# Inductive reasoning explained

Inductive reasoning, also known as induction, is a kind of reasoning that constructs or evaluates propositions that are abstractions of observations of individual instances of members of the same class. Inductive reasoning contrasts with deductive reasoning in that a general conclusion is arrived at by specific examples.

## Definition

However, philosophically the definition is much more nuanced than simple progression from particular / individual instances to wider 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 generalizations to individual instances. Inductive reasoning consists of inferring general principles or rules from specific facts. A well-known laboratory example of inductive reasoning works like a guessing game. The participates are shown cards that contain figures differing in several dimensions, such as shape,number, and colour. On each trail, they are given two cards and asked to choose the one that represents a particular concept. After they choose a card, the researcher says "right" or "wrong."[1]

Though many dictionaries define inductive reasoning as reasoning that derives general principles from specific observations, this usage is outdated.

### Examples

This is an example of inductive reasoning:

1. 90% of humans are right-handed.
2. Joe is a human.
3. Therefore, the probability that Joe is right-handed is 90%. (See section on Statistical syllogism.)

Probability is employed, for example, in the following argument:

Every life form we know of depends on liquid water to exist.

All life depends on liquid water to exist.

However, induction is employed in the following argument:

Every life form that everyone knows of depends on liquid water to exist.

Therefore, all known life depends on liquid water to exist.

### Inductive vs. deductive reasoning

Inductive reasoning allows for the possibility that the conclusion be false, even where all of the premises are true.[2] The previous deduction was a false assertion of inductive reasoning based on the weak inductive conjecture of John Vickers.

His example is as follows:

All of the swans we have seen are white.

All swans are white.

The previous statement is an example of probabilistic reasoning, which is a weak type of induction. It is not an example of Strong Inductive Reasoning.

A proper example of inductive reasoning is as follows:

All of the swans that all living beings have ever seen are white

Therefore, all swans are white.Note that this definition of inductive reasoning excludes mathematical induction, which is considered to be a form of deductive reasoning.

## Strong and weak induction

The words 'strong' and 'weak' are sometimes used to praise or demean the quality of an inductive argument. The idea is that you say "this is an example of strong induction" when you would decide to believe the conclusion if presented with the premises. Alternatively, you say "that is weak induction" when your particular world view does not allow you to see that the conclusions are likely given the premises.

### Strong induction

The equation

"the gravitational force between two objects equals the gravitational constant times the product of the masses divided by the distance between them squared,"

has allowed us to describe the rate of fall of all objects we have observed.

Therefore:

The gravitational force between two objects equals the gravitational constant times the product of the masses divided by the distance between them squared.

The conclusion of this argument is not absolutely certain, even given the premise. At speeds we normally experience, Newtonian mechanics holds quite well. But at speeds approaching that of light, the Newtonian system is not accurate and the conclusion in that case would be false. However, since, in most cases that we experience, the premise as stated would usually lead to the conclusion given, we are logical in calling this argument an instance of strong induction.

Even very strong inductions are potentially flawed interpretations of the truth, however reasonable and logical they might appear.

### Weak induction

Consider this example:

I always hang pictures on nails.

Therefore:

All pictures hang from nails.

Here, the link between the premise and the conclusion is very weak. Not only is it possible for the conclusion to be false given the premise, it is even fairly likely that the conclusion is false. Not all pictures are hung from nails; moreover, not all pictures are hung. Thus we say that this argument is an instance of weak induction.

The previous is an example of probabilistic reasoning which employs weak induction. Therefore the previous example is closer to an example of probabilistic reasoning rather than Induction. Weak Induction is merely a type of conjecture, not a proof.

## Induction

See main article: Problem of induction.

Inductive reasoning has been attacked for millennia by thinkers as diverse as Sextus Empiricus[3] and Karl Popper.[4]

The classic philosophical treatment of the problem of induction was given by the Scottish philosopher David Hume. Hume highlighted the fact that our every day habits of mind depend on drawing uncertain conclusions from our relatively limited experiences rather than on deductively valid arguments. For example, we believe that bread will nourish us because it has done so in the past, despite no guarantee that it will do so. Hume argued that it is impossible to justify inductive reasoning. Inductive reasoning certainly cannot be justified deductively, and so our only option is to justify it inductively. However, to justify induction inductively is circular. Therefore, it is impossible to justify induction.[5]

However, Hume immediately argued that even if induction were proved unreliable, we would have to rely on it. So he took a middle road. Rather than approach everything with severe skepticism, Hume advocated a practical skepticism based on common sense, where the inevitability of induction is accepted.[6]

## Bias

Inductive reasoning is also known as hypothesis construction because any conclusions made are based on educated predictions. There are three biases that could distort the proper application of induction, thereby preventing the reasoner from forming the best, most logical conclusion based on the clues. These biases include the availability bias, the confirmation bias, and the predictable-world bias.

The availability bias causes the reasoner to depend primarily upon information that is readily available to him/her. 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 him/her.

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. A major aspect of this bias is superstition, which is derived from the inability to acknowledge that coincidences are merely coincidences. Gambling, for example, is one of the most obvious forms of predictable-world bias. Gamblers often begin to think that they see 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 always entirely random. There is no order. Since people constantly seek some type of order to explain human experiences, it is difficult for people to acknowledge that order may be nonexistent.[7]

## 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.

ExampleThere 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 compared to the number in the population and (b) 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

See main article: 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.

#### Argument from analogy

See main article: Argument from Analogy.

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

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. For more information on inferences by analogy, see Juthe, 2005.

### 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 (really any) hypothesis, 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 mathematically formalized Occam's razor.Fundamental ingredients of the theory are the concepts of algorithmic probability and Kolmogorov complexity.