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Jump to navigationJump to searchNew page: '''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. h...
'''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.
* [[Statistics|Statisticians]] have developed formal rules for inference from quantitative data.
* [[Artificial intelligence]] researchers develop automated inference systems.
==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.
The field of [[half-truths]] as they relate to the truth of observations, is another area of concern impacting inference based on observations.
==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 premises, 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 system]]s that embody [[Aristotelian logic]] (or variants thereof).
===An example: the classic syllogism===
[[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:
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 [[Aristotle|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".
∴ is the [[therefore symbol]] which denotes the conclusion.
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 [[cognitive bias|biases in human reasoning]] that favor incorrect reasoning.
==Automatic logical inference==
AI systems first provided automated logical inference and these were once extremely popular research topics, leading to industrial applications under the form of [[expert system]]s and later [[business rule engine]]s.
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 [[relevance|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:
<code>
mortal(X) :- man(X).
man(socrates). </code>
This states that all men are mortal and that Socrates is a man. Now we can ask [[Prolog]] about Socrates.
<code>
?- mortal(socrates).
Yes </code>
On the other hand :
<code>
?- mortal(plato).
No </code>
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.
===Automatic inference and the semantic web===
Recently automatic reasoners found in [[semantic web]] a new field of application. As [[Web Ontology Language|OWL]] is based upon [[first-order logic]], knowledge expressed using it can be logically processed, i.e. inference can be made upon it.
==Inference and uncertainty==
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.
* 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 inference|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===
===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 premises 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 premises.
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.
==Three types of logical inference==
There are three types of inference:
*[[Deductive reasoning]], finding the effect with the cause and the rule.
*[[Abductive reasoning]], finding the cause with the rule and the effect.
*[[Inductive reasoning]], finding the rule with the cause and the effect.
===An example===
[[Hooke's law]] is the rule that gives the elongation of a beam (that's an effect) when a force (that's the cause) is acting on a beam.
*If the force and Hooke's law are known, the elongation of the beam can be deduced.
*If the elongation and Hooke's law are known, the force acting on the beam can be abduced.
*If the elongation and the force are known, Hooke's law can be induced.
==Infer vs imply==
Inference is not the same as implication. As a verb, to infer means to deduce the meaning of a message one receives. Conversely, to imply is to deliberately communicate a particular meaning through a message.
==References==
* 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.
* 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: [http://www.uni-konstanz.de/FuF/Philo/Philosophie/Fuhrmann/papers/nomoLog.pdf Nonmonotonic Logic].
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.
* [[Statistics|Statisticians]] have developed formal rules for inference from quantitative data.
* [[Artificial intelligence]] researchers develop automated inference systems.
==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.
The field of [[half-truths]] as they relate to the truth of observations, is another area of concern impacting inference based on observations.
==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 premises, 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 system]]s that embody [[Aristotelian logic]] (or variants thereof).
===An example: the classic syllogism===
[[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:
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 [[Aristotle|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".
∴ is the [[therefore symbol]] which denotes the conclusion.
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 [[cognitive bias|biases in human reasoning]] that favor incorrect reasoning.
==Automatic logical inference==
AI systems first provided automated logical inference and these were once extremely popular research topics, leading to industrial applications under the form of [[expert system]]s and later [[business rule engine]]s.
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 [[relevance|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:
<code>
mortal(X) :- man(X).
man(socrates). </code>
This states that all men are mortal and that Socrates is a man. Now we can ask [[Prolog]] about Socrates.
<code>
?- mortal(socrates).
Yes </code>
On the other hand :
<code>
?- mortal(plato).
No </code>
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.
===Automatic inference and the semantic web===
Recently automatic reasoners found in [[semantic web]] a new field of application. As [[Web Ontology Language|OWL]] is based upon [[first-order logic]], knowledge expressed using it can be logically processed, i.e. inference can be made upon it.
==Inference and uncertainty==
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.
* 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 inference|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===
===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 premises 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 premises.
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.
==Three types of logical inference==
There are three types of inference:
*[[Deductive reasoning]], finding the effect with the cause and the rule.
*[[Abductive reasoning]], finding the cause with the rule and the effect.
*[[Inductive reasoning]], finding the rule with the cause and the effect.
===An example===
[[Hooke's law]] is the rule that gives the elongation of a beam (that's an effect) when a force (that's the cause) is acting on a beam.
*If the force and Hooke's law are known, the elongation of the beam can be deduced.
*If the elongation and Hooke's law are known, the force acting on the beam can be abduced.
*If the elongation and the force are known, Hooke's law can be induced.
==Infer vs imply==
Inference is not the same as implication. As a verb, to infer means to deduce the meaning of a message one receives. Conversely, to imply is to deliberately communicate a particular meaning through a message.
==References==
* 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.
* 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: [http://www.uni-konstanz.de/FuF/Philo/Philosophie/Fuhrmann/papers/nomoLog.pdf Nonmonotonic Logic].
