllogism

A statistical syllogism proceeds from a generalization to a conclusion about an individual.

A proportion Q of population P has attribute A.
An individual X is a member of P.
Therefore:
There is a probability which corresponds to Q that X has A.

The proportion in the first premise would be something like "3/5ths of", "all", "few", etc. Two dicto simpliciter fallacies can occur in statistical syllogisms: "accident" and "converse accident".

### Simple induction

Simple induction proceeds from a premise about a sample group to a conclusion about another individual.

Proportion Q of the known instances of population P has attribute A.
Individual I is another member of P.
Therefore:
There is a probability corresponding to Q that I has A.

This is a combination of a generalization and a statistical syllogism, where the conclusion of the generalization is also the first premise of the statistical syllogism.

### Enumerative induction

The basic form of inductive inference, simply induction, reasons from particular instances to all instances, and is thus an unrestricted generalization.[28] If one observes 100 swans, and all 100 were white, one might infer a universal categorical proposition of the form All swans are white. As this reasoning form's premises, even if true, do not entail the conclusion's truth, this is a form of inductive inference. The conclusion might be true, and might be thought probably true, yet it can be false. Questions regarding the justification and form of enumerative inductions have been central in philosophy of science, as enumerative induction has a pivotal role in the traditional model of the scientific method.

### Argument from analogy

The process of analogical inference involves noting the shared properties of two or more things, and from this basis inferring that they also share some further property:[29]

P and Q are similar in respect to properties a, b, and c.
Object P has been observed to have further property x.
Therefore, Q probably has property x also.

Analogical reasoning is very frequent in common sense, science, philosophy and the humanities, but sometimes it is accepted only as an auxiliary method. A refined approach is case-based reasoning.[30]

### Causal inference

A causal inference draws a conclusion about a causal connection based on the conditions of the occurrence of an effect. Premises about the correlation of two things can indicate a causal relationship between them, but additional factors must be confirmed to establish the exact form of the causal relationship.

### Prediction

A prediction draws a conclusion about a future individual from a past sample.

Proportion Q of observed members of group G have had attribute A.
Therefore:
There is a probability corresponding to Q that other members of group G will have attribute A when next observed.

## Bayesian inference

As a logic of induction rather than a theory of belief, Bayesian inference does not determine which beliefs are a priori rational, but rather determines how we should rationally change the beliefs we have when presented with evidence. We begin by committing to a prior probability for a hypothesis based on logic or previous experience, and when faced with evidence, we adjust the strength of our belief in that hypothesis in a precise manner using Bayesian logic.

## Inductive inference

Around 1960, Ray Solomonoff founded the theory of universal inductive inference, the theory of prediction based on observations; for example, predicting the next symbol based upon a given series of symbols. This is a formal inductive framework that combines algorithmic information theory with the Bayesian framework. Universal inductive inference is based on solid philosophical foundations,[31] and can be considered as a mathematically formalized Occam's razor. Fundamental ingredients of the theory are the concepts of algorithmic probability and Kolmogorov complexity.

