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Inference is the act or process of deriving a conclusion based solely on what one already knows.
Inference is studied within several different fields.
 Human inference (i.e. how humans draw conclusions) is traditionally studied within the field of cognitive psychology.
 Logic studies the laws of valid inference.
 Statisticians have developed formal rules for inference from quantitative data.
 Artificial intelligence researchers develop automated inference systems.
Contents
The accuracy of inductive and deductive inferences
The conclusion inferred from multiple observations is made by the process of inductive reasoning. The conclusion may be correct or incorrect, and may be tested by additional observations. In contrast, the conclusion of a valid deductive inference is true if the premises are true. The conclusion is inferred using the process of deductive reasoning. A valid deductive inference is never false. This is because the validity of a deductive inference is formal. The inferred conclusion of a valid deductive inference is necessarily true if the premises it is based on are true.
Valid inferences
Inferences are either valid or invalid, but not both. Philosophical logic has attempted to define the rules of proper inference, i.e. the formal rules that, when correctly applied to true premisses, lead to true conclusions. Aristotle has given one of the most famous statements of those rules in his Organon. Modern mathematical logic, beginning in the 19th century, has built numerous formal systems that embody Aristotelian logic (or variants thereof).
An example: the classic syllogism
Greek philosophers defined a number of syllogisms, correct threepart inferences, that can be used as building blocks for more complex reasoning. We'll begin with the most famous of them all:
All men are mortal Socrates is a man  Therefore Socrates is mortal.
The reader can check that the premises and conclusion are true. The validity of the inference may not be true. The validity of the inference depends on the form of the inference. That is, a valid inference does not depend on the truth of the premises and conclusion, but on the formal rules of inference being used. In traditional logic, the form of the syllogism is:
All A is B All C is A  All C is B
Since the syllogism fits this form, then the inference is valid. And if the premises are true, then the conclusion is necessarily true.
In predicate logic (a simple but useful formalization of Aristotelician logic), this syllogism can be stated as follows:
∀ X, man(X) → mortal(X) man(Socrates)  ∴mortal(Socrates)
Or in its general form:
∀ X, A(X) → B(X) A(x)  ∴B(x)
∀, the universal quantifier, is pronounced "for all". It allows us to state a general property. Here it is used to say that "if any X is a man, X is also mortal". Socrates is a man, and the conclusion follows.
Consider the following:
All fat people are musicians John Lennon was fat  Therefore John Lennon was a musician
In this case we have two false premises that implies a true conclusion. The inference is valid because it follows the form of a correct inference.
An incorrect inference is known as a fallacy. Philosophers who study informal logic have compiled large lists of them, and cognitive psychologists have documented many biases in human reasoning that favor incorrect reasoning.
Automatic logical inference
Although now somewhat past their heyday, AI systems for automated logical inference once were extremely popular research topics, and have known industrial applications under the form of expert systems.
An inference system's job is to extend a knowledge base automatically. The knowledge base (KB) is a set of propositions that represent what the system knows about the world. Several techniques can be used by that system to extend KB by means of valid inferences. An additional requirement is that the conclusions the system arrives at are relevant to its task.
An example: inference using Prolog
Prolog (Programming in Logic) is a programming language based on a subset of predicate calculus. Its main job is to check whether a certain proposition can be inferred from the KB using an algorithm called backward chaining.
Let us return to our Socrates syllogism. We enter into our Knowledge Base the following piece of code:
mortal(X) : man(X).
man(socrates).
This states that all men are mortal and that Socrates is a man. Now we can ask Prolog about Socrates.
? mortal(socrates).
Yes
On the other hand :
? mortal(plato).
No
This is because Prolog does not know anything about Plato, and hence defaults to any property about Plato being false (the socalled closed world assumption). Prolog can be used for vastly more complicated inference tasks. See the corresponding article for further examples.
Inference and uncertainty
Traditional logic is only concerned with certainty  one progresses from certain premises to certain conclusions. There are several motivations for extending logic to deal with uncertain propositions and weaker modes of reasoning.
