Inductive reasoning

Inductive reasoning is a method of reasoning in which a general principle is derived from a body of observations. It consists of making broad generalizations based on specific observations. Inductive reasoning is distinct from ''deductive'' reasoning. If the premises are correct, the conclusion of a deductive argument is ''certain''; in contrast, the truth of the conclusion of an inductive argument is '' probable'', based upon the evidence given.

Types

The types of inductive reasoning include generalization, prediction, statistical syllogism, argument from analogy, and causal inference.

Inductive generalization

A generalization (more accurately, an ''inductive generalization'') proceeds from a premise about a sample to a conclusion about the population. The observation obtained from this sample is projected onto the broader population.

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

For example, say 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. An 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 (1) the number in the sample group, (2) the number in the population, and (3) the degree to which the sample represents the population (which may be achieved by taking a random sample). The greater the sample size relative to the population and the more closely the sample represents the population, the stronger the generalization is. The hasty generalization and the biased sample are generalization fallacies.

Statistical generalization

A statistical generalization is a type of inductive argument in which a conclusion about a population is inferred using a statistically-representative sample. For example:

:Of a sizeable random sample of voters surveyed, 66% support Measure Z.
:Therefore, approximately 66% of voters support Measure Z.

The measure is highly reliable within a well-defined margin of error provided the sample is large and random. It is readily quantifiable. Compare the preceding argument with the following. "Six of the ten people in my book club are Libertarians. Therefore, about 60% of people are Libertarians." The argument is weak because the sample is non-random and the sample size is very small.

Statistical generalizations are also called ''statistical projections'' and ''sample projections''.

Anecdotal generalization

An anecdotal generalization is a type of inductive argument in which a conclusion about a population is inferred using a non-statistical sample. In other words, the generalization is based on anecdotal evidence. For example:

:So far, this year his son's Little League team has won 6 of 10 games.
:Therefore, by season's end, they will have won about 60% of the games.

This inference is less reliable (and thus more likely to commit the fallacy of hasty generalization) than a statistical generalization, first, because the sample events are non-random, and second because it is not reducible to mathematical expression. Statistically speaking, there is simply no way to know, measure and calculate the circumstances affecting performance that will occur in the future. On a philosophical level, the argument relies on the presupposition that the operation of future events will mirror the past. In other words, it takes for granted a uniformity of nature, an unproven principle that cannot be derived from the empirical data itself. Arguments that tacitly presuppose this uniformity are sometimes called ''Humean'' after the philosopher who was first to subject them to philosophical scrutiny.

Prediction

An inductive prediction draws a conclusion about a future, current, or past instance from a sample of other instances. Like an inductive generalization, an inductive prediction relies on a data set consisting of specific instances of a phenomenon. But rather than conclude with a general statement, the inductive prediction concludes with a specific statement about the probability that a single instance will (or will not) have an attribute shared (or not shared) by the other instances.

: 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.

Statistical syllogism

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

For example:
:90% of graduates from Excelsior Preparatory school go on to University.
:Bob is a graduate of Excelsior Preparatory school.
:Therefore, Bob will go on to University.

This is a ''statistical syllogism''.Introduction to Logic. Harry J. Gensler, Rutledge, 2002. p. 268 Even though one cannot be sure Bob will attend university, we can be fully assured of the exact probability of this outcome (given no further information). Arguably the argument is too strong and might be accused of "cheating". After all, the probability is given in the premise. Typically, inductive reasoning seeks to ''formulate'' a probability. Two dicto simpliciter fallacies can occur in statistical syllogisms: " accident" and " converse accident".

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:

:P and Q are similar with 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, law, and the humanities, but sometimes it is accepted only as an auxiliary method. A refined approach is case-based reasoning.

:Mineral A and Mineral B are both igneous rocks often containing veins of quartz and are most commonly found in South America in areas of ancient volcanic activity.
:Mineral A is also a soft stone suitable for carving into jewelry.
:Therefore, mineral B is probably a soft stone suitable for carving into jewelry.

