Inferences are steps in

This states that all men are mortal and that Socrates is a man. Now we can ask the Prolog system about Socrates: ?- mortal(socrates). (where ''?-'' signifies a query: Can ''mortal(socrates).'' be deduced from the KB using the rules) gives the answer "Yes". On the other hand, asking the Prolog system the following: ?- mortal(plato). gives the answer "No". This is because

Inference example and definition

* {{Authority control Concepts in epistemology Concepts in logic Concepts in metaphilosophy Concepts in metaphysics Concepts in the philosophy of mind History of logic Intellectual history Logic Logic and statistics Logical consequence Metaphysics of mind Reasoning Sources of knowledge Thought

reasoning
Reason is the capacity of consciously applying logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: ...

, moving from premise
A premise or premiss is a true or false statement that helps form the body of an argument
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, apply ...

s to logical consequence
Logical consequence (also entailment) is a fundamental concept
Concepts are defined as abstract ideas or general notions that occur in the mind, in speech, or in thought. They are understood to be the fundamental building blocks of thoughts ...

s; etymologically, the word ''infer
Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word ''wikt:infer, infer'' means to "carry forward". Inference is theoretically traditionally divided into deductive reasoning, deduction and ind ...

'' means to "carry forward". Inference is theoretically traditionally divided into deduction and induction
Induction may refer to:
Philosophy
* Inductive reasoning, in logic, inferences from particular cases to the general case
Biology and chemistry
* Labor induction (birth/pregnancy)
* Induction chemotherapy, in medicine
* Induction period, the t ...

, a distinction that in Europe dates at least to Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher
A philosopher is someone who practices philosophy
Philosophy (from , ) is the study of general and fundamental quest ...

(300s BCE). Deduction is inference deriving logical conclusions from premises known or assumed to be true
True most commonly refers to truth
Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherw ...

, with the laws of valid inference being studied in logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

. Induction is inference from particular In metaphysics
Metaphysics is the branch of philosophy
Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of ...

premises to a universal
Universal is the adjective for universe.
Universal may also refer to:
Companies
* NBCUniversal, a media and entertainment company
** Universal Animation Studios, an American Animation studio, and a subsidiary of NBCUniversal
** Universal TV, a te ...

conclusion. A third type of inference is sometimes distinguished, notably by Charles Sanders Peirce
Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, ian, mathematician and scientist who is sometimes known as "the father of ". He was known as a somewhat unusual character.
Educated as a chemist an ...

, contradistinguishing abduction
Abduction may refer to:
Of a person or people
* Alien abduction, memories of being taken by apparently nonhuman entities from a different planet
* Bride kidnapping, a practice in which a man abducts the woman he wishes to marry
* Child abducti ...

from induction.
Various fields study how inference is done in practice. Human inference (i.e. how humans draw conclusions) is traditionally studied within the fields of logic, argumentation studies, and cognitive psychology
Cognitive psychology is the scientific study of mental process
Cognition () refers to "the mental action or process of acquiring knowledge and understanding through thought, experience, and the senses". It encompasses many aspects of intelle ...

; artificial intelligence
Artificial intelligence (AI) is intelligence
Intelligence has been defined in many ways: the capacity for abstraction
Abstraction in its main sense is a conceptual process where general rules and concept
Concepts are defined as abstra ...

researchers develop automated inference systems to emulate human inference. Statistical inference
Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution, distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical ...

uses mathematics to draw conclusions in the presence of uncertainty. This generalizes deterministic reasoning, with the absence of uncertainty as a special case. Statistical inference uses quantitative or qualitative (categorical) data which may be subject to random variations.
Definition

The process by which a conclusion is inferred from multipleobservations
Observation is the active acquisition of information
Information can be thought of as the resolution of uncertainty; it answers the question of "What an entity is" and thus defines both its essence and the nature of its characteristics. Th ...

is called inductive reasoning
Inductive reasoning is a method of reasoning
Reason is the capacity of Consciousness, consciously making sense of things, applying logic, and adapting or justifying practices, institutions, and beliefs based on new or existing information. It ...