 Philosophical motivations
 A large part of our everyday reasoning does not follow the strict rules of logic, but is nevertheless effective in many cases
 Science itself is not deductive, but largely inductive, and its process cannot be captured by standard logic (see problem of induction).
 Technical motivations
 Statisticians and scientists wish to be able to infer parameters or test hypothesis on statistical data in a rigorous, quantified way.
 Artificial intelligence systems need to reason efficiently about uncertain quantities.
Common sense and uncertain reasoning
The reason most examples of applying deductive logic, such as the one above, seem artificial is because they are rarely encountered outside fields such as mathematics. Most of our everyday reasoning is of a less "pure" nature.
To take an example: suppose you live in a flat. Late at night, you are awoken by creaking sounds in the ceiling. You infer from these sounds that your neighbour upstairs is having another bout of insomnia and is pacing in his room, sleepless.
Although that reasoning seems sound, it does not fit in the logical framework described above. First, the reasoning is based on uncertain facts: what you heard were creaks, not necessarily footsteps. But even if those facts were certain, the inference is of an inductive nature: perhaps you have often heard your neighbour at night, and the best explanation you have found is that he or she is an insomniac. Hence tonight's footsteps.
It is easy to see that this line of reasoning does not necessarily lead to true conclusions: perhaps your neighbour had a very early plane to catch, which would explain the footsteps just as well. Uncertain reasoning can only find the best explanation among many alternatives.
Bayesian statistics and probability logic
Philosophers and scientists who follow the Bayesian framework for inference use the mathematical rules of probability to find this best explanation. The Bayesian view has a number of desirable features  one of them is that it embeds deductive (certain) logic as a subset (this prompts some writers to call Bayesian probability "probability logic", following E. T. Jaynes).
Bayesianists identify probabilities with degrees of beliefs, with certainly true propositions having probability 1, and certainly false propositions having probability 0. To say that "it's going to rain tomorrow" has a 0.9 probability is to say that you consider the possibility of rain tomorrow as extremely likely.
Through the rules of probability, the probability of a conclusion and of alternatives can be calculated. The best explanation is most often identified with the most probable (see Bayesian decision theory). A central rule of Bayesian inference is Bayes' theorem, which gave its name to the field.
See Bayesian inference for examples.
===Frequentist statistical inference=== (to be written)
===Fuzzy logic=== (to be written)
Nonmonotonic logic
Source: Article of André Fuhrmann about "Nonmonotonic Logic"
A relation of inference is monotonic if the addition of premisses does not undermine previously reached conclusions; otherwise the relation is nonmonotonic. Deductive inference, at least according to the canons of classical logic, is monotonic: if a conclusion is reached on the basis of a certain set of premisses, then that conclusion still holds if more premisses are added.
By contrast, everyday reasoning is mostly nonmonotonic because it involves risk: we jump to conclusions from deductively insufficient premisses. We know when it is worth or even necessary (e.g. in medical diagnosis) to take the risk. Yet we are also aware that such inference is defeasible—that new information may undermine old conclusions. Various kinds of defeasible but remarkably successful inference have traditionally captured the attention of philosophers (theories of induction, Peirce’s theory of abduction, inference to the best explanation, etc.). More recently logicians have begun to approach the phenomenon from a formal point of view. The result is a large body of theories at the interface of philosophy, logic and artificial intelligence.
References
 Ian Hacking. An Introduction to Probability and Inductive Logic. Cambridge University Press, (2000).
 Edwin Thompson Jaynes. Probability Theory: The Logic of Science. Cambridge University Press, (2003). ISBN 0521592712.
 David J.C. McKay. Information Theory, Inference, and Learning Algorithms. Cambridge University Press, (2003).
 Stuart Russell, Peter Norvig. Artificial Intelligence: A Modern Approach. Prentice Hall, (2002).
 Henk Tijms. Understanding Probability. Cambridge University Press, (2004).
 André Fuhrmann: Nonmonotonic Logic.
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