Inductive reasoning

# Inductive reasoning

Discussion

Encyclopedia
Inductive reasoning, also known as induction or inductive logic, is a kind of reasoning
Logical reasoning
In logic, three kinds of logical reasoning can be distinguished: deduction, induction and abduction. Given a precondition, a conclusion, and a rule that the precondition implies the conclusion, they can be explained in the following way:...

that constructs or evaluates proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...

s that are abstractions of observations. It is commonly construed as a form of reasoning that makes generalizations based on individual instances. In this sense it is often contrasted with deductive reasoning
Deductive reasoning
Deductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis...

.

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
Entailment
In logic, entailment is a relation between a set of sentences and a sentence. Let Γ be a set of one or more sentences; let S1 be the conjunction of the elements of Γ, and let S2 be a sentence: then, Γ entails S2 if and only if S1 and not-S2 are logically inconsistent...

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.

The following is an example of probabilistic reasoning, which is a type of weak induction:
1. 90% of humans are right-handed.
2. Joe is a human.

Therefore, Joe is probably right-handed.

This is an example of inductive reasoning:
1. 90% of humans are right-handed.
2. Joe is a human.

Therefore, the probability that Joe is right-handed is 90%. (See section on Statistical syllogism
Statistical syllogism
A statistical syllogism is a non-deductive syllogism. It argues from a generalization true for the most part to a particular case .-Introduction:Statistical syllogisms may use qualifying words like "most", "frequently", "almost never", "rarely",...

.)

Probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

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 reasoning allows for the possibility that the conclusion is false, even where all of the premises are true. 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
Mathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...

, which is considered to be a form of deductive
Deductive reasoning
Deductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis...

reasoning.

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

## 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
World view
A comprehensive world view is the fundamental cognitive orientation of an individual or society encompassing the entirety of the individual or society's knowledge and point-of-view, including natural philosophy; fundamental, existential, and normative postulates; or themes, values, emotions, and...

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.

## Is induction reliable?

Inductive reasoning has been attacked for millennia by thinkers as diverse as Sextus Empiricus
Sextus Empiricus
Sextus Empiricus , was a physician and philosopher, and has been variously reported to have lived in Alexandria, Rome, or Athens. His philosophical work is the most complete surviving account of ancient Greek and Roman skepticism....

and Karl Popper
Karl Popper
Sir Karl Raimund Popper, CH FRS FBA was an Austro-British philosopher and a professor at the London School of Economics...

.

The classic philosophical treatment of the problem of induction
Problem of induction
The problem of induction is the philosophical question of whether inductive reasoning leads to knowledge. That is, what is the justification for either:...

was given by the Scottish
Scottish people
The Scottish people , or Scots, are a nation and ethnic group native to Scotland. Historically they emerged from an amalgamation of the Picts and Gaels, incorporating neighbouring Britons to the south as well as invading Germanic peoples such as the Anglo-Saxons and the Norse.In modern use,...

philosopher David Hume
David Hume
David Hume was a Scottish philosopher, historian, economist, and essayist, known especially for his philosophical empiricism and skepticism. He was one of the most important figures in the history of Western philosophy and the Scottish Enlightenment...

. Hume highlighted the fact that our everyday functioning depends 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.

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
Philosophical skepticism
Philosophical skepticism is both a philosophical school of thought and a method that crosses disciplines and cultures. Many skeptics critically examine the meaning systems of their times, and this examination often results in a position of ambiguity or doubt...

, Hume advocated a practical skepticism
Scientific skepticism
Scientific skepticism is the practice of questioning the veracity of claims lacking empirical evidence or reproducibility, as part of a methodological norm pursuing "the extension of certified knowledge". For example, Robert K...

based on common sense
Common sense
Common sense is defined by Merriam-Webster as, "sound and prudent judgment based on a simple perception of the situation or facts." Thus, "common sense" equates to the knowledge and experience which most people already have, or which the person using the term believes that they do or should have...

, where the inevitability of induction is accepted.