. The conclusion may be correct or incorrect, or correct to within a certain degree of accuracy, or correct in certain situations. Conclusions inferred from multiple observations may be tested by additional observations.
This definition is disputable (due to its lack of clarity. Ref: Oxford English dictionary: "induction ... 3. Logic the inference of a general law from particular instances." ) The definition given thus applies only when the "conclusion" is general.
Two possible definitions of "inference" are:
# A conclusion reached on the basis of evidence and reasoning.
# The process of reaching such a conclusion.
Examples

Example for definition #1

Ancient Greek philosophers
Ancient Greek philosophy arose in the 6th century BC, at a time when the inhabitants of ancient Greece were struggling to repel devastating invasions from the east. Greek philosophy continued throughout the Hellenistic period
The Hellenistic p ...

defined a number of syllogism
A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument
In logic and philosophy, an argument is a series of statements (in a natural language), called the premises or premisses (bo ...

s, correct three part inferences, that can be used as building blocks for more complex reasoning. We begin with a famous example:
# All humans are mortal.
# All Greeks are humans.
# All Greeks are mortal.
The reader can check that the premises and conclusion are true, but logic is concerned with inference: does the truth of the conclusion follow from that of the premises?
The validity of an inference depends on the form of the inference. That is, the word "valid" does not refer to the truth of the premises or the conclusion, but rather to the form of the inference. An inference can be valid even if the parts are false, and can be invalid even if some parts are true. But a valid form with true premises will always have a true conclusion.
For example, consider the form of the following symbological track:
#All meat comes from animals.
#All beef is meat.
#Therefore, all beef comes from animals.
If the premises are true, then the conclusion is necessarily true, too.
Now we turn to an invalid form.
#All A are B.
#All C are B.
#Therefore, all C are A.
To show that this form is invalid, we demonstrate how it can lead from true premises to 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, (this and the following examples do not follow the Greek syllogism):
#All tall people are French. (False)
#John Lennon was tall. (True)
#Therefore, John Lennon was French. (False)
When a valid argument is used to derive a false conclusion from a false premise, the inference is valid because it follows the form of a correct inference.
A valid argument can also be used to derive a true conclusion from a false premise:
#All tall people are musicians. (Valid, False)
#John Lennon was tall. (Valid, True)
#Therefore, John Lennon was a musician. (Valid, True)
In this case we have one false premise and one true premise where a true conclusion has been inferred.
Example for definition #2

Evidence: It is the early 1950s and you are an American stationed in theSoviet Union
The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a that spanned during its existence from 1922 to 1991. It was nominally a of multiple national ; in practice and were highly until its final years. The ...

. You read in the Moscow
Moscow ( , American English, US chiefly ; rus, links=no, Москва, r=Moskva, p=mɐˈskva, a=Москва.ogg) is the Capital city, capital and List of cities and towns in Russia by population, largest city of Russia. The city stands on the ...

newspaper that a soccer
Association football, more commonly known as simply football or soccer, is a team sport
A team sport includes any sport
Sport pertains to any form of Competition, competitive physical activity or game that aims to use, maintain ...

team from a small city in Siberia
Siberia (; rus, Сибирь, r=Sibir', p=sʲɪˈbʲirʲ, a=Ru-Сибирь.ogg) is an extensive geographical region, constituting all of North Asia, from the Ural Mountains in the west to the Pacific Ocean in the east. It has been a part of R ...

starts winning game after game. The team even defeats the Moscow team. Inference: The small city in Siberia is not a small city anymore. The Soviets are working on their own nuclear or high-value secret weapons program.
Knowns: The Soviet Union is a command economy
A planned economy is a type of economic system
An economic system, or economic order, is a system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
A ...

: people and material are told where to go and what to do. The small city was remote and historically had never distinguished itself; its soccer season was typically short because of the weather.
Explanation: In a command economy
A planned economy is a type of economic system
An economic system, or economic order, is a system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
A ...

, people and material are moved where they are needed. Large cities might field good teams due to the greater availability of high quality players; and teams that can practice longer (weather, facilities) can reasonably be expected to be better. In addition, you put your best and brightest in places where they can do the most good—such as on high-value weapons programs. It is an anomaly for a small city to field such a good team. The anomaly (i.e. the soccer scores and great soccer team) indirectly described a condition by which the observer inferred a new meaningful pattern—that the small city was no longer small. Why would you put a large city of your best and brightest in the middle of nowhere? To hide them, of course.
Incorrect inference

An incorrect inference is known as afallacy
A fallacy is the use of invalid or otherwise faulty reason
Reason is the capacity of consciously applying logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of ...

