Inferences are steps in logical reasoning that lead from starting points to conclusions. Inference is usually divided into two types: deduction and induction. This idea has been around since at least the time of Aristotle in the 300s BC. A third type of inference, called abduction, was later suggested by Charles Sanders Peirce. Deduction is a type of inference that draws conclusions from starting points that are known or assumed to be true. The rules for valid deduction are studied in the field of logic. Induction is a type of inference that moves from specific examples to general conclusions. Abduction does not aim for certain conclusions or universal rules. Instead, it looks for the most likely explanation based on the available evidence.

Many areas of study examine how inference works in practice. Human inference, or how people make conclusions, is traditionally studied in logic, argumentation studies, and cognitive psychology. Researchers in artificial intelligence create computer programs that mimic human inference. Statistical inference uses math to draw conclusions when there is uncertainty. This method expands on reasoning that is certain, with certainty being a special case of statistical inference. Statistical inference uses data that can be either numerical or categorical, and this data may include random changes.

Definition

Inductive reasoning is the process of forming a general conclusion based on many observations. The conclusion might be correct, incorrect, partially correct, or correct only in specific situations. These conclusions can be checked by making more observations.

This definition is sometimes debated because it is not always clear. According to the Oxford English Dictionary, "induction" refers to "the inference of a general law from particular instances," meaning the conclusion must be general.

There are two possible meanings of "inference":
1. A conclusion made using evidence and reasoning.
2. The process of making such a conclusion.

Examples

Ancient Greek philosophers created a number of syllogisms, which are three-part arguments that can help build more complex reasoning. Here is a well-known example:

• All humans are mortal.
• All Greeks are humans.
• All Greeks are mortal.

The reader can see that the premises and conclusion are true, but logic focuses on inference: does the conclusion logically follow from the premises?

The validity of an inference depends on its structure, not the truth of the premises or conclusion. An inference can be valid even if the parts are false, and invalid even if some parts are true. However, a valid form with true premises will always result in a true conclusion.

For example, consider the structure of the following argument:

• All meat comes from animals.
• All beef is meat.
• Therefore, all beef comes from animals.

If the premises are true, the conclusion must also be true.

Now consider an invalid form:

• All A are B.
• All C are B.
• Therefore, all C are A.

To show this form is invalid, we can use true premises to reach a false conclusion:

• All apples are fruit. (True)
• All bananas are fruit. (True)
• Therefore, all bananas are apples. (False)

A valid argument with a false premise may lead to a false conclusion:

• All tall people are French. (False)
• John Lennon was tall. (True)
• Therefore, John Lennon was French. (False)

When a valid argument leads to a false conclusion from a false premise, the inference is still valid because it follows a correct structure.

A valid argument can also lead to a true conclusion from a false premise:

• All tall people are musicians. (False)
• John Lennon was tall. (True)
• Therefore, John Lennon was a musician. (True)

In this case, one premise is false, one is true, and the conclusion is true.

Evidence: In the early 1950s, an American stationed in the Soviet Union reads in a Moscow newspaper that a soccer team from a remote Siberian city is winning many games, even defeating the Moscow team. The inference is that the city is no longer small and that the Soviets are developing secret weapons.

Knowns: The Soviet Union is a command economy, meaning the government controls where people and resources go. The city was remote and historically unremarkable, with short soccer seasons due to harsh weather.

Explanation: In a command economy, people and resources are moved where they are needed. Large cities often have better teams because of more skilled players and better facilities. However, a remote city fielding a strong team is unusual. This suggests the city might have been chosen to hide a group of skilled individuals working on a secret project, such as weapons development. The unusual success of the team indirectly indicates a hidden effort.

Incorrect inference

An incorrect conclusion is called a fallacy. Philosophers who study informal logic have made long lists of these fallacies. Cognitive psychologists have found many biases in how people think that lead to wrong reasoning. They explain these biases using mental shortcuts called heuristics.

One example of a reasoning bias is confirmation bias. This happens when people look for information that supports what they already believe, instead of looking for information that might prove their belief wrong. This is shown in an experiment called the Watson selection task. Another example is the conjunction fallacy. This occurs when people think that a situation where two things happen together (A and B) is more likely than just one of those things (A). This is because the second part (B) seems more familiar or typical. This idea is shown in an example called the "Linda problem" and explained using the representativeness heuristic.

Applications

AI systems first provided automated logical inference, which was a popular research topic. These systems led to industrial uses, such as expert systems and business rule engines. More recent work on automated theorem proving has a stronger connection to formal logic.

An inference system's job is to add new information to a knowledge base automatically. The knowledge base is a collection of statements that describe what the system knows about the world. The system can use various methods to add new information to the knowledge base through valid reasoning. It is important that the conclusions the system reaches are related to its task.

The term "inference" is also used to describe the process of making predictions using trained neural networks. In this case, an "inference engine" refers to the system or hardware that performs these tasks. This type of inference is used in many applications, such as image recognition and natural language processing.

Prolog (short for "Programming in Logic") is a programming language based on a type of logic called predicate calculus. Its main purpose is to determine whether a specific statement can be logically derived from a knowledge base using a method called backward chaining.

Consider the example of the Socrates syllogism. If we add the following information to the knowledge base:
– All men are mortal.
– Socrates is a man.

We can then ask the Prolog system:
– "Is Socrates mortal?"
The system would answer "Yes."

If we ask:
– "Is Plato mortal?"
The system would answer "No," because it has no information about Plato. This is due to the closed world assumption, which means the system assumes anything not stated is false. If we ask:
– "Who is mortal?"
The system would respond "Yes" and identify Socrates as the answer. Prolog can handle complex inference tasks.

Recently, automatic reasoners have been used in the semantic web. These systems use description logic to process information written in a language called OWL. This allows computers to make logical inferences from the information.

Philosophers and scientists who use the Bayesian framework apply probability rules to find the best explanation. This approach includes deductive logic as part of its structure. Bayesians treat probabilities as measures of belief, with certain truths having a probability of 1 and certain falsehoods having a probability of 0. For example, saying an event has a 0.9 probability means it is very likely to happen.

Using probability rules, the likelihood of a conclusion and its alternatives can be calculated. The best explanation is often the one with the highest probability. A key rule in Bayesian inference is Bayes' theorem.

For example, logicians have developed a type of logic called non-monotonic logic to help reason about causes and effects.