This is ''analogical induction'', according to which things alike in certain ways are more prone to be alike in other ways. This form of induction was explored in detail by philosopher John Stuart Mill in his ''System of Logic'', where he states, " ere can be no doubt that every resemblance ot known to be irrelevantaffords some degree of probability, beyond what would otherwise exist, in favor of the conclusion." See Mill's Methods.

Some thinkers contend that analogical induction is a subcategory of inductive generalization because it assumes a pre-established uniformity governing events. Analogical induction requires an auxiliary examination of the ''relevancy'' of the characteristics cited as common to the pair. In the preceding example, if a premise were added stating that both stones were mentioned in the records of early Spanish explorers, this common attribute is extraneous to the stones and does not contribute to their probable affinity.

A pitfall of analogy is that features can be cherry-picked: while objects may show striking similarities, two things juxtaposed may respectively possess other characteristics not identified in the analogy that are characteristics sharply ''dis''similar. Thus, analogy can mislead if not all relevant comparisons are made.

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.

Methods

The two principal methods used to reach inductive conclusions are ''enumerative induction'' and ''eliminative induction.''

Enumerative induction

Enumerative induction is an inductive method in which a conclusion is constructed based on the ''number'' of instances that support it. The more supporting instances, the stronger the conclusion.

The most basic form of enumerative induction reasons from particular instances to all instances, and is thus an unrestricted generalization. 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.

:All life forms so far discovered are composed of cells.
:Therefore, all life forms are composed of cells.

This is ''enumerative induction'', also known as ''simple induction'' or ''simple predictive induction''. It is a subcategory of inductive generalization. In everyday practice, this is perhaps the most common form of induction. For the preceding argument, the conclusion is tempting but makes a prediction well in excess of the evidence. First, it assumes that life forms observed until now can tell us how future cases will be: an appeal to uniformity. Second, the conclusion ''All'' is a bold assertion. A single contrary instance foils the argument. And last, quantifying the level of probability in any mathematical form is problematic. By what standard do we measure our Earthly sample of known life against all (possible) life? Suppose we do discover some new organism—such as some microorganism floating in the mesosphere or an asteroid—and it is cellular. Does the addition of this corroborating evidence oblige us to raise our probability assessment for the subject proposition? It is generally deemed reasonable to answer this question "yes," and for a good many this "yes" is not only reasonable but incontrovertible. So then just ''how much'' should this new data change our probability assessment? Here, consensus melts away, and in its place arises a question about whether we can talk of probability coherently at all without numerical quantification.

:All life forms so far discovered have been composed of cells.
:Therefore, the ''next'' life form discovered will be composed of cells.

This is enumerative induction in its ''weak form''. It truncates "all" to a mere single instance and, by making a far weaker claim, considerably strengthens the probability of its conclusion. Otherwise, it has the same shortcomings as the strong form: its sample population is non-random, and quantification methods are elusive.

Eliminative induction

Eliminative induction, also called variative induction, is an inductive method in which a conclusion is constructed based on the ''variety'' of instances that support it. Unlike enumerative induction, eliminative induction reasons based on the various kinds of instances that support a conclusion, rather than the number of instances that support it. As the variety of instances increases, the more possible conclusions based on those instances can be identified as incompatible and eliminated. This, in turn, increases the strength of any conclusion that remains consistent with the various instances. This type of induction may use different methodologies such as quasi-experimentation, which tests and where possible eliminates rival hypotheses. Different evidential tests may also be employed to eliminate possibilities that are entertained.

Eliminative induction is crucial to the scientific method and is used to eliminate hypotheses that are inconsistent with observations and experiments. It focuses on possible causes instead of observed actual instances of causal connections.

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''.Stefano Gattei, ''Karl Popper's Philosophy of Science: Rationality without Foundations'' (New York: Routledge, 2009), ch. 2 "Science and philosophy"
pp. 28–30

Aristotle and the Peripatetic School

Aristotle's ''Posterior Analytics'' covers the methods of inductive proof in natural philosophy and in the social sciences. The first book of Posterior Analytics describes the nature and science of demonstration and its elements: including definition, division, intuitive reason of first principles, particular and universal demonstration, affirmative and negative demonstration, the difference between science and opinion, etc.

Pyrrhonism

The ancient Pyrrhonists were the first Western philosophers to point out the