. Philosophers who study informal logic
Informal logic encompasses the principles of logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: g ...

have compiled large lists of them, and cognitive psychologists have documented many biases in human reasoning that favor incorrect reasoning.
Applications

Inference engines

AI systems first provided automated logical inference and these were once extremely popular research topics, leading to industrial applications under the form ofexpert system
In artificial intelligence
Artificial intelligence (AI) is intelligence
Intelligence has been defined in many ways: the capacity for logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=posse ...

s and later business rule engines. More recent work on automated theorem provingAutomated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic
Mathematical logic, also called formal logic, is a subfield of mathematics
Mathematics (from Ancient Greek, Gree ...

has had a stronger
basis in formal logic.
An inference system's job is to extend a knowledge base automatically. The knowledge base
A knowledge base (KB) is a technology used to information storage, store complex structured data, structured and unstructured information used by a computer system. The initial use of the term was in connection with expert systems; which were the ...

(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 to its task.
Prolog engine

Prolog
Prolog is a logic programming
Logic programming is a programming paradigm which is largely based on formal logic. Any program written in a logic programming language is a set of sentences in logical form, expressing facts and rules about some ...

(for "Programming in Logic") is a programming language
A programming language is a formal language
In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science) ...

based on a subset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of predicate calculus
Predicate or predication may refer to:
Computer science
*Syntactic predicateA syntactic predicate specifies the syntactic validity of applying a Formal grammar#The syntax of grammars, production in a formal grammar and is analogous to a semantic P ...

. Its main job is to check whether a certain proposition can be inferred from a KB (knowledge base) using an algorithm called backward chaining
Backward chaining (or backward reasoning) is an inference method described colloquially as working backward from the goal. It is used in automated theorem provers, inference engines, proof assistants, and other artificial intelligence applications. ...

.
Let us return to our Socrates
Socrates (; ; –399 BC) was a Greek philosopher from Athens
Athens ( ; el, Αθήνα, Athína ; grc, Ἀθῆναι, Athênai (pl.) ) is the capital city, capital and List of cities in Greece, largest city of Greece. Athens domi ...

syllogism
A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument
In logic and philosophy, an argument is a series of statements (in a natural language), called the premises or premisses (bo ...

. We enter into our Knowledge Base the following piece of code:
mortal(X) :- man(X).
man(socrates).
( Here '':-'' can be read as "if". Generally, if ''P Q'' (if P then Q) then in Prolog we would code ''Q:-P'' (Q if P).)This states that all men are mortal and that Socrates is a man. Now we can ask the Prolog system about Socrates: ?- mortal(socrates). (where ''?-'' signifies a query: Can ''mortal(socrates).'' be deduced from the KB using the rules) gives the answer "Yes". On the other hand, asking the Prolog system the following: ?- mortal(plato). gives the answer "No". This is because

Prolog
Prolog is a logic programming
Logic programming is a programming paradigm which is largely based on formal logic. Any program written in a logic programming language is a set of sentences in logical form, expressing facts and rules about some ...

does not know anything about Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was an Classical Athens, Athenian philosopher during the Classical Greece, Classical period in Ancient Greece, founder of the Platonist school of thought and the Platoni ...

, and hence defaults to any property about Plato being false (the so-called closed world assumptionThe closed-world assumption (CWA), in a Mathematical logic, formal system of logic used for knowledge representation, is the presumption that a statement that is true is also known to be true. Therefore, conversely, what is not currently known to be ...

). Finally
?- mortal(X) (Is anything mortal) would result in "Yes" (and in some implementations: "Yes": X=socrates)Prolog
Prolog is a logic programming
Logic programming is a programming paradigm which is largely based on formal logic. Any program written in a logic programming language is a set of sentences in logical form, expressing facts and rules about some ...

can be used for vastly more complicated inference tasks. See the corresponding article for further examples.
Semantic web

Recently automatic reasoners found insemantic web
The Semantic Web (sometimes known as Web 3.0) is an extension of the World Wide Web
The World Wide Web (WWW), commonly known as the Web, is an information system
An information system (IS) is a formal, sociotechnical
Sociotechnica ...

a new field of application. Being based upon description logic
Description logics (DL) are a family of formal knowledge representation
Knowledge representation and reasoning (KR², KR&R) is the field of artificial intelligence
Artificial intelligence (AI) is intelligence demonstrated by machines, unlike ...