## 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. 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.

### Generalization

A generalization (more accurately, an inductive generalization) proceeds from a premise about a sample to a conclusion about the population
Statistical population
A statistical population is a set of entities concerning which statistical inferences are to be drawn, often based on a random sample taken from the population. For example, if we were interested in generalizations about crows, then we would describe the set of crows that is of interest...

.
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 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
Hasty generalization
Hasty generalization is a logical fallacy of faulty generalization by reaching an inductive generalization based on insufficient evidence essentially making a hasty conclusion without considering all of the variables...

and the biased sample
Biased sample
In statistics, sampling bias is when a sample is collected in such a way that some members of the intended population are less likely to be included than others. It results in a biased sample, a non-random sample of a population in which all individuals, or instances, were not equally likely to...

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
Dicto simpliciter
A dicto simpliciter or ad dictum simpliciter are Latin phrases for a type of logical fallacy.A dicto simpliciter fallacies are deductive fallacies that occur in statistical syllogisms...

fallacies can occur in statistical syllogisms: "accident
Accident (fallacy)
The logical fallacy of accident is a deductive fallacy occurring in statistical syllogisms when an exception to a rule of thumb is ignored. It is one of the thirteen fallacies originally identified by Aristotle...

" and "converse accident
Converse accident
The logical fallacy of converse accident is a deductive fallacy that can occur in a statistical syllogism when an exception to a generalization is wrongly called for.For example:The inductive version of this fallacy is called hasty generalization...

".

### 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

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
Common sense
Common sense is defined by Merriam-Webster as, "sound and prudent judgment based on a simple perception of the situation or facts." Thus, "common sense" equates to the knowledge and experience which most people already have, or which the person using the term believes that they do or should have...

, science
Science
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...

, philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...

and the humanities
Humanities
The humanities are academic disciplines that study the human condition, using methods that are primarily analytical, critical, or speculative, as distinguished from the mainly empirical approaches of the natural sciences....

, but sometimes it is accepted only as an auxiliary method. A refined approach is case-based reasoning
Case-based reasoning
Case-based reasoning , broadly construed, is the process of solving new problems based on the solutions of similar past problems. An auto mechanic who fixes an engine by recalling another car that exhibited similar symptoms is using case-based reasoning...

### 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

Of the candidate systems for an inductive logic, the most influential is Bayesianism. As a logic of induction rather than a theory of belief, Bayesianism 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
Conditional probability
In probability theory, the "conditional probability of A given B" is the probability of A if B is known to occur. It is commonly notated P, and sometimes P_B. P can be visualised as the probability of event A when the sample space is restricted to event B...

.

## Inductive inference

Around 1960, Ray Solomonoff
Ray Solomonoff
Ray Solomonoff was the inventor of algorithmic probability, and founder of algorithmic information theory, He was an originator of the branch of artificial intelligence based on machine learning, prediction and probability...

founded the theory of universal inductive inference
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...

, 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
Occam's razor
Occam's razor, also known as Ockham's razor, and sometimes expressed in Latin as lex parsimoniae , is a principle that generally recommends from among competing hypotheses selecting the one that makes the fewest new assumptions.-Overview:The principle is often summarized as "simpler explanations...

.
Fundamental ingredients of the theory are the concepts of algorithmic probability
Algorithmic probability
In algorithmic information theory, algorithmic probability is a method of assigning a probability to each hypothesis that explains a given observation, with the simplest hypothesis having the highest probability and the increasingly complex hypotheses receiving increasingly small probabilities...

and Kolmogorov complexity
Kolmogorov complexity
In algorithmic information theory , the Kolmogorov complexity of an object, such as a piece of text, is a measure of the computational resources needed to specify the object...

.

• Abductive reasoning
Abductive reasoning
Abduction is a kind of logical inference described by Charles Sanders Peirce as "guessing". The term refers to the process of arriving at an explanatory hypothesis. Peirce said that to abduce a hypothetical explanation a from an observed surprising circumstance b is to surmise that a may be true...

• Analogy
Analogy
Analogy is a cognitive process of transferring information or meaning from a particular subject to another particular subject , and a linguistic expression corresponding to such a process...

• Deductive reasoning
Deductive reasoning
Deductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis...

