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'''Inference''' is the act or process of deriving a [[conclusion]] based solely on what one already knows.
'''Inference''' is the act or process of deriving a [[conclusion]] based solely on what one already knows.
* [[Statistics|Statisticians]] have developed formal rules for inference from quantitative data.
* [[Statistics|Statisticians]] have developed formal rules for inference from quantitative data.
* [[Artificial intelligence]] researchers develop automated inference systems.
* [[Artificial intelligence]] researchers develop automated inference systems.
==The accuracy of inductive and deductive inferences==
==The accuracy of inductive and deductive inferences==
[[Greek philosophy|Greek philosophers]] defined a number of [[syllogism]]s, correct three-part inferences, that can be used as building blocks for more complex reasoning. We'll begin with the most famous of them all:
[[Greek philosophy|Greek philosophers]] defined a number of [[syllogism]]s, correct three-part 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 [[term logic|traditional logic]], the form of the syllogism is:
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 [[term logic|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.
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 [[Aristotle|Aristotelician logic]]), this syllogism can be stated as follows:
In predicate logic (a simple but useful formalization of [[Aristotle|Aristotelician logic]]), this syllogism can be stated as follows:
∀ X, man(X) → mortal(X)
man(Socrates)
-------------------------------
∴mortal(Socrates)
Or in its general form:
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".
∀, 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".
Consider the following:
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.
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 [[cognitive bias|biases in human reasoning]] that favor incorrect reasoning.
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 [[cognitive bias|biases in human reasoning]] that favor incorrect reasoning.
[[Category: General Reference]]
==Automatic logical inference==
==Automatic logical inference==
Let us return to our [[Socrates]] [[syllogism]]. We enter into our Knowledge Base the following piece of code:
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.
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 :
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 so-called [[closed world assumption]]). [[Prolog]] can be used for vastly more complicated inference tasks. See the corresponding article for further examples.
This is because [[Prolog]] does not know anything about [[Plato]], and hence defaults to any property about Plato being false (the so-called [[closed world assumption]]). [[Prolog]] can be used for vastly more complicated inference tasks. See the corresponding article for further examples.
[[Category: General Reference]]
===Automatic inference and the semantic web===
===Automatic inference and the semantic web===
==Inference and uncertainty==
==Inference and uncertainty==
[[Image:Inference_model.jpg|right|frame]]
Traditional logic is only concerned with [[certainty]] - one progresses from certain [[Premise_%28argument%29|premises]] to certain conclusions. There are several motivations for extending logic to deal with uncertain propositions and weaker modes of reasoning.
Traditional logic is only concerned with [[certainty]] - one progresses from certain [[Premise_%28argument%29|premises]] to certain conclusions. There are several motivations for extending logic to deal with uncertain propositions and weaker modes of reasoning.
See [[Bayesian inference]] for examples.
See [[Bayesian inference]] for examples.
==Frequentist statistical inference==
=== Nonmonotonic logic ===
=== Nonmonotonic logic ===
* Ian Hacking. ''An Introduction to Probability and Inductive Logic''. Cambridge University Press, (2000).
* Ian Hacking. ''An Introduction to Probability and Inductive Logic''. Cambridge University Press, (2000).
* Edwin Thompson Jaynes. [http://titles.cambridge.org/catalogue.asp?isbn=0521592712 ''Probability Theory: The Logic of Science''.] Cambridge University Press, (2003). ISBN 0-521-59271-2.
* Edwin Thompson Jaynes. [https://titles.cambridge.org/catalogue.asp?isbn=0521592712 ''Probability Theory: The Logic of Science''.] Cambridge University Press, (2003). ISBN 0-521-59271-2.
* David J.C. McKay. ''Information Theory, Inference, and Learning Algorithms''. Cambridge University Press, (2003).
* 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).
* Stuart Russell, Peter Norvig. ''Artificial Intelligence: A Modern Approach''. Prentice Hall, (2002).
* Henk Tijms. ''Understanding Probability''. Cambridge University Press, (2004).
* Henk Tijms. ''Understanding Probability''. Cambridge University Press, (2004).
* André Fuhrmann: [http://www.uni-konstanz.de/FuF/Philo/Philosophie/Fuhrmann/papers/nomoLog.pdf Nonmonotonic Logic].
* André Fuhrmann: [https://www.uni-konstanz.de/FuF/Philo/Philosophie/Fuhrmann/papers/nomoLog.pdf Nonmonotonic Logic].