, knowledge expressed using one variant of OWL
Owls are bird
Birds are a group of warm-blooded vertebrate
Vertebrates () comprise all species of animal
Animals (also called Metazoa) are multicellular eukaryotic organisms that form the Kingdom (biology), biological k ...

can be logically processed, i.e., inferences can be made upon it.
Bayesian statistics and probability logic

Philosophers and scientists who follow the Bayesian framework for inference use the mathematical rules ofprobability
Probability is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

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).
Bayesians 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
Thomas Bayes
Thomas Bayes (; 1701 7 April 1761) was an English people, English statistician, philosopher and Presbyterian minister who is known for formulating a specific case of the theorem that bears his name: Bayes' theorem. Bayes never publ ...

). A central rule of Bayesian inference is Bayes' theorem
In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule; recently Bayes–Price theorem), named after the Reverend Thomas Bayes, describes the probability of an event (probability theory), event, based on p ...

.
Fuzzy logic

Non-monotonic logic

A relation of inference ismonotonic
Figure 3. A function that is ''not'' monotonic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calc ...

if the addition of premises does not undermine previously reached conclusions; otherwise the relation is non-monotonic.
Deductive inference is monotonic: if a conclusion is reached on the basis of a certain set of premises, then that conclusion still holds if more premises are added.
By contrast, everyday reasoning is mostly non-monotonic 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
Abduction may refer to:
Of a person or people
* Alien abduction, memories of being taken by apparently nonhuman entities from a different planet
* Bride kidnapping, a practice in which a man abducts the woman he wishes to marry
* Child abducti ...

, 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.
See also

* ''A priori and a posteriori
''A priori'' and ''a posteriori'' ('from the earlier' and 'from the later', respectively) are Latin phrases used in philosophy
Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaph ...

''
* Abductive reasoning
Abductive reasoning (also called abduction,For example: abductive inference, or retroduction) is a form of logical inference
Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word ''wikt:inf ...

* Deductive reasoning
Deductive reasoning, also deductive logic, is the process of reasoning
Reason is the capacity of consciously applying logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making ...

* Inductive reasoning
Inductive reasoning is a method of reasoning
Reason is the capacity of Consciousness, consciously making sense of things, applying logic, and adapting or justifying practices, institutions, and beliefs based on new or existing information. It ...

* Entailment
Logical consequence (also entailment) is a fundamental concept
Concepts are defined as abstract ideas
A mental representation (or cognitive representation), in philosophy of mind
Philosophy of mind is a branch of philosophy that studies th ...

* Epilogism
Epilogism, also known as epilogismos, is a style of inference used by the ancient Empiric school of medicine and Pyrrhonism. It is a theory-free method that looks at history through the accumulation of facts without major generalization and with co ...

* Analogy
Analogy (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximate ...

* Axiom system
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

** Axiom
An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

* Immediate inferenceAn immediate inference is an inference
Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word ''wikt:infer, infer'' means to "carry forward". Inference is theoretically traditionally divided int ...

* Inferential programming
In ordinary computer programming, the programmer keeps the program's intended results in mind and painstakingly constructs a computer program to achieve those results. Inferential programming refers to (still mostly hypothetical) techniques and tech ...

* Inquiry
An inquiry (also spelled as enquiry in British English
British English (BrE) is the standard dialect of the English language
English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon E ...

* Logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

* Logic of information
The logic of information, or the logical theory of information, considers the information content of logical sign
A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of s ...

* Logical assertion
In mathematical logic
Mathematical logic, also called formal logic, is a subfield of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alg ...

* Logical graph
* Rule of inference
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their Syntax (logic), syntax, and returns a conclusion (or multiple-conclusion logic, ...

* List of rules of inference
This is a list of rules of inference
A rule of inference, inference rule or transformation rule is a logical form
In logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, inte ...

* Theorem
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

* Transduction (machine learning)
In logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ...

References

Further reading

* * * * * Inductive inference: * * * * * * * Abductive inference: * * * Psychological investigations about human reasoning: * deductive: ** ** ** ** ** * statistical: ** **, * analogical: ** * spatial: ** ** ** * moral: **External links

*Inference example and definition

* {{Authority control Concepts in epistemology Concepts in logic Concepts in metaphilosophy Concepts in metaphysics Concepts in the philosophy of mind History of logic Intellectual history Logic Logic and statistics Logical consequence Metaphysics of mind Reasoning Sources of knowledge Thought