• Explanation
Explanation
An explanation is a set of statements constructed to describe a set of facts which clarifies the causes, context, and consequencesof those facts....

• Falsifiability
Falsifiability
Falsifiability or refutability of an assertion, hypothesis or theory is the logical possibility that it can be contradicted by an observation or the outcome of a physical experiment...

• Inductive inference
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...

• Inductive reasoning aptitude
Inductive reasoning aptitude
Inductive reasoning is a measurable aptitude for how well a person can identify a pattern within a large amount of data. Measurement is generally done in a timed test by showing four pictures or words and asking the test taker to identify which of the pictures or words does not belong in the set....

• Inductive Logic Programming
Inductive logic programming
Inductive logic programming is a subfield of machine learning which uses logic programming as a uniform representation for examples, background knowledge and hypotheses...

• Inferential statistics
• Inquiry
Inquiry
An inquiry is any process that has the aim of augmenting knowledge, resolving doubt, or solving a problem. A theory of inquiry is an account of the various types of inquiry and a treatment of the ways that each type of inquiry achieves its aim.-Deduction:...

• Lateral thinking
Lateral thinking
Lateral thinking is solving problems through an indirect and creative approach, using reasoning that is not immediately obvious and involving ideas that may not be obtainable by using only traditional step-by-step logic...

• Logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

• Machine learning
Machine learning
Machine learning, a branch of artificial intelligence, is a scientific discipline concerned with the design and development of algorithms that allow computers to evolve behaviors based on empirical data, such as from sensor data or databases...

• Mathematical induction
Mathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...

• Mill's Methods
Mill's Methods
Mill's Methods are five methods of induction described by philosopher John Stuart Mill in his 1843 book A System of Logic. They are intended to illuminate issues of causation.-Direct method of agreement:...

The Raven paradox, also known as Hempel's paradox or Hempel's ravens is a paradox proposed by the German logician Carl Gustav Hempel in the 1940s to illustrate a problem where inductive logic violates intuition...

• Retroduction
Retroduction
Retroduction is similar to induction, but it is predicated on known or assumed relationary rules and observations that contain at least one of the predicates or predictors of the rules in question. Another predicate of the relationary rule is then generalized to the observation due to the...

• Laurence Jonathan Cohen
Laurence Jonathan Cohen
Jonathan Cohen FBA was a British philosopher. He was Fellow and Praelector in Philosophy, 1957–90 and Senior Tutor, 1985–90 at The Queen's College, Oxford and British Academy Reader in Humanities, University of Oxford, 1982–84.Education: St...

• Counterinduction
Counterinduction
Counterinduction is the rule of inference that one should assume the opposite of what induction suggests. For example:In most references to counterinduction, it is not suggested that counterinduction is valid...

• Four Varieties of Inductive Argument from the Department of Philosophy, University of North Carolina at Greensboro
University of North Carolina at Greensboro
The University of North Carolina at Greensboro , also known as UNC Greensboro, is a public university in Greensboro, North Carolina, United States and is a constituent institution of the University of North Carolina system. The university offers more than 100 undergraduate, 61 master's and 26...

.
• Inductive Logic from the Stanford Encyclopedia of Philosophy
Stanford Encyclopedia of Philosophy
The Stanford Encyclopedia of Philosophy is a freely-accessible online encyclopedia of philosophy maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from over 65 academic institutions worldwide...

., a psychological review by Evan Heit of the University of California, Merced
University of California, Merced
The University of California, Merced, commonly referred to as UC Merced or UCM, is the tenth and newest of the University of California campuses. Located in the San Joaquin Valley in unincorporated Merced County, California, near Merced, UC Merced was the first American research university to...

.
• The Mind, Limber An article which employs the film The Big Lebowski
The Big Lebowski
The Big Lebowski is a 1998 comedy film written and directed by Joel and Ethan Coen. Jeff Bridges stars as Jeff Lebowski, an unemployed Los Angeles slacker and avid bowler, who is referred to as "The Dude". After a case of mistaken identity, The Dude is introduced to a millionaire also named...

to explain the value of inductive reasoning.