## References

1. ^ Copi, I. M.; Cohen, C.; Flage, D. E. (2006). Essentials of Logic (Second ed.). Upper Saddle River, NJ: Pearson Education. ISBN 978-0-13-238034-8.
2. ^ "Deductive and Inductive Arguments", Internet Encyclopedia of Philosophy, Some dictionaries define "deduction" as reasoning from the general to specific and "induction" as reasoning from the specific to the general. While this usage is still sometimes found even in philosophical and mathematical contexts, for the most part, it is outdated.
3. ^ Kosko, Bart (1990). "Fuzziness vs. Probability". International Journal of General Systems. 17 (1): 211-240. doi:10.1080/03081079008935108.
4. ^ John Vickers. The Problem of Induction. The Stanford Encyclopedia of Philosophy.
5. ^ Herms, D. "Logical Basis of Hypothesis Testing in Scientific Research" (pdf).
6. ^
7. ^ Chowdhry, K.R. (January 2, 2015). Fundamentals of Discrete Mathematical Structures (3rd ed.). PHI Learning Pvt. Ltd. p. 26. ISBN 9788120350748. Retrieved 2016.
8. ^ a b Stefano Gattei, Karl Popper's Philosophy of Science: Rationality without Foundations (New York: Routledge, 2009), ch. 2 "Science and philosophy", pp 28-30.
9. ^ a b Wesley C Salmon, "The uniformity of Nature", Philosophy and Phenomenological Research, 1953 Sep;14(1):39-48, p 39.
10. ^ a b c d Roberto Torretti, The Philosophy of Physics (Cambridge: Cambridge University Press, 1999), pp 216, 219-21.
11. ^ Roberto Torretti, The Philosophy of Physics (Cambridge: Cambridge University Press, 1999), pp 226, 228-29.
12. ^ a b c Ted Poston "Foundationalism", § b "Theories of proper inference", §§ iii "Liberal inductivism", Internet Encyclopedia of Philosophy, 10 Jun 2010 (last updated): "Strict inductivism is motivated by the thought that we have some kind of inferential knowledge of the world that cannot be accommodated by deductive inference from epistemically basic beliefs. A fairly recent debate has arisen over the merits of strict inductivism. Some philosophers have argued that there are other forms of nondeductive inference that do not fit the model of enumerative induction. C. S. Peirce describes a form of inference called 'abduction' or 'inference to the best explanation'. This form of inference appeals to explanatory considerations to justify belief. One infers, for example, that two students copied answers from a third because this is the best explanation of the available data--they each make the same mistakes and the two sat in view of the third. Alternatively, in a more theoretical context, one infers that there are very small unobservable particles because this is the best explanation of Brownian motion. Let us call 'liberal inductivism' any view that accepts the legitimacy of a form of inference to the best explanation that is distinct from enumerative induction. For a defense of liberal inductivism, see Gilbert Harman's classic (1965) paper. Harman defends a strong version of liberal inductivism according to which enumerative induction is just a disguised form of inference to the best explanation".
13. ^ David Andrews, Keynes and the British Humanist Tradition: The Moral Purpose of the Market (New York: Routledge, 2010), pp 63-65.
14. ^ Bertrand Russell, The Basic Writings of Bertrand Russell (New York: Routledge, 2009), "The validity of inference"], pp 157-64, quote on p 159.
15. ^ Gregory Landini, Russell (New York: Routledge, 2011), p 230.
16. ^ a b Bertrand Russell, A History of Western Philosophy (London: George Allen and Unwin, 1945 / New York: Simon and Schuster, 1945), pp 673-74.
17. ^ Stathis Psillos, "On Van Fraassen's critique of abductive reasoning", Philosophical Quarterly, 1996 Jan;46(182):31-47, p 31.
18. ^ Sextus Empiricus, Outlines of Pyrrhonism. Trans. R.G. Bury, Harvard University Press, Cambridge, Massachusetts, 1933, p. 283.
19. ^ David Hume (1910) [1748]. An Enquiry concerning Human Understanding. P.F. Collier & Son. ISBN 0-19-825060-6. Archived from the original on 31 December 2007. Retrieved 2007.
20. ^ Vickers, John. "The Problem of Induction" (Section 2). Stanford Encyclopedia of Philosophy. 21 June 2010
21. ^ Vickers, John. "The Problem of Induction" (Section 2.1). Stanford Encyclopedia of Philosophy. 21 June 2010.
22. ^ The story by Russell is found in Alan Chalmers, What is this thing Called Science, Open University Press, Milton Keynes, 1982, p. 14
23. ^ Popper, Karl R.; Miller, David W. (1983). "A proof of the impossibility of inductive probability". Nature. 302 (5910): 687-688. Bibcode:1983Natur.302..687P. doi:10.1038/302687a0.
24. Donald Gillies, "Problem-solving and the problem of induction", in Rethinking Popper (Dordrecht: Springer, 2009), Zuzana Parusniková & Robert S Cohen, eds, pp 103-05.
25. ^ Ch 5 "The controversy around inductive logic" in Richard Mattessich, ed, Instrumental Reasoning and Systems Methodology: An Epistemology of the Applied and Social Sciences (Dordrecht: D. Reidel Publishing, 1978), pp 141-43.
26. ^ a b c Donald Gillies, "Problem-solving and the problem of induction", in Rethinking Popper (Dordrecht: Springer, 2009), Zuzana Parusniková & Robert S Cohen, eds, p 111: "I argued earlier that there are some exceptions to Popper's claim that rules of inductive inference do not exist. However, these exceptions are relatively rare. They occur, for example, in the machine learning programs of AI. For the vast bulk of human science both past and present, rules of inductive inference do not exist. For such science, Popper's model of conjectures which are freely invented and then tested out seems to me more accurate than any model based on inductive inferences. Admittedly, there is talk nowadays in the context of science carried out by humans of 'inference to the best explanation' or 'abductive inference', but such so-called inferences are not at all inferences based on precisely formulated rules like the deductive rules of inference. Those who talk of 'inference to the best explanation' or 'abductive inference', for example, never formulate any precise rules according to which these so-called inferences take place. In reality, the 'inferences' which they describe in their examples involve conjectures thought up by human ingenuity and creativity, and by no means inferred in any mechanical fashion, or according to precisely specified rules".
27. ^ Gray, Peter (2011). Psychology (Sixth ed.). New York: Worth. ISBN 978-1-4292-1947-1.
28. ^ Churchill, Robert Paul (1990). Logic: An Introduction (2nd ed.). New York: St. Martin's Press. p. 355. ISBN 0-312-02353-7. OCLC 21216829. In a typical enumerative induction, the premises list the individuals observed to have a common property, and the conclusion claims that all individuals of the same population have that property.
29. ^ Baronett, Stan (2008). Logic. Upper Saddle River, NJ: Pearson Prentice Hall. pp. 321-325.
30. ^ For more information on inferences by analogy, see Juthe, 2005.
31. ^ Rathmanner, Samuel; Hutter, Marcus (2011). "A Philosophical Treatise of Universal Induction". Entropy. 13 (6): 1076-1136. arXiv:. Bibcode:2011Entrp..13.1076R. doi:10.3390/e